The algebraic L-groups

by Andrew Ranicki

Introduction

These notes present an account of the definition of the algebraic L-groups L_{m}(A) (m(mod4)) of a ring with involution A. The surgery obstruction of an m-dimensional degree 1 normal map (f, b) : M^{m}⟶X is an element

σ_{∗}(f, b) ∈ L_{m}(ℤ[π_{1}(X)])

such that σ_{∗}(f, b) = 0 if (and for m ≥ 5 only if) (f, b) is normal bordant to a homotopy equivalence. The algebraic L-groups were defined in the book “Surgery on compact manifolds” (1971) of C.T.C.Wall. The original definition was for the category of based f.g. free A-modules and simple isomorphisms, for surgery up to simple homotopy equivalence. Here, f.g. stands for finitely generated and simple means that the Whitehead torsion is trivial, as in the hypothesis of the s-cobordism theorem. In these notes we shall only deal with the free L-groups L_{∗}(A) = L_{∗}^{h}(A), the obstruction groups for surgery up to homotopy equivalence. This spares the reader any acquaintance with simple homotopy theory.

The algebraic theory of ϵ-quadratic forms (K, λ, μ) over a ring A with an involution A⟶A; a⟶ā is developed in §§1,2, with ϵ = ±1, K an A-module,

λ : K × K ⟶ A

an ϵ-symmetric pairing such that

λ(x, y) = ϵ\overline{λ(y, x)} ∈ A (x, y ∈ K)

and

μ : K ⟶ Q_{ϵ}(A) = A/{a − ϵā|a ∈ A}

an ϵ-quadratic refinement of λ such that

λ(x, x) = μ(x) + ϵ\overline{μ(x)} ∈ A (x ∈ K) .

An (n − 1)-connected 2n-dimensional normal map (f, b) : M^{2n}⟶X has a kernel (−)^{n}-quadratic form (K_{n}(M), λ, μ) on the stably f.g. free ℤ[π_{1}(X)]-module

K_{n}(M) = ker(f̃_{∗} : H_{n}(M̃)⟶H_{n}(X̃)) ,

with X̃ the universal cover of X and M̃ the pullback of X̃ to M.

The hyperbolic ϵ-quadratic forms on f.g. free A-modules

H_{ϵ}(A^{m}) = (A^{2m}, λ, μ)

are defined by

λ : A^{2m} × A^{2m} ⟶ A ;

((a_{1}, a_{2}, …, a_{2m}), (b_{1}, b_{2}, …, b_{2m})) ⟶ ∑_{i=1}^{m}(b_{2i−1}ā_{2i} + ϵb_{2i}ā_{2i−1})

μ : A^{2m} ⟶ Q_{ϵ}(A) ; (a_{1}, a_{2}, …, a_{2m}) ⟶ ∑_{i=1}^{m} a_{2i−1}ā_{2i} .

The even-dimensional L-group L_{2n}(A) is defined in §3 to be the abelian group of stable isomorphism classes of nonsingular (−)^{n}-quadratic forms on (stably) f.g. free A-modules, where stabilization is with respect to the hyperbolic forms H_{(−)^{n}}(A^{m}). A (−)^{n}-quadratic form (K, λ, μ) represents 0 in L_{2n}(A) if and only if there exists an isomorphism

(K, λ, μ) ⊕ H_{(−)^{n}}(A^{m}) ≅ H_{(−)^{n}}(A^{m′})

for some integers m, m′ ≥ 0. The surgery obstruction of an (n − 1)-connected 2ndimensional normal map (f, b) : M^{2n}⟶X is defined by

σ_{∗}(f, b) = (K_{n}(M), λ, μ) ∈ L_{2n}(ℤ[π_{1}(X)]) .

The algebraic effect of geometric surgery on an (n − 1)-connected 2n-dimensional normal map (f, b) is given in §5. Assuming that the result of the surgery is still (n−1)- connected the effect on the kernel form is to add on a copy of the hyperbolic (−)^{n}- quadratic form H_{(−)^{n}}(ℤ[π_{1}(X)]) for a surgery of type (n − 1, n) on S^{n−1} × D^{n+1} ⊂ M, and to split off a copy of H_{(−)^{n}}(ℤ[π_{1}(X)]) for a surgery of type (n, n − 1) on S^{n} × D^{n} ⊂ M.

In §6 we introduce the notion of a “(2n + 1)-complex” (C, ψ), which is a f.g. free A-module chain complex of the type

C : … ⟶ 0 ⟶ C_{n+1} ⟶^{d} C_{n} ⟶ 0 ⟶ …

together with a quadratic structure ψ inducing Poincaré duality isomorphisms (1 + T)ψ : H^{2n+1−∗}(C)⟶H_{∗}(C). An (n − 1)-connected (2n + 1)-dimensional normal map (f, b) : M^{2n+1}⟶X determines a kernel (2n + 1)-complex (C, ψ) (or rather a homotopy equivalence class of such complexes) with

H_{∗}(C) = K_{∗}(M) = ker(f̃_{∗} : H_{∗}(M̃)⟶H_{∗}(X̃)) .

The algebraic effect of geometric surgery on (f, b) is given in §7. The odd-dimensional L-group L_{2n+1}(A) is defined in §8 as the cobordism group of (2n+1)-complexes. The surgery obstruction of an (n − 1)-connected (f, b) : M^{2n+1}⟶X is the cobordism class of the kernel complex

σ_{∗}(f, b) = (C, ψ) ∈ L_{2n+1}(ℤ[π_{1}(X)]) .

The odd-dimensional L-group L_{2n+1}(A) was originally defined by Wall as a potentially non-abelian quotient of the stable group U_{(−)^{n}}(A) of automorphisms of hyperbolic (−)^{n}-quadratic forms H_{(−)^{n}}(A^{m}) by the normal subgroup generated by the

stable elementary (−)^{n}-unitary group EU_{(−)^{n}}(A) and the element σ = ( \begin{matrix}0 & 1 \\ (−)^{n} & 0 \end{matrix} ),

where stability means passing to the limit as m⟶∞. An automorphism of a hyperbolic form is represented by a unitary matrix. The group law is given by the composition of automorphisms, which is nonabelian. Wall verified that L_{2n+1}(A) is in fact abelian using a somewhat complicated matrix identity.

In §9 and §10 we relate the other ways of defining L_{2n+1}(A), using formations and automorphisms of hyperbolic forms.

The three equivalent ways of defining L_{2n+1}(A), using complexes, formations and matrices all have their own advantages, which should become apparent with time. The surgery obstruction σ_{∗}(f, b) ∈ L_{2n+1}(ℤ[π_{1}(X)]) of an (n − 1)-connected (2n + 1)- dimensional normal map (f, b) : M^{2n+1}⟶X is represented by an automorphism of a hyperbolic (−)^{n}-quadratic form, which may be obtained from a formation, which in turn may be obtained from a (2n + 1)-complex. In each case it is necessary to make some choices in passing from the geometry to the algebra, and to verify that the equivalence class in the L-group is independent of the choices. Of course, it must also be verified that the equivalence class is invariant under surgeries on the normal map, and that in fact it determines whether surgery to a homotopy equivalence is possible, at least for n ≥ 2.

The definition of L_{2n+1}(A) using complexes given in §8 is a special case of a general theory in which the m-dimensional L-groups L_{m}(A) for any m ≥ 0 are viewed as the groups of equivalence classes of “m-dimensional quadratic Poincaré complexes” (C, ψ) with C an m-dimensional f.g. free A-module chain complex

C : C_{m} ⟶^{d}C_{m−1} ⟶^{d}C_{m−2}⟶ … ⟶C_{1} ⟶^{d}C_{0}

and ψ a quadratic structure inducing m-dimensional Poincaré duality isomorphisms (1 + T)ψ_{0} : H^{∗}(C)⟶H_{m−∗}(C). The equivalence relation used to define L_{m}(A) is modelled on the cobordism of manifolds, and the group law is given by the direct sum. The surgery obstruction of an m-dimensional normal map (f, b) : M^{m}⟶X is an element σ_{∗}(f, b) ∈ L_{m}(ℤ[π_{1}(X)]) which is represented by a quadratic Poincaré complex (C, ψ) with

H_{∗}(C) = K_{∗}(M) = H_{∗+1}(f̃: M̃⟶X̃) .

By the algebraic analogue of surgery below the middle dimension it is possible to represent every element of L_{m}(A) by a quadratic Poincaré complex (C, ψ) which is “highly-connected”, meaning that

C_{r} = 0 for \{ \begin{matrix}r ≠ n & if m = 2n \\ r ≠ n, n + 1 & if m = 2n + 1 \end{matrix} .

In these notes we shall only be considering highly-connected (2n + 1)-dimensional quadratic Poincaré complexes, namely the “(2n + 1)-complexes”. The general theory works equally well for f.g. projective modules, but the geometric applications are most immediate for f.g. free modules.

The titles of the sections are:

§1. Duality

§2. Quadratic forms

§3. The even-dimensional L-groups §4. Split forms §5. Surgery on forms §6. Short odd complexes §7. Complex cobordism §8. The odd-dimensional L-groups §9. Formations §10. Automorphisms

§1. Duality

Taking the fundamental group π = π_{1}(X) of a connected space X to act on the left of the universal cover X̃

π × X̃ ⟶ X̃ ; (g, x) ⟶ gx

there is induced a left ℤ[π]-structure on the homology H_{∗}(X̃)

ℤ[π] × H_{∗}(X̃) ⟶ H_{∗}(X̃) ; (∑_{g∈π} n_{g}g, x) ⟶ ∑_{g∈π} n_{g}gx .

Dually, there is induced a right ℤ[π]-module structure on the cohomology H^{∗}(X̃)

H^{∗}(X̃) × ℤ[π] ⟶ H^{∗}(X̃) ; (x, ∑_{g∈π} n_{g}g) ⟶ ∑_{g∈π} n_{g}xg .

If X is an oriented m-dimensional geometric Poincaré complex with fundamental class [X] ∈ H_{m}(X) (e.g. a compact m-manifold) the Poincaré duality isomorphisms are defined by the cap product

ϕ = [X] ∩ − : H^{∗}(X̃) ⟶ H_{m−∗}(X̃)

with H^{∗} ≡ H_{cpt}^{∗} the compactly supported cohomology. The right module structure of the cohomology is related to the left module structure of the homology by

ϕ(xg) = g^{−1}x ∈ H_{m−∗}(X̃) (x ∈ H^{∗}(X̃), g ∈ π) .

It is inconvenient to simultaneously work with both left and right modules. This can be avoided by defining a left ℤ[π]-action on H^{∗}(X̃) by

ℤ[π] × H^{∗}(X̃) ⟶ H^{∗}(X̃) ; (∑_{g∈π} n_{g}g, x) ⟶ ∑_{g∈π} n_{g}xg^{−1} ,

with respect to which the Poincaré duality ϕ : H^{∗}(X̃)⟶H_{m−∗}(X̃) is an isomorphism of left ℤ[π]-modules. This formal procedure of converting right to left actions depends on the automorphism of the additive structure of the group ring ℤ[π]

ℤ[π] ⟶ ℤ[π] ; ∑_{g∈π} n_{g}g ⟶ ∑_{g∈π} n_{g}g^{−1}

which reverses the order of multiplication.

In §1 we consider rings A equipped with an “involution” reversing the order of multiplication. This allows right A-modules to be regarded as A-modules, especially the right A-modules which arise as the duals of left A-modules.

Definition 1.1 Let A be an associative ring with 1. An involution on A is a function

A ⟶ A ; a ⟶ \overline{a}

satisfying

\overline{(a + b)} = \overline{a} + \overline{b} , \overline{(ab)} = \overline{b}.\overline{a} , \overline{\overline{a}} = a , \overline{1} = 1 ∈ A .

Example 1.2 A commutative ring A admits the identity involution

A ⟶ A ; a ⟶ \overline{a} = a .

Example 1.3 Given a group π and a group morphism w : π⟶ℤ_{2} = {±1} define the w-twisted involution on the integral group ring A = ℤ[π]

A ⟶ A ; ∑_{g∈π} n_{g}g ⟶ ∑_{g∈π} w(g)n_{g}g^{−1} (n_{g} ∈ ℤ) .

In the topological application π is the fundamental group of a space with a bundle and w is the orientation character of the bundle, with w(g) = +1 if and only if g is a homotopy class of orientation-preserving loops. In the oriented case discussed above w(g) = +1 for all g ∈ π.

Example 1.4 Complex conjugation defines an involution on the ring of complex numbers A = ℂ

ℂ ⟶ ℂ ; z = a + ib ⟶ \overline{z} = a − ib .

A “hermitian” form is a symmetric form on a (finite-dimensional) vector space over ℂ with respect to this involution. The study of forms over rings with involution is sometimes called “hermitian K-theory”, although “algebraic L-theory” seems preferable.

The dual of a left A-module M is the right A-module

M^{∗} = Hom_{A}(M, A)

with

M^{∗} × A ⟶ M^{∗} ; (f, a) ⟶ (x ⟶ f(x).a) .

An involution A⟶A defines an isomorphism of categories

{right A − modules} ⟶ {left A − modules} ; M ⟶ M^{op} ,

with M^{op} the left A-module with the same additive group as the right A-module M and A acting by

A × M^{op} ⟶ M^{op} ; (a, x) ⟶ x\overline{a} .

From now on we work with a ring A which is equipped with a particular choice of involution A⟶A. We shall only be working with left A-modules M, so “A-module” always means “left A-module”. The A-module (M^{∗})^{op} is written as M^{∗}. Here is the definition of M^{∗} all at once:

Definition 1.5 The dual of an A-module M is the A-module M^{∗} = Hom_{A}(M, A), with A acting by

A × M^{∗} ⟶ M^{∗} ; (a, f) ⟶ (x ⟶ f(x).\overline{a})

for all a ∈ A, f ∈ M^{∗}, x ∈ M.

There is a corresponding notion for morphisms:

Definition 1.6 The dual of an A-module morphism f : M⟶N is the A-module morphism

f^{∗} : N^{∗} ⟶ M^{∗} ; g ⟶ (x ⟶ g(f(x))) .

Thus duality is a contravariant functor

∗ : {A − modules} ⟶ {A − modules} ; M ⟶ M^{∗} .

Definition 1.7 For any A-module M define the A-module morphism

e_{M} : M ⟶ M^{∗∗} ; x ⟶ (f ⟶ \overline{f(x)}) .

The morphism e_{M} is natural in the sense that for any A-module morphism f : M⟶N there is defined a commutative diagram

\begin{matrix}M  w f u e_{M} & Nue_{N} \\ M^{∗∗}  w f^{∗∗} & N^{∗∗} \end{matrix}

Definition 1.8 i) An A-module M is f.g. projective if there exists an A-module N such that M ⊕ N is isomorphic to the f.g. free A-module A^{n}, for some n ≥ 0. ii) An A-module M is stably f.g. free if M ⊕ A^{m} is isomorphic to A^{n}, for some m, n ≥ 0.

In particular, f.g. free A-modules are stably f.g. free, and stably f.g. free A-modules are f.g. projective.

Proposition 1.9 The dual of a f.g. projective A-module M is a f.g. projective A-module M^{∗}, and e_{M} : M⟶M^{∗∗} is an isomorphism. Moreover, if M is stably f.g. free then so is M^{∗}. Proof: For any A-modules M, N there are evident identifications

\begin{matrix}(M ⊕ N)^{∗} = M^{∗} ⊕ N^{∗} , \\ e_{M⊕N} = e_{M} ⊕ e_{N} : M ⊕ N ⟶ (M ⊕ N)^{∗∗} = M^{∗∗} ⊕ N^{∗∗} , \end{matrix}

so it suffices to consider the special case M = A. Here we have an explicit isomorphism of A-modules

f : A ⟶ A^{∗} ; a ⟶ (b ⟶ b\overline{a}) ,

and also an explicit inverse for e_{A}

(e_{A})^{−1} : A^{∗∗} ⟶ A ; g ⟶ g(f(1)) .

In dealing with f.g. projective A-modules M we shall use the natural isomorphism e_{M} : M⟶M^{∗∗} to identify M^{∗∗} = M . For any morphism f : M⟶N of f.g. projective A-modules there is a corresponding identification

f^{∗∗} = f : M^{∗∗} = M ⟶ N^{∗∗} = N .

Remark 1.10 The additive group Hom_{A}(A^{m}, A^{n}) of the morphisms A^{m}⟶A^{n} between f.g. free A-modules A^{m}, A^{n} may be identified with the additive group M_{m,n}(A) of m × n matrices (a_{ij})_{1≤i≤m,1≤j≤n} with entries a_{ij} ∈ A, using the isomorphism

M_{m,n}(A) ⟶ Hom_{A}(A^{m}, A^{n}) ;

(a_{ij}) ⟶ ((x_{1}, x_{2}, …, x_{m}) ⟶ (∑_{i=1}^{m} x_{i}a_{i1}, ∑_{i=1}^{m} x_{i}a_{i2}, …, ∑_{i=1}^{m} x_{i}a_{in})) .

The composition of morphisms

Hom_{A}(A^{m}, A^{n}) × Hom_{A}(A^{n}, A^{p}) ⟶ Hom_{A}(A^{m}, A^{p}) ;

(f, g) ⟶ (gf : x⟶(gf)(x) = g(f(x)))

corresponds to the multiplication of matrices

M_{m,n}(A) × M_{n,p}(A) ⟶ M_{m,p}(A) ; ((a_{ij}), (b_{jk})) ⟶ (c_{ik})

c_{ik} = ∑_{j=1}^{n} a_{ij}b_{jk} (1 ≤ i ≤ m, 1 ≤ k ≤ p) .

Use the isomorphism of f.g. free A-modules

A^{m} ⟶ (A^{m})^{∗} ; (x_{1}, x_{2}, …, x_{m}) ⟶ ((y_{1}, y_{2}, …, y_{m}) ⟶ ∑_{i=1}^{m} y_{i}\overline{x}_{i})

to identify

(A^{m})^{∗} = A^{m} .

The duality isomorphism

∗ : Hom_{A}(A^{m}, A^{n}) ⟶ Hom_{A}((A^{n})^{∗}, (A^{m})^{∗}) = Hom_{A}(A^{n}, A^{m}) ; f ⟶ f^{∗}

to the isomorphism defined by conjugate transposition of matrices

M_{m,n}(A) ⟶ M_{n,m}(A) ; α = (a_{ij}) ⟶ α^{∗} = (b_{ji}) , b_{ji} = \overline{a}_{ij} .

Example 1.11 A 2 × 2 matrix

α = ( \begin{matrix}a & b \\ c & d \end{matrix} ) ∈ M_{2,2}(A)

corresponds to the A-module morphism

( \begin{matrix}a & b \\ c & d \end{matrix} ) : A ⊕ A ⟶ A ⊕ A ; (x, y) ⟶ (xa + yb, xc + yd) .

The conjugate transpose matrix

α^{∗} = ( \begin{matrix}\overline{a} & \overline{c} \\ \overline{b} & \overline{d} \end{matrix} ) ∈ M_{2,2}(A)

corresponds to the dual A-module morphism

\begin{aligned}( \begin{matrix}\overline{a} & \overline{c} \\ \overline{b} & \overline{d} \end{matrix} ) : (A ⊕ A)^{∗} = & A ⊕ A ⟶ (A ⊕ A)^{∗} = A ⊕ A ; \\ & (x, y) ⟶ (x\overline{a} + y\overline{c}, x\overline{b} + y\overline{d}) . \end{aligned}

§2. Quadratic forms

In the first instance suppose that the ground ring A is commutative, with the identity involution ā = a (1.2). A symmetric form (M, λ) over A is an A-module M together with a bilinear pairing

λ : M × M ⟶ A ; (x, y) ⟶ λ(x, y)

such that for all x, y, z ∈ M and a ∈ A

\begin{matrix}λ(x, ay) = aλ(x, y) , λ(x, y + z) = λ(x, y) + λ(x, z) \\ λ(x, y) = λ(y, x) ∈ A . \end{matrix}

A quadratic form (M, λ, μ) over A is a symmetric form (M, λ) together with a function

μ : M ⟶ Q_{+}(A) = A ; x ⟶ μ(x)

such that for all x, y ∈ M and a ∈ A

μ(ax) = a^{2}μ(x) , μ(x + y) = μ(x) + μ(y) + λ(x, y) ∈ Q_{+}(A) .

In particular, for every x ∈ M

2μ(x) = λ(x, x) ∈ Q^{+}(A) = A .

If 2 ∈ A is invertible (e.g. if A is a field of characteristic ≠ 2, such as ℝ, ℂ, ℚ) there is no difference between symmetric and quadratic forms, with μ determined by λ according to μ(x) = λ(x, x)/2.

A symplectic form (M, λ) over a commutative ring A is an A-module M together with a bilinear pairing λ : M × M⟶A such that for all x, y, z ∈ M and a ∈ A

\begin{matrix}λ(x, ay) = aλ(x, y) , λ(x, y + z) = λ(x, y) + λ(x, z), \\ λ(x, y) = −λ(y, x) ∈ A . \end{matrix}

A skew-quadratic form (M, λ, μ) over A is a symplectic form (M, λ) together with a function

μ : M ⟶ Q_{−}(A) = A/{2a|a ∈ A} ; x ⟶ μ(x)

such that for all x, y ∈ M and a ∈ A

μ(ax) = a^{2}μ(x) , μ(x + y) = μ(x) + μ(y) + λ(x, y) ∈ Q_{−}(A) . In particular, for every x ∈ M

2μ(x) = λ(x, x) ∈ Q^{−}(A) = {a ∈ A|2a = 0} .

If 2 ∈ A is invertible then Q_{−}(A) = 0 and there is no difference between symplectic and skew-quadratic forms, with μ = 0.

In the applications of forms to surgery theory it is necessary to work with quadratic and skew-quadratic forms over noncommutative group rings with the involution as in 1.3. In §2 we develop the general theory of forms over rings with involution, taking account of these differences.

Let X be an m-dimensional geometric Poincaré complex with universal cover X̃ and fundamental group ring A = ℤ[π_{1}(X)]. The Poincaré duality isomorphism

ϕ = [X] ∩ − : H^{m−r}(X̃) ⟶ H_{r}(X̃)

and the evaluation pairing

H^{r}(X̃) ⟶ H_{r}(X̃)^{∗} = Hom_{A}(H_{r}(X̃), A) ; y ⟶ (x ⟶ < y, x >) can be combined to define a sesquilinear pairing

λ : H_{r}(X̃) × H_{m−r}(X̃) ⟶ A ; (x, ϕ(y)) ⟶ < y, x >

such that

λ(y, x) = (−)^{r(m−r)}\overline{λ(x, y)}

with A⟶A; a⟶ā the involution of 1.3.

If M is an m-manifold with fundamental group ring A = ℤ[π_{1}(M)] the pairing λ : H_{r}(M̃) × H_{m−r}(M̃)⟶A can be interpreted geometrically using the geometric intersection numbers of cycles. For any two immersions x : S^{r}⟶M̃, y : S^{m−r}⟶M̃ in general position

λ(x, y) = ∑_{g∈π_{1}(X)} n_{g}g ∈ A

with n_{g} ∈ ℤ the algebraic number of intersections in M̃ of x and gy. In particular, for m = 2n there is defined a (−)^{n}-symmetric pairing

λ : H_{n}(M̃) × H_{n}(M̃) ⟶ A

which is relevant to surgery in the middle dimension n. In order to kill an element in π_{n}(M) by a type (n, n − 1) surgery on a framed embedding S^{n} × D^{n} ⊂ M^{2n} it is necessary that its Hurewicz image x ∈ H_{n}(M̃) be such that λ(x, x) = 0 ∈ A. The condition is necessary but not sufficient to represent an element x ∈ π_{n}(M) by a framed embedding x : S^{n} × D^{n}⟶M.

The theory of forms developed in §2 is required for an algebraic formulation of the necessary and sufficient condition for an element in the kernel K_{n}(M) of an (n−1)-connected 2n-dimensional normal map (f, b) : M⟶X to be killed by surgery, assuming n ≥ 3.

As in §1 let A be a ring with involution, not necessarily a group ring.

Definition 2.1 A sesquilinear pairing (M, N, λ) on A-modules M, N is a function

λ : M × N ⟶ A ; (x, y) ⟶ λ(x, y)

such that for all x, y, z ∈ M, a, b ∈ A

{i) λ(x, y + z) = λ(x, y) + λ(x, z) ∈ A , \atop ii) λ(ax, by) = bλ(x, y)\overline{a} ∈ A .}

The dual (or transpose ) sesquilinear pairing is

T λ : N × M ⟶ A ; (y, x) ⟶ T λ(y, x) = \overline{λ(x, y)} .

Let S(M, N) be the additive group of sesquilinear pairings λ : M ×N⟶A. Transposition defines an isomorphism T : S(M, N)⟶S(N, M) such that

T^{2} = id : S(M, N) ⟶ S(N, M) ⟶ S(M, N) .

Proposition 2.2 For any A-modules M, N there is a natural isomorphism of additive groups

\begin{matrix}S(M, N) ⟶ Hom_{A}(M, N^{∗}) ; \\ (λ : M × N⟶A) ⟶ (λ : M⟶N^{∗}; x⟶(y⟶λ(x)(y) = λ(x, y))) . \end{matrix}

For f.g. projective M, N the transposition isomorphism T : S(M, N)⟶S(N, M) corresponds to the duality isomorphism

\begin{aligned}∗ : Hom_{A} & (M, N^{∗}) ⟶ Hom_{A}(N, M^{∗}) ; \\ & (λ : M⟶N^{∗}) ⟶ (λ^{∗} : N⟶M^{∗}; x⟶(y⟶λ(y)(x))) . \end{aligned}

We use 2.2 to identify

S(M, N) = Hom_{A}(M, N^{∗}) , S(M) = Hom_{A}(M, M^{∗}) ,
Q^{ϵ}(M) = ker(1 − T_{ϵ} : Hom_{A}(M, M^{∗})⟶Hom_{A}(M, M^{∗})) ,
Q_{ϵ}(M) = coker(1 − T_{ϵ} : Hom_{A}(M, M^{∗})⟶Hom_{A}(M, M^{∗}))

for any f.g. projective A-modules M, N.

Remark 2.3 For f.g. free A-modules A^{m}, A^{n} it is possible to identify S(A^{m}, A^{n}) with the additive group M_{m,n}(A) of m × n matrices (a_{ij}) with entries a_{ij} ∈ A, using the isomorphism

M_{m,n}(A) ⟶ S(A^{m}, A^{n}) ; (a_{ij}) ⟶ λ

defined by

λ((x_{1}, x_{2}, …, x_{m}), (y_{1}, y_{2}, …, y_{n})) = ∑_{i=1}^{m} ∑_{j=1}^{n} y_{j}a_{ij}\overline{x}_{i} .

The transposition isomorphism T : S(A^{m}, A^{n})⟶S(A^{n}, A^{m}) corresponds to the isomorphism defined by conjugate transposition of matrices

T : M_{m,n}(A) ⟶ M_{n,m}(A) ; (a_{ij}) ⟶ (b_{ji}) , b_{ji} = \overline{a}_{ij} .

We shall be particularly concerned with S(M, N) in the case M = N, writing

S(M) = S(M, M) .

Definition 2.4 The ϵ-transposition involution is given for ϵ = ±1 by

T_{ϵ} : S(M) ⟶ S(M) ; λ ⟶ T_{ϵ}λ = ϵ(T λ) ,

such that

T_{ϵ}λ(x, y) = ϵ\overline{λ(y, x)} ∈ A , (T_{ϵ})^{2} = id : S(M) ⟶ S(M) .

Definition 2.5 The ϵ-symmetric group of an A-module M is the additive group

Q^{ϵ}(M) = ker(1 − T_{ϵ} : S(M)⟶S(M)) .

The ϵ-quadratic group of M is the additive group

Q_{ϵ}(M) = coker(1 − T_{ϵ} : S(M)⟶S(M)) .

The ϵ-symmetrization morphism is given by

1 + T_{ϵ} : Q_{ϵ}(M) ⟶ Q^{ϵ}(M) ; ψ ⟶ ψ + T_{ϵ}ψ .

For ϵ = +1 it is customary to refer to ϵ-symmetric and ϵ-quadratic objects as symmetric and quadratic, as in the commutative case. Similarly, for ϵ = −1 the customary terminology is skew-symmetric (or symplectic) and skew-quadratic.

For M = A there is an isomorphism of additive groups with involution

A ⟶ S(A) ; a ⟶ ((x, y) ⟶ ya\overline{x})

and we can identify

Q^{ϵ}(A) = \{a ∈ A|ϵ\overline{a} = a\} ,
Q_{ϵ}(A) = A/\{a − ϵ\overline{a}|a ∈ A\} ,
1 + T_{ϵ} : Q_{ϵ}(A) ⟶ Q^{ϵ}(A) ; a ⟶ a + ϵ\overline{a} .

Example 2.6 Let A = ℤ. The ϵ-symmetric and ϵ-quadratic groups of M = ℤ are given by

Q^{ϵ}(ℤ) = \{ \begin{matrix}ℤ & if ϵ = +1 \\ 0 & if ϵ = −1 \end{matrix} ,

Q_{ϵ}(ℤ) = \{ \begin{matrix}ℤ & if ϵ = +1 \\ ℤ/2 & if ϵ = −1 \end{matrix}

with generators represented by 1 ∈ ℤ, and with

1 + T_{+} = 2 : Q_{+}(ℤ) = ℤ ⟶ Q^{+}(ℤ) = ℤ .

Definition 2.7 An ϵ-symmetric form (M, λ) over A is an A-module M together with an element λ ∈ Q^{ϵ}(M). Thus λ is a sesquilinear pairing

λ : M × M ⟶ A ; (x, y) ⟶ λ(x, y)

such that for all x, y ∈ M

λ(x, y) = ϵ\overline{λ(y, x)} ∈ A .

The adjoint of (M, λ) is the A-module morphism

M ⟶ M^{∗} ; x ⟶ (y ⟶ λ(x, y))

which is also denoted by λ. The form is nonsingular if λ : M⟶M^{∗} is an isomorphism of A-modules.

Example 2.8 i) The symmetric form (A, λ) defined by

λ = 1 : A ⟶ A^{∗} ; a ⟶ (b ⟶ b\overline{a})

is nonsingular. ii) The hyperbolic ϵ-symmetric form H^{ϵ}(P) = (P ⊕ P^{∗}, λ) defined for any f.g. projective A-module P by

\begin{aligned}λ = ( \begin{matrix}0 & 1 \\ ϵ & 0 \end{matrix} ) : P ⊕ P^{∗} & ⟶ (P ⊕ P^{∗})^{∗} = P^{∗} ⊕ P ; \\ & (x, f) ⟶ ((y, g) ⟶ f(y) + ϵ\overline{g(x)}) \end{aligned}

is nonsingular.

Example 2.9 Given an oriented m-dimensional manifold (or geometric Poincaré complex) M^{m} and a regular covering M̃ with group of covering translations π cap product with the fundamental class [M] ∈ H_{m}(M) defines the Poincaré duality ℤ[π]-module isomorphisms [M]∩− : H^{m−∗}(M̃)⟶H_{∗}(M̃). If m = 2n geometric intersection numbers define a (−)^{n}-symmetric form (H_{n}(M̃), λ) over ℤ[π] with adjoint the composite

λ : H_{n}(M̃)⟶^{([M]∩−)^{−1}}H^{n}(M̃)⟶^{evaluation} H_{n}(M̃)^{∗} .

If H_{n}(M̃) is a f.g. projective ℤ[π]-module then the evaluation map is an isomorphism, and (H_{n}(M̃), λ) is a nonsingular form. In particular, for M^{2n} = S^{n} × S^{n} this is the hyperbolic form H^{(−)^{n}}(ℤ[π]) of 2.8 ii). An element x ∈ im(π_{n}(M)⟶π_{n}(M̃) can be killed by surgery if and only if it can be represented by a framed embedding x : S^{n}×D^{n} ⊂ M, in which case the homology class x ∈ H_{n}(M̃) is such that λ(x)(x) =. However, the condition λ(x)(x) = 0 given by the symmetric structure alone is not sufficient for the existence of such a framed embedding. It is also necessary to consider quadratic structure.

Definition 2.10 An ϵ-quadratic form (M, λ, μ) over A is an ϵ-symmetric form (M, λ) together with a function

μ : M ⟶ Q_{ϵ}(A) ; x ⟶ μ(x)

such that for all x, y ∈ M, a ∈ A

i) μ(x + y) − μ(x) − μ(y) = λ(x, y) ∈ Q_{ϵ}(A) ,
ii) μ(x) + ϵ\overline{μ(x)} = λ(x, x) ∈ im(1 + T_{ϵ} : Q_{ϵ}(A)⟶Q^{ϵ}(A)) ,
iii) μ(ax) = aμ(x)\overline{a} ∈ Q_{ϵ}(A) .

Example 2.11 An (n − 1)-connected 2n-dimensional normal map (f, b) : M^{2n}⟶X determines a (−)^{n}-quadratic form (K_{n}(M), λ, μ) over A = ℤ[π_{1}(X)], with

K_{n}(M) = π_{n+1}(f) = H_{n+1}(f̃) = ker(f̃_{∗} : H_{n}(M̃)⟶H_{n}(X̃))

the stably f.g. free kernel A-module, and f̃: M̃⟶X̃ a π_{1}(X)-equivariant lift of f to the universal covers. The pairing λ : K_{n}(M) × K_{n}(M)⟶A is defined by geometric intersection numbers. Every element x ∈ K_{n}(M) can be represented by a framed immersion x : S^{n} × D^{n}⟶M with a nullhomotopy in X, and μ(x) ∈ Q_{(−)^{n}}(A) is the geometric self-intersection, such that μ(x) = 0 if (and for n ≥ 3 only if) it can be killed by surgery, i.e. represented by a framed embedding S^{n} × D^{n} ⊂ M with a nullhomotopy in X. In general, the effect of the surgery

(f′, b′) : M^{′2n} = cl(M − S^{n} × D^{n}) ∪ D^{n+1} × S^{n−1} ⟶ X

is not (n − 1)-connected - see 4.2 for further discussion. Note that K_{n}(M) = 0 if (and for n ≥ 2 only if) f : M⟶X is a homotopy equivalence, by the theorem of J.H.C.Whitehead.

Definition 2.12 For any f.g. projective A-module P define the nonsingular hyperbolic ϵ-quadratic form over A by

H_{ϵ}(P) = (P ⊕ P^{∗}, λ, μ)

with

λ : ( \begin{matrix}0 & 1 \\ ϵ & 0 \end{matrix} ) : P ⊕ P^{∗} ⟶ (P ⊕ P^{∗})^{∗} = P^{∗} ⊕ P ;

(x, f) ⟶ ((y, g) ⟶ f(y) + ϵ\overline{g(x)}) ,

μ : P ⊕ P^{∗} ⟶ Q_{ϵ}(A) ; (x, f) ⟶ f(x) .

(P ⊕ P^{∗}, λ) is the hyperbolic (−)^{n}-symmetric form H^{(−)^{n}}(P) of 2.8 ii).

Remark 2.13 The main result of even-dimensional surgery obstruction theory is that for n ≥ 3 an (n − 1)-connected 2n-dimensional normal map (f, b) : M^{2n}⟶X is normal bordant to a homotopy equivalence if and only if there exists an isomorphism of (−)^{n}-quadratic forms over ℤ[π_{1}(X)] of the type

(K_{n}(M), λ, μ) ⊕ H_{(−)^{n}}(ℤ[π_{1}(X)]^{m}) ⟶ H_{(−)^{n}}(ℤ[π_{1}(X)]^{m′})

for some m, m′ ≥ 0.

Example 2.14 There is also a relative version of 2.11. An (n − 1)-connected 2ndimensional normal map of pairs (M^{2n}, ∂M)⟶(X, ∂X) with ∂M⟶∂X a homotopy equivalence has a kernel nonsingular (−)^{n}-quadratic form (K_{n}(M), λ, μ) over ℤ[π_{1}(X)] .

Remark 2.15 (Realization of forms) For every (2n − 1)-dimensional manifold X^{2n−1} with n ≥ 3 and every nonsingular (−)^{n}-quadratic form (K, λ, μ) over ℤ[π_{1}(X)] with K f.g. free there exists an (n − 1)- connected normal map

(M^{2n}; ∂_{0}M, ∂_{1}M) ⟶ X^{2n−1} × ([0, 1]; {0}, {1})

with ∂_{0}M = X⟶X the identity and ∂_{1}M⟶X a homotopy equivalence, with kernel form (K_{n}(M) = K, λ, μ). Such a normal map is the trace of type (n − 1, n − 1) surgeries on the identity X⟶X by m disjoint trivial embeddings S^{n−1} ×D^{n} ⊂ ∂_{0}M, with a handle decomposition

M = ∂_{0}M × [0, 1] ∪ ∪_{m}D^{n} × D^{n} .

Definition 2.16 An isomorphism of ϵ-symmetric forms f : (M, λ)⟶(M′, λ′) is an A-module isomorphism f : M⟶M′ such that

λ′(f(x), f(y)) = λ(x, y) ∈ A .

An isomorphism of ϵ-quadratic forms f : (M, λ, μ)⟶(M′, λ′, μ′) is an isomorphism of the underlying ϵ-symmetric forms f : (M, λ)⟶(M′, λ′) such that

μ′(f(x)) = μ(x) ∈ Q_{ϵ}(A) .

Proposition 2.17 If there exists a central element s ∈ A such that s + \overline{s} = 1 ∈ A there is an identification of categories

{ϵ−quadratic forms over A} = {ϵ−symmetric forms over A} .

Proof: The ϵ-symmetrization map 1 + T_{ϵ} : Q_{ϵ}(M)⟶Q^{ϵ}(M) is an isomorphism for any A-module M, with inverse

Q^{ϵ}(M) ⟶ Q_{ϵ}(M) ; λ ⟶ ((x, y) ⟶ sλ(x, y)) .

For any ϵ-quadratic form (M, λ, μ) the ϵ-quadratic function μ is determined by the ϵ-symmetric pairing λ, with

μ(x) = sλ(x, x) ∈ Q_{ϵ}(A) .

In particular, 2.17 applies if 2 ∈ A is invertible, with s = 1/2 ∈ A.

For any ϵ-symmetric form (M, λ) and x ∈ M

λ(x, x) ∈ Q^{ϵ}(A) .

Definition 2.18 An ϵ-symmetric form (M, λ) is even if for all x ∈ M

λ(x, x) ∈ im(1 + T_{ϵ} : Q_{ϵ}(A)⟶Q^{ϵ}(A)) .

Proposition 2.19 If for one of the two possible values of ϵ = ±1 the map 1 + T_{ϵ} : Q_{ϵ}(A)⟶Q^{ϵ}(A) is an injection there is an identification of categories

{ϵ−quadratic forms over A} = {even ϵ−symmetric forms over A} .

Proof: Given an even ϵ-symmetric form (M, λ) over A there is a unique function μ : M⟶Q_{ϵ}(A) such that for all x ∈ M

(1 + T_{ϵ})(μ(x)) = λ(x, x) ∈ Q^{ϵ}(A) ,

which then automatically satisfies the conditions of 2.10 for (M, λ, μ) to be an ϵquadratic form.

Example 2.20 The symmetrization map 1 + T = 2 : Q_{+}(ℤ) = ℤ⟶Q^{+}(ℤ) = ℤ is an injection, so that quadratic forms over ℤcoincide with the even symmetric forms.

Definition 2.21 If the additive group of the ring A with the ℤ[ℤ_{2}]-module structure determined by the involution splits as

A = ℤ ⊕ Ã

then the Q-groups of A split as

Q^{ϵ}(A) = Q^{ϵ}(ℤ) ⊕ Q̃^{ϵ}(A) ,

Q_{ϵ}(A) = Q_{ϵ}(ℤ) ⊕ Q̃_{ϵ}(A) ,

with

Q̃^{ϵ}(A) = \{a ∈ Ã|ϵ\overline{a} = a\} ,
Q̃_{ϵ}(A) = Ã/\{a − ϵ\overline{a}|a ∈ Ã\} .

In the circumstances of 2.21 we write the components of elements a ∈ A as

(a_{0}, ã) ∈ ℤ ⊕ Ã ,

calling a_{0} the integral component of a and ã the reduced component of a. More generally, if (M, λ, μ) is an ϵ-quadratic form we write

λ = λ_{0} ⊕ λ̃ , μ = μ_{0} ⊕ μ̃ .

Proposition 2.22 If the involution A⟶A is such that there is a ℤ[ℤ_{2}]-module splitting

A = ℤ ⊕ (B ⊕ B̄)

then the morphisms of additive groups

{B ⟶ Q̃_{ϵ}(A) ; b ⟶ (b, 0) = (0, ϵb̄), \atop 1 + T_{ϵ} : Q̃_{ϵ}(A) ⟶ Q̃^{ϵ}(A) ; (b, 0) ⟶ (b, ϵb̄)}

are isomorphisms. The reduced component μ̃ of the function μ in any ϵ-quadratic form (M, λ, μ) over A is determined by the reduced component λ̃ of λ, with

(1 + T_{ϵ})μ̃(x) = λ̃(x, x) ∈ Q̃^{ϵ}(A) (x ∈ M) .

Proof: For any ring with involution A define the ϵ-symmetric Q̂-groups

Q̂^{ϵ}(A) = {a ∈ A|ϵā = a}/{b + ϵb̄|b ∈ A}

to fit into an exact sequence

0 ⟶ Q̂^{−ϵ}(A) ⟶ Q_{ϵ}(A)⟶^{1+T_{ϵ}}Q^{ϵ}(A) ⟶ Q̂^{ϵ}(A) ⟶ 0 .

For example,

Q̂^{+}(ℤ) = ℤ/2 , Q̂^{−}(ℤ) = 0 .

If A = ℤ ⊕ Ã the Q̂-groups split as

Q̂^{ϵ}(A) = Q̂^{ϵ}(ℤ) ⊕ Q̂^{ϵ}(Ã) ,

and the exact sequence is the direct sum of an integral and a reduced exact sequence. If also Ã = B ⊕ B̄ then

Q̂^{ϵ}(Ã) = 0 ,

so the reduced exact sequence collapses to an isomorphism 1 + T_{ϵ} : Q̃_{ϵ}(A)⟶Q̃^{ϵ}(A) .

A group ring A = ℤ[π] with the w-twisted involution (1.3) has a ℤ[ℤ_{2}]-module splitting

ℤ[π] = ℤ ⊕ ℤ[π − {1}] .

The integral components of an ϵ-quadratic form (M, λ, μ) over ℤ[π] are invariant under the group action

λ_{0}(gx, gy) = λ_{0}(x, y) , μ_{0}(gx) = μ_{0}(x) (x, y ∈ M, g ∈ π) ,

and so may be regarded as defining an ϵ-quadratic form (M_{0}, λ_{0}, μ_{0}) over ℤon the induced ℤ-module M_{0} = ℤ ⊗_{ℤ[π]} M .

Example 2.23 If π is a group without 2-torsion and A = ℤ[π] with the oriented involution ḡ = g^{−1} (g ∈ π) then a decomposition π = {1} ∪ S ∪ S^{−1} determines a ℤ[ℤ_{2}]-module splitting

ℤ[π] = ℤ ⊕ (ℤ[S] ⊕ ℤ[S^{−1}])

satisfying the hypothesis of 2.22, so that up to isomorphism

Q̃_{ϵ}(ℤ[π]) = Q̃^{ϵ}(ℤ[π]) = ℤ[S] .

A quadratic form over ℤ[π] is the same as an even symmetric form. A skew-quadratic form (M, λ, μ) over ℤ[π] is the same as a skew-symmetric form (M, λ) over ℤ[π] together with a function μ_{0} : M_{0}⟶Q_{−}(ℤ) = ℤ/2 such that (M_{0}, λ_{0}, μ_{0}) is a skewquadratic form ober ℤ. There exists such μ_{0} for every skew-symmetric form (M, λ) over ℤ[π] with M f.g. projective, but it is not unique.

Example 2.24 The skew-symmetric form (M, λ) over ℤdefined by

M = ℤ ⊕ ℤ , λ : M × M ⟶ ℤ ; ((a, b), (c, d)) ⟶ ad − bc

admits two distinct skew-quadratic refinements (M, λ, μ_{0}), (M, λ, μ_{1}), with

{μ_{0} : M ⟶ Q_{−}(ℤ) = ℤ/2 ; (a, b) ⟶ ab , \atop μ_{1} : M ⟶ Q_{−}(ℤ) = ℤ/2 ; (a, b) ⟶ a^{2} + ab + b^{2} .}

§3. The even-dimensional L-groups

We define the even-dimensional surgery obstruction groups L_{2n}(A), using the following preliminary result.

Lemma 3.1 For any nonsingular ϵ-quadratic form (M, λ, μ) on a f.g. projective Amodule M the sum (M, λ, μ) ⊕ (M, −λ, −μ) is isomorphic to to the hyperbolic ϵquadratic form H_{ϵ}(M). Proof: Let N be a f.g. projective A-module such that M ⊕ N is f.g. free, with base elements {x_{1}, x_{2}, …, x_{m}} say. Let

λ_{ij} = (λ ⊕ 0)(x_{j})(x_{i}) ∈ A (1 ≤ i < j ≤ m)

and choose representatives μ_{i} ∈ A of μ(x_{i}) ∈ Q_{ϵ}(A) (1 ≤ i ≤ m). Define the A-module morphism

\begin{aligned}ψ_{M⊕N} & : M ⊕ N ⟶ (M ⊕ N)^{∗} ; \\ & ∑_{i=1}^{m} a_{i}x_{i} ⟶ (∑_{j=1}^{m} b_{j}x_{j} ⟶ ∑_{i=1}^{m} b_{i}μ_{i}\overline{a}_{i} + ∑_{1≤i<j≤m} b_{j}λ_{ij}\overline{a}_{i}) . \end{aligned}

The A-module morphism defined by

ψ : M ⟶^{inclusion}M ⊕ N ⟶^{ψ_{M⊕N}}(M ⊕ N)^{∗} = M^{∗} ⊕ N^{∗} ⟶^{projection}M^{∗}

is such that

\begin{matrix}λ = ψ + ϵψ^{∗} : M ⟶ M^{∗}, \\ μ(x) = ψ(x)(x) ∈ Q_{ϵ}(A) (x ∈ M) . \end{matrix}

As (M, λ, μ) is nonsingular ψ + ϵψ^{∗} : M⟶M^{∗} is an isomorphism. The A-module morphism defined by

ψ̃ = (ψ + ϵψ^{∗})^{−1}ψ(ψ + ϵψ^{∗})^{−1} : M^{∗} ⟶ M

is such that

(ψ + ϵψ^{∗})^{−1} = ψ̃ + ϵψ̃^{∗} : M^{∗} ⟶ M .

Define an isomorphism of ϵ-quadratic forms

f : H_{ϵ}(M) ⟶ (M ⊕ M, λ ⊕ −λ, μ ⊕ −μ)

by

f = ( \begin{matrix}1 & −ϵψ̃^{∗} \\ 1 & ψ̃ \end{matrix} ) : M ⊕ M^{∗} ⟶ M ⊕ M .

Definition 3.2 The 2n-dimensional L-group L_{2n}(A) is the group of equivalence classes of nonsingular (−)^{n}-quadratic forms (M, λ, μ) on stably f.g. free A-modules, subject to the equivalence relation (M, λ, μ) ∼ (M′, λ′, μ′) if there exists an isomorphism of (−)^{n}-quadratic forms

(M, λ, μ) ⊕ H_{(−)^{n}}(A^{m}) ⟶ (M′, λ′, μ′) ⊕ H_{(−)^{n}}(A^{m′})

for some f.g. free A-modules A^{m}, A^{m′} .

The addition and inverses in L_{2n}(A) are given by

(M_{1}, λ_{1}, μ_{1}) + (M_{2}, λ_{2}, μ_{2}) = (M_{1} ⊕ M_{2}, λ_{1} ⊕ λ_{2}, μ_{1} ⊕ μ_{2}) ,
−(M, λ, μ) = (M, −λ, −μ) ∈ L_{2n}(A) .

It should be clear that L_{2n}(A) depends only on the residue n(mod 2), so that only two L-groups have actually been defined, L_{0}(A) and L_{2}(A). Note that 3.2 uses Lemma 3.1 to justify (M, λ, μ) ⊕ (M, −λ, −μ) ∼ 0.

Remark 3.3 The surgery obstruction of an (n−1)-connected 2n-dimensional normal map (f, b) : M^{2n}⟶X is an element

σ_{∗}(f, b) = (K_{n}(M), λ, μ) ∈ L_{2n}(ℤ[π_{1}(X)])

such that σ_{∗}(f, b) = 0 if (and for n ≥ 3 only if) (f, b) is normal bordant to a homotopy equivalence.

Example 3.4 Let M = M_{g}^{2} be the orientable 2-manifold (=surface) of genus g, with degree 1 Gauss map f : M⟶S^{2}. A choice of framing of the stable normal bundle of an embedding M ⊂ℝ^{3} determines a 0-connected 2-dimensional normal map

(f, b) : M ⟶ S^{2} .

For a standard choice of framing the kernel form and the surgery obstruction are given by

σ_{∗}(f, b) = H_{−}(ℤ^{g}) = 0 ∈ L_{2}(ℤ)

and (f, b) is normal bordant to a homotopy equivalence, i.e. M is framed nullcobordant.

Example 3.5 The even-dimensional L-groups of the ring A = ℝ of real numbers with the identity involution are given by

L_{2n}(ℝ) = \{ \begin{matrix}ℤ & if n is even \\ 0 & if n is odd . \end{matrix}

Since 1/2 ∈ ℝ there is no difference between symmetric and quadratic forms over ℝ. The signature (alias index) of a nonsingular symmetric form (M, λ) over ℝ is defined by

σ(M, λ) = no. of positive eigenvalues of λ − no. of negative eigenvalues of λ ∈ ℤ .

Here, we are identifying the symmetric form λ ∈ Q^{+}(M) with the symmetric n × n matrix (λ(x_{i}, x_{j})_{1≤i,j≤n}) ∈ M_{n,n}(ℝ) determined by any choice of base x_{1}, x_{2}, …, x_{n} for M. By Sylvester’s law of inertia the rank of M and the signature define a complete set of invariants for the isomorphism classification of nonsingular symmetric forms (M, λ) on finite-dimensional vector spaces M over

ℝ

, meaning that two forms are isomorphic if and only if they have the same rank and signature. A nonsingular quadratic form (M, ψ) over ℝ is isomorphic to a hyperbolic form if and only if it has signature 0. Two such forms (M, λ), (M′, λ′) are related by an isomorphism

(M, λ) ⊕ H_{+}(ℝ^{m}) ⟶ (M′, λ′) ⊕ H_{+}(ℝ^{m′})

if and only if they have the same signature

σ(M, λ) = σ(M′, λ′) ∈ ℤ .

Moreover, every integer is the signature of a form, since 1 ∈ ℤ is the signature of the nonsingular symmetric form (ℝ,1) with

1 : ℝ⟶ℝ^{∗} ; x ⟶ (y ⟶ xy)

and for any nonsingular symmetric forms (M, λ), (M′, λ′) over ℝ

σ((M, λ) ⊕ (M′, λ′)) = σ(M, λ) + σ(M′, λ′) , σ(M, −λ) = −σ(M, λ) ∈ ℤ .

The isomorphism of 3.7 in the case n ≡ 0(mod2) is defined by

L_{0}(ℝ) ⟶ ℤ ; (M, λ) ⟶ σ(M, λ) .

L_{2}(ℝ) = 0 because every nonsingular skew-symmetric (alias symplectic) form over ℝ admits a lagrangian, and is thus isomorphic to a hyperbolic form.

Example 3.6 The even-dimensional L-groups of A = ℤ_{2} are given by

L_{2n}(ℤ_{2}) = ℤ_{2} .

It is clear that L_{0}(A) = L_{2}(A) for a ring A of characteristic 2, since +1 = −1 ∈ A, so we need only consider L_{0}(ℤ_{2}). By a classical theorem for any nonsingular quadratic form (M, λ, μ) over ℤ_{2} there exists a symplectic base x_{1}, …, x_{2m} for M, such that

λ(x_{i})(x_{j}) = \{ \begin{matrix}1 & if |i − j| = m \\ 0 & otherwise . \end{matrix}

The Arf invariant of (M, λ, μ) is defined using such a base to be

c(M, λ, μ) = ∑_{i=1}^{m} μ(x_{i})μ(x_{i+m}) ∈ ℤ_{2} .

The isomorphism of 3.6 is defined by

L_{2n}(ℤ_{2}) ⟶ ℤ_{2} ; (M, λ, μ) ⟶ c(M, λ, μ) .

The nonsingular quadratic form (ℤ_{2} ⊕ ℤ_{2}, ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )) over ℤ_{2} has Arf invariant 1.

Example 3.7 The even-dimensional L-groups of A = ℤ with (necessarily) the identity involution are given by

L_{2n}(ℤ) = \{ \begin{matrix}ℤ & if n is even \\ ℤ_{2} & if n is odd . \end{matrix}

See Chapter II of the book “Symmetric bilinear forms” by Milnor and Husemoller for what is known about the isomorphism classification of nonsingular symmetric and quadratic forms over ℤ. Fortunately, it is much easier to decide if two forms become isomorphic after adding hyperbolics then whether they are actually isomorphic. Define the signature of a nonsingular symmetric form (M, λ) over ℤ to be the signature of the induced nonsingular symmetric form over ℝ

σ(M, λ) = σ(ℝ ⊗ M, 1 ⊗ λ) ∈ ℤ .

It is a non-trivial theorem that two nonsingular even symmetric forms (M, λ), (M′, λ′) are related by an isomorphism

(M, λ) ⊕ H_{+}(ℤ^{m}) ⟶ (M′, λ′) ⊕ H_{+}(ℤ^{m′})

if and only if they have the same signature

σ(M, λ) = σ(M′, λ′) ∈ ℤ .

Moreover, not every integer arises as the signature of an even symmetric form, only those divisible by 8. The isomorphism of 3.7 in the case n ≡ 0(mod2) is given by

L_{0}(ℤ) ⟶ ℤ ; (M, λ) ⟶ σ(M, λ)/8 .

The nonsingular even symmetric form (ℤ^{8}, λ) associated to the “E_{8}-lattice”

λ = ( \begin{matrix}2 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 2 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 2 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 & 2 & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 2 \end{matrix} ) : ℤ^{8} ⟶ (ℤ^{8})^{∗}

has signature

σ(ℤ^{8}, λ) = 8 ∈ ℤ .

Example 3.8 The result of 8 type (3,3) surgeries on the identity normal map (f, b) = id. : S^{7}⟶S^{7} with the trace kernel form (ℤ^{8}, λ) is a normal map (f′, b′) : Σ^{7}⟶S^{7} from one of the exotic 7-spheres Σ^{7} originally constructed by Milnor.

The isomorphism of 3.7 in the case n ≡ 1(mod2) is given by the Arf invariant of the induced form over ℤ_{2}

L_{2}(ℤ) ⟶ L_{2}(ℤ_{2}) = ℤ_{2} ; (M, λ, μ) ⟶ c(ℤ_{2} ⊗ (M, λ, μ)) .

The nonsingular skew-quadratic form (ℤ ⊕ ℤ, λ, μ_{1}) over ℤ defined in 2.24 has Arf invariant 1.

Example 3.9 The trace of a type (4,4) surgery on the identity normal map (f, b) = id. : S^{9}⟶S^{9} with the kernel form (ℤ⊕ℤ, λ, μ_{1}) is a smooth 10-dimensional manifold with two boundary components which can be coned off to obtain the closed 10dimensional PL manifold without differentiable structure of Kervaire.

§4. Split forms

A “split form” on an A-module M is an element ψ ∈ S(M) = Hom_{A}(M, M^{∗}). Split forms are more convenient to deal with than ϵ-quadratic forms in describing the algebraic effects of even-dimensional surgery (in §5 below), and are closer to the geometric applications. From now on we shall be only concerned with forms on f.g. projective A-modules.

The main result of §4 is that the ϵ-quadratic stuctures (λ, μ) on a f.g. projective A-module M correspond to the elements of the ϵ-quadratic group of 2.5

Q_{ϵ}(M) = coker(1 − T_{ϵ} : S(M)⟶S(M)) .

The pair of functions (λ, μ) used to define an ϵ-quadratic form (M, λ, μ) can thus be replaced by an equivalence class of A-module morphisms ψ : M⟶M^{∗} such that

λ(x, y) = ψ(x)(y) + ϵ\overline{ψ(y)(x)} ∈ A , μ(x) = ψ(x)(x) ∈ Q_{ϵ}(A) ,

i.e. by an equivalence class of split forms.

Definition 4.1 i) A split ϵ-quadratic form (M, ψ) over A is a f.g. projective Amodule M together with an element ψ ∈ S(M). ii) An equivalence of split ϵ-quadratic forms (M, ψ), (M, ψ′) on the same f.g. projective A-module M is an element χ ∈ Q_{−ϵ}(M) such that

ψ′ − ψ = χ − ϵχ^{∗} : M ⟶ M^{∗} .

iii) A morphism of split ϵ-quadratic forms

(f, χ) : (M, ψ) ⟶ (M′, ψ′)

is an A-module morphism f ∈ Hom_{A}(M, M′) together with an element χ ∈ Q_{−ϵ}(M) such that

f^{∗}ψ′f − ψ = χ − ϵχ^{∗} : M ⟶ M^{∗} .

Proposition 4.2 i) The ϵ-quadratic structures (λ, μ) on a f.g. projective A-module M are in one-one correspondence with the equivalence classes [ψ] ∈ Q_{ϵ}(M) of split ϵ-quadratic forms (M, ψ) over A. ii) Every morphism of ϵ-quadratic forms f : (M, λ, μ)⟶(M′, λ′, μ′) lifts to a morphism of split ϵ-quadratic forms (f, χ) : (M, ψ)⟶(M′, ψ′). Proof: i) Given a split ϵ-quadratic form (M, ψ) define an ϵ-quadratic form (M, λ, μ) by

λ = (1 + T_{ϵ})ψ : M ⟶ M^{∗} ;
x ⟶ (y ⟶ ψ(y)(x) + ϵ\overline{ψ(y)(x)}) ,
μ : M ⟶ Q_{ϵ}(A) ; x ⟶ ψ(x)(x) .

Conversely, given an ϵ-quadratic form (M, λ, μ) let (M, ψ) be the split ϵ-quadratic form defined as in 3.1, such that (λ, μ) = (0, 0) if and only if [ψ] = 0 ∈ Q_{ϵ}(M). ii) Immediate from i).

Thus Q_{ϵ}(M) is both the group of isomorphism classes of ϵ-quadratic and split ϵ-quadratic structures on a f.g. projective A-module M.

For f.g. free A-modules A^{m} we have:

Proposition 4.3 An ϵ-quadratic form (A^{m}, λ, μ) over A is determined by an m × mmatrix λ = {λ_{ij} ∈ A|1 ≤ i, j ≤ m} such that

\overline{λ_{ij}} = ϵλ_{ji} ∈ A

and a collection of elements μ = {μ_{i} ∈ Q_{ϵ}(A)|1 ≤ i ≤ m} such that

μ_{i} + ϵ\overline{μ_{i}} = λ_{ii} ∈ Q^{ϵ}(A) .

Choosing any representatives μ_{i} ∈ A of μ_{i} ∈ Q_{ϵ}(A^{m}) there is defined a split ϵquadratic form (A^{m}, ψ) with ψ = {ψ_{ij} ∈ A|1 ≤ i, j ≤ m} the m × m matrix defined by

ψ_{ij} = \{ \begin{matrix}λ_{ij} & if i < j \\ μ_{i} & if i = j \\ 0 & otherwise . \end{matrix}

Remark 4.4 If A = ℤ ⊕ B ⊕ B̄ (as in 2.22) the element ψ ∈ Q_{ϵ}(A^{m}) associated to an ϵ-quadratic form (A^{m}, λ, μ) is determined by λ and the integral components μ_{0} = {(μ_{i})_{0} ∈ Q_{ϵ}(ℤ)|1 ≤ i ≤ m} .

Remark 4.5 Historically, the kernel of an (n − 1)-connected 2n-dimensional normal map (f, b) : M⟶X was defined to be the (−)^{n}-quadratic form (K_{n}(M), λ, μ) over ℤ[π_{1}(X)] defined via (intersection, self-intersection) numbers, rather than as a split form (K_{n}(M), ψ). However, consider the following. Define a presentation of (f, b) to be an (n − 1)-connected normal bordism

((g, c); (f, b), (f′, b′) : (W^{2n+1}; M^{2n}, M^{′2n}) ⟶ X × ([0, 1]; 0, 1) obtained from the product

(f, b) × id : M × ([0, 1]; 0, 1)⟶X × ([0, 1]; 0, 1)

by surgery on a base of the (stably) f.g. free ℤ[π_{1}(X)]-module

K_{n}(M × [0, 1]) = K_{n}(M) ,

so that (f′, b′) is a copy of (f, b). The identification (2n + 1)-dimensional normal map

(g, c)/((f, b) = (f′, b′)) : W/(M = M′) ⟶ X × ([0, 1]/0 = 1) = X × S^{1} is an (n − 1)-connected ℤ[π_{1}(X)]-homology equivalence. The ℤ[π_{1}(X)]-module morphisms induced by the inclusions M⟶W, M′⟶W

h = inclusion_{∗} : K_{n}(M) ⟶ K_{n}(W) ,
h′ = inclusion_{∗} : K_{n}(M) = K_{n}(M′) ⟶ K_{n}(W)

are such that h − h′ : K_{n}(M)⟶K_{n}(W) is a ℤ[π_{1}(X)]-module isomorphism. The ℤ[π_{1}(X)]-module morphism

ψ : K_{n}(M)⟶^{(h−h′)^{−1}h}K_{n}(M)⟶^{adjoint(λ)}K_{n}(M)^{∗}

defines a split (−)^{n}-quadratic form (K_{n}(M), ψ) corresponding to (K_{n}(M), λ, μ).

Example 4.6 Presentations as in 4.5 arise naturally in knot theory. Given an embedding k : S^{2n−1} ⊂ S^{2n+1} let M be a simple Seifert surface of the knot k, an (n − 1)-connected framed codimension 1 submanifold M^{2n} ⊂ S^{2n+1} with boundary ∂M = k(S^{2n−1}). The inclusion defines an (n − 1)-connected normal map

(f, b) : (M^{2n}, ∂M) ⟶ (D^{2n+2}, k(S^{2n−1}))

which is the identity on the boundary. The knot complement is the (2n + 1)- dimensional manifold with boundary defined by

(X^{2n+1}, ∂X) = (cl(S^{2n+1} − nbhd.(k(S^{2n−1}))), S^{2n−1} × S^{1})

such that there is defined an (n − 1)-connected (2n + 1)-dimensional normal map

(e, a) : (X, ∂X) ⟶ (D^{2n+2}, k(S^{2n−1}))

which is the identity on the boundary. Cutting (e, a) along (f, b) results in a presentation

((g, c); (f, b), (f′, b′) : (W^{2n+1}; M^{2n}, M^{′2n}) ⟶ D^{2n+2} × ([0, 1]; 0, 1) .

The Seifert matrix of the knot is the matrix with respect to any base of the f.g. free ℤ-module K_{n}(M) of the ℤ-module morphism ψ : K_{n}(M)⟶K_{n}(M)^{∗} given by 4.5.

§5. Surgery on forms

In §5 we develop algebraic surgery on forms. The effect of geometric surgery on an (n − 1)-connected 2n-dimensional normal map is an algebraic surgery on a (−)^{n}- quadratic form. Moreover, geometric surgery is possible if and only if algebraic surgery is possible.

Given an ϵ-quadratic form (M, λ, μ) it is possible to kill an element x ∈ M by algebraic surgery if and only if μ(x) = 0 ∈ Q_{ϵ}(A) and x generates a direct summand < x >= Ax ⊂ M. The effect of the surgery is the ϵ-quadratic form (M′, λ′, μ′) defined on the subquotient M′ =< x >^{⊥} / < x > of M, with < x >^{⊥} = {y ∈ M|λ(y)(x) = 0}.

Definition 5.1 i) Given an ϵ-symmetric form (M, λ) and a submodule L ⊆ M define the orthogonal submodule

L^{⊥} = {y ∈ M|λ(y)(x) = 0 ∈ A for all x ∈ L} = ker(i^{∗}λ : M⟶L^{∗})

with i : L⟶M the inclusion. If (M, λ) is nonsingular and L is a direct summand of M then so is L^{⊥}. ii) A sublagrangian of a nonsingular ϵ-quadratic form (M, λ, μ) over A is a direct summand L ⊆ M such that

μ(L) = {0} ⊆ Q_{ϵ}(A) ,

so that

λ(L)(L) = {0} , L ⊆ L^{⊥} .

iii) A lagrangian of (M, λ, μ) is a sublagrangian L such that L^{⊥} = L.

The main result of §5 is that the inclusion of a sublagrangian is a morphism of ϵ-quadratic forms

i : (L, 0, 0) ⟶ (M, λ, μ)

which extends to an isomorphism

f : H_{ϵ}(L) ⊕ (L^{⊥}/L, [λ], [μ]) ⟶ (M, λ, μ)

with H_{ϵ}(L) the hyperbolic ϵ-quadratic form (2.11)

Example 5.2 Let (f, b) : M^{2n}⟶X be an (n − 1)-connected 2n-dimensional normal map with kernel (−)^{n}-quadratic form (K_{n}(M), λ, μ) over ℤ[π_{1}(X)], and n ≥ 3. An element x ∈ K_{n}(M) generates a sublagrangian L =< x >⊂ K_{n}(M) if and only if it can be killed by a type (n, n − 1)-surgery on S^{n} × D^{n} ⊂ M with an (n − 1)-connected trace normal bordism

((g, c); (f, b), (f′, b′)) : (W^{2n+1}; M^{2n}, M^{′2n}) ⟶ X × ([0, 1]; {0}, {1})

such that K_{n+1}(W, M′) = 0. The kernel form of the effect of such a surgery

(f′, b′) : M′ = cl(M − S^{n} × D^{n}) ∪ D^{n+1} × S^{n−1} ⟶ X

is given by

(K_{n}(M′), λ′, μ′) = (L^{⊥}/L, [λ], [μ]) .

There exists an (n − 1)-connected normal bordism (g, c) of (f, b) to a homotopy equivalence (f′, b′) with K_{n+1}(W, M′) = 0 if and only if (K_{n}(M), λ, μ) admits a lagrangian.

Remark 5.3 There are other terminologies. In the classical theory of quadratic forms over fields a lagrangian is a “maximal isotropic subspace”. Wall called hyperbolic forms “kernels” and the lagrangians “subkernels”. Novikov called hyperbolic forms “hamiltonian”, and introduced the name “lagrangian”, because of the analogy with the hamiltonian formulation of physics in which the lagrangian expresses a minimum condition.

Example 5.4 An (n − 1)-connected (2n + 1)-dimensional normal bordism

((g, c); (f, b), (f′, b′)) : (W^{2n+1}; M^{2n}, M^{′2n}) ⟶ X × ([0, 1]; {0}, {1})

with K_{n+1}(W, M′) = 0 determines a sublagrangian

L = im(K_{n+1}(W, M)⟶K_{n}(M)) ⊂ K_{n}(M)

of the kernel (−)^{n}-quadratic form (K, λ, μ) of (f, b), with K = K_{n}(M). L is a lagrangian if and only if (f′, b′) is a homotopy equivalence. W has a handle decomposition on M of the type

W = M × I ∪ ∪_{m}(n + 1)−handles D^{n+1} × D^{n} ,

and L ≅ K_{n+1}(W, M) ≅ ℤ[π_{1}(X)]^{m} is a f.g. free ℤ[π_{1}(X)]-module with rank the number m of (n + 1)-handles. The exact sequences of stably f.g. free ℤ[π_{1}(X)]- modules

\begin{matrix}0 ⟶ K_{n+1}(W, M) ⟶ K_{n}(M) ⟶ K_{n}(W) ⟶ 0 , \\ 0 ⟶ K_{n}(M′) ⟶ K_{n}(W) ⟶ K_{n}(W, M′) ⟶ 0 \end{matrix}

are isomorphic to

\begin{matrix}0 ⟶ L ⟶^{i} K ⟶ K/L ⟶ 0 , \\ 0 ⟶ L^{⊥}/L ⟶ K/L ⟶^{[i^{∗}λ]} L^{∗} ⟶ 0 . \end{matrix}

It is actually more convenient to work with the split forms of §4.

Definition 5.5 i) A sublagrangian of a nonsingular split ϵ-quadratic form (M, ψ) is a morphism of split forms

(i, θ) : (L, 0) ⟶ (M, ψ)

with i : L⟶M a split injection. ii) A lagrangian of (M, ψ) is a sublagrangian such that the sequence

0 ⟶ L ⟶^{i} M ⟶^{i^{∗}(ψ+ϵψ^{∗})} L^{∗} ⟶ 0

is exact.

A nonsingular split ϵ-quadratic form (M, ψ) admits a (sub)lagrangian if and only if the associated ϵ-quadratic form (M, λ, μ) admits a (sub)lagrangian. (Sub)lagrangians in split ϵ-quadratic forms are thus (sub)lagrangians in ϵ-quadratic forms with the (−ϵ)-quadratic structure θ, which (following Novikov) is sometimes called the “hessian” form.

Definition 5.6 The nonsingular hyperbolic split ϵ-quadratic form H_{ϵ}(L) is given for any f.g. projective A-module L by

H_{ϵ}(L) = (L ⊕ L^{∗}, ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) : L ⊕ L^{∗} ⟶ (L ⊕ L^{∗})^{∗} = L^{∗} ⊕ L) ,

with lagrangian

(i = ( \begin{matrix}1 \\ 0 \end{matrix} ) , 0) : (L, 0) ⟶ H_{ϵ}(L) .

Theorem 5.7 A nonsingular split ϵ-quadratic form (M, ψ) admits a lagrangian if and only if it is isomorphic to the hyperbolic form H_{ϵ}(L). Moreover, the inclusion (i, θ) : (L, 0)⟶(M, ψ) of a lagrangian extends to an isomorphism of split ϵ-quadratic forms (f, χ) : H_{ϵ}(L)⟶(M, ψ). Proof: An isomorphism of forms sends lagrangians to lagrangians, so any form isomorphic to a hyperbolic has at least one lagrangian. Conversely suppose that (M, ψ) has a lagrangian (i, θ) : (L, 0)⟶(M, ψ). An extension of (i, θ) to an isomorphism (f, χ) : H_{ϵ}(L)⟶(M, ψ) determines a lagrangian f(L^{∗}) ⊂ M complementary to L. We shall construct an isomorphism f by choosing a complementary lagrangian to L in (M, ψ). Let i ∈ Hom_{A}(L, M) be the inclusion, and choose a splitting j′ ∈ Hom_{A}(L^{∗}, M) of i^{∗}(ψ + ϵψ^{∗}) ∈ Hom_{A}(M, L^{∗}), so that

i^{∗}(ψ + ϵψ^{∗})j′ = 1 ∈ Hom_{A}(L^{∗}, L^{∗}) .

In general, j′ : L^{∗}⟶M is not the inclusion of a lagrangian, with j^{′∗}ψj′ ≠ 0 ∈ Q_{ϵ}(L^{∗}). Given any k ∈ Hom_{A}(L^{∗}, L) there is defined another splitting

j = j′ + ik : L^{∗} ⟶ M

such that

j^{∗}ψj = j^{′∗}ψj′ + k^{∗}i^{∗}ψik + k^{∗}i^{∗}ψj′ + j^{′∗}ψik = j^{′∗}ψj′ + k ∈ Q_{ϵ}(L^{∗}) . Choosing a representative ψ ∈ Hom_{A}(M, M^{∗}) of ψ ∈ Q_{ϵ}(M) and setting

k = −j^{′∗}ψj′ : L^{∗} ⟶ L^{∗}

there is obtained a splitting j : L^{∗}⟶M which is the inclusion of a lagrangian (j, ν) : (L^{∗}, 0) ⟶ (M, ψ) .

The isomorphism of split ϵ-quadratic forms

((i j) = ( \begin{matrix}θ & 0 \\ j^{∗}ψi & ψ \end{matrix} ) : H_{ϵ}(L) ⟶ (M, ψ)

is an isomorphism of split ϵ-quadratic forms.

Remark 5.8 Theorem 5.7 is a generalization of Witt’s theorem on the extension to isomorphism of an isometry of quadratic forms over fields. The procedure for modifying the choice of complement to be a lagrangian is a generalization of the Gram-Schmidt method of constructing orthonormal bases in an inner product space. Ignoring the split structure 5.7 shows that a f.g. projective nonsingular ϵ-quadratic form admits a lagrangian (in the sense of 5.1 iii)) if and only if it is isomorphic to a hyperbolic form.

Corollary 5.9 For any nonsingular split ϵ-quadratic form (M, ψ) the diagonal inclusion

Δ : M ⟶ M ⊕ M ; x ⟶ (x, x)

extends to an isomorphism of ϵ-quadratic forms

H_{ϵ}(M) ⟶ (M, ψ) ⊕ (M, −ψ) .

Proof: Apply 5.7 to the inclusion of the lagrangian

(Δ, 0) : (M, 0) ⟶ (M ⊕ M, ψ ⊕ −ψ) .

(We have already used this result in 3.1).

Proposition 5.10 The inclusion (i, θ) : (L, 0)⟶(M, ψ) of a sublagrangian in a nonsingular split ϵ-quadratic form (M, ψ) extends to an isomorphism of forms

(f, χ) : H_{ϵ}(L) ⊕ (L^{⊥}/L, [ψ]) ⟶ (M, ψ) .

Proof: For any direct complement L_{1} to L^{⊥} in M there is defined an A-module isomorphism

h : L_{1} ⟶ L^{∗} ; x ⟶ (y ⟶ (1 + T_{ϵ})ψ(x)(y)) .

Define an A-module morphism

j : L^{∗} ⟶^{h^{−1}} L_{1} ⟶^{inclusion}M .

The nonsingular split ϵ-quadratic form defined by

(H, ϕ) = (L ⊕ L^{∗}, ( \begin{matrix}0 & 1 \\ 0 & j^{∗}ψj \end{matrix} ))

has lagrangian L, so that it is isomorphic to the hyperbolic form H_{ϵ}(L) by 5.7. Also, there is defined a morphism of split ϵ-quadratic forms

(g = (i j) , ( \begin{matrix}θ & i^{∗}ψj \\ 0 & 0 \end{matrix} )) : (H, ϕ) ⟶ (M, ψ)

with g : H⟶M an injection split by

h = ((1 + T_{ϵ})ϕ)^{−1}g^{∗}(1 + T_{ϵ})ψ : M ⟶ H .

The direct summand of M defined by

H^{⊥} = {x ∈ M|(1 + T_{ϵ})ψ(x)(gy) = 0 for all y ∈ H}
= ker(g^{∗}(1 + T_{ϵ})ψ : M⟶H^{∗}) = ker(h : M⟶H)

is such that

M = g(H) ⊕ H^{⊥} .

It follows from the factorization

i^{∗}(1 + T_{ϵ})ψ : M ⟶^{h} H = L ⊕ L^{∗} ⟶^{projection} L^{∗}

that

L^{⊥} = ker(i^{∗}(1 + T_{ϵ})ψ : M⟶L^{∗}) = L ⊕ H^{⊥} .

The restriction of ψ ∈ S(M) to H^{⊥} defines a nonsingular split ϵ-quadratic form (H^{⊥}, ϕ^{⊥}) . Define an extension of g to an isomorphism of split forms

(f = (g g^{⊥}), χ) : (H, ϕ) ⊕ (H^{⊥}, ϕ^{⊥}) ⟶ (M, ψ)

with g^{⊥} : H^{⊥}⟶M the inclusion, and with (H^{⊥}, ϕ^{⊥}) isomorphic to (L^{⊥}/L, [ψ]).

Example 5.12 An (n − 1)-connected (2n + 1)-dimensional normal bordism

((g, c); (f, b), (f′, b′)) : (W^{2n+1}; M^{2n}, M^{′2n}) ⟶ X × ([0, 1]; {0}, {1})

is such that W has a handle decomposition on M of the type

W = M × I ∪ ∪_{m}n−handles D^{n} × D^{n+1} ∪_{m′} (n + 1)−handles D^{n+1} × D^{n} .

Let

(W; M, M′) = (W′; M, M′′) ∪_{M′′} (W′′; M′′, M′)

with

W′ = M × [0, 1] ∪ ∪_{m} n−handles D^{n} × D^{n+1} ,
W′′ = M′′ × [0, 1] ∪ ∪_{m′} (n + 1)−handles D^{n+1} × D^{n} .

The restriction of (g, c) to M′′ is an (n − 1)-connected 2n-dimensional normal map

(f′′, b′′) : M′′ ≅ M#(#_{m}S^{n} × S^{n}) ≅ M′#(#_{m′}S^{n} × S^{n}) ⟶ X

with kernel (−)^{n}-quadratic form

\begin{aligned}(K_{n}(M′′), λ′′, μ′′) & ≅ (K_{n}(M), λ, μ) ⊕ H_{(−)^{n}}(ℤ[π_{1}(X)]^{m}) \\ & ≅ (K_{n}(M′), λ′, μ′) ⊕ H_{(−)^{n}}(ℤ[π_{1}(X)]^{m′}) . \end{aligned}

Thus (K_{n}(M′′), λ′′, μ′′) has sublagrangians

\begin{matrix}L = im(K_{n+1}(W′, M′′)⟶K_{n}(M′′)) ≅ ℤ[π_{1}(X)]^{m} , \\ L′ = im(K_{n+1}(W′′, M′′)⟶K_{n}(M′′)) ≅ ℤ[π_{1}(X)]^{m′} \end{matrix}

such that

(L^{⊥}/L, [λ′′], [μ′′]) ≅ (K_{n}(M), λ, μ) , (L^{′⊥}/L′, [λ′′]′, [μ′′]′) ≅ (K_{n}(M′), λ′, μ′) .

Note that L is a lagrangian if and only if (f, b) : M⟶X is a homotopy equivalence. Similarly for L′ and (f′, b′) : M′⟶X.

§6. Short odd complexes

A “(2n+1)-complex” is the algebraic structure best suited to describing the surgery obstruction of an (n − 1)-connected (2n + 1)-dimensional normal map. In essence it is a 1-dimensional chain complex with (−)^{n}-quadratic Poincaré duality.

As before, let A be a ring with involution.

Definition 6.1 A (2n + 1)-complex over A (C, ψ) is a f.g. free A-module chain complex of the type

C : … ⟶ 0 ⟶ C_{n+1} ⟶^{d} C_{n} ⟶ 0 ⟶ …

together with two A-module morphisms

ψ_{0} : C^{n} = (C_{n})^{∗} ⟶ C_{n+1} , ψ_{1} : C^{n} ⟶ C_{n}

such that

dψ_{0} + ψ_{1} + (−)^{n+1}ψ_{1}^{∗} = 0 : C^{n} ⟶ C_{n} ,

and such that the chain map

(1 + T)ψ_{0} : C^{2n+1−∗} ⟶ C

defined by

d_{C^{2n+1−∗}} = (−)^{n+1}d^{∗} :

(C^{2n+1−∗})_{n+1} = C^{n} ⟶ (C^{2n+1−∗})_{n} = C^{n+1} ,

(1 + T)ψ_{0} = \{ \begin{matrix}ψ_{0} : (C^{2n+1−∗})_{n+1} = C^{n} ⟶ C_{n+1} \\ ψ_{0}^{∗} : (C^{2n+1−∗})_{n} = C^{n+1} ⟶ C_{n} , \end{matrix}

(C^{2n+1−∗})_{r} = C^{2n+1−r} = 0 for r ≠ n, n + 1 ,

is a chain equivalence.

Remark 6.2 A (2n + 1)-complex is essentially the inclusion of a lagrangian in a hyperbolic split (−)^{n}-quadratic form

(( \begin{matrix}ψ_{0} \\ d^{∗} \end{matrix} ) , −ψ_{1}) : (C^{n}, 0) ⟶ H_{(−)^{n}}(C_{n+1}) .

The chain map (1 + T)ψ_{0} : C^{2n+1−∗}⟶C is a chain equivalence if and only if the algebraic mapping cone

0 ⟶ C^{n} \underrightarrow{( \begin{matrix}ψ_{0} \\ d^{∗} \end{matrix} )}C_{n+1} ⊕ C^{n+1} \underrightarrow{( d (−)^{n}ψ_{0}^{∗})}C_{n} ⟶ 0

is contractible, which is just the lagrangian condition. The triple

(form; lagrangian, lagrangian) = (H_{(−)^{n}}(C_{n+1}); C_{n+1}, im ( \begin{matrix}ψ_{0} \\ d^{∗} \end{matrix} ))

is an example of a “(−)^{n}-quadratic formation”. We shall come back to formations in §9 below.

Example 6.3 Define a presentation of an (n − 1)-connected (2n + 1)-dimensional normal map (f, b) : M^{2n+1}⟶X to be a normal bordism

((g, c); (f, b), (f′, b′)) : (W^{2n+2}; M^{2n+1}, M^{′2n+1}) ⟶ X × ([0, 1]; {0}, {1})

such that W ⟶X × [0, 1] is n-connected, with

K_{r}(W) = 0 for r ≠ n + 1 .

Then K_{n+1}(W) a f.g. free ℤ[π_{1}(X)]-module and W has a handle decomposition on M of the type

W = M × I ∪ ∪_{m}(n + 1)−handles D^{n+1} × D^{n+1} ,

and K_{n+1}(W, M) ≅ ℤ[π_{1}(X)]^{m} is a f.g. free ℤ[π_{1}(X)]-module with rank the number m of (n+1)-handles. Thus (W; M, M′) is the trace of type (n, n) surgeries on m disjoint framed embeddings S^{n} × D^{n+1} ⊂ M^{2n+1} with null-homotopy in X representing a set of ℤ[π_{1}(X)]-module generators of K_{n}(M) . For every (n − 1)-connected (2n + 1)- dimensional normal map (f, b) : M^{2n+1}⟶X the kernel ℤ[π_{1}(X)]-module K_{n}(M) is f.g., so that there exists a presentation (g, c) : (W; M, M′)⟶X × ([0, 1]; {0}, {1}). Poincaré duality and the universal coefficient theorem give natural identifications of f.g. free ℤ[π_{1}(X)]-modules

K_{n+1}(W) = K^{n+1}(W, ∂W) = K_{n+1}(W, ∂W)^{∗} , (∂W = M ∪ M′) ,
K_{n+1}(W, M) = K^{n+1}(W, M′) = K_{n+1}(W, M′)^{∗} .

The presentation determines a (2n + 1)-complex (C, ψ) such that

H_{∗}(C) = K_{∗}(M) ,

with

d = (inclusion)_{∗} : C_{n+1} = K_{n+1}(W, M′) ⟶ C_{n} = K_{n+1}(W, ∂W) ,
ψ_{0} = (inclusion)_{∗} : C^{n} = K_{n+1}(W) ⟶ C_{n+1} = K_{n+1}(W, M′) .

The hessian (C^{n}, −ψ_{1} ∈ Q_{(−)^{n+1}}(C^{n})) is the geometric self-intersection (−)^{n+1}- quadratic form on the kernel C^{n} = K_{n+1}(W) of W^{2n+2}⟶X × [0, 1], such that

−(ψ_{1} + (−)^{n+1}ψ_{1}^{∗}) = dψ_{0} = inclusion_{∗} :

C^{n} = K_{n+1}(W) ⟶ C_{n} = K_{n+1}(W, ∂W) = K_{n+1}(W)^{∗} .

The chain equivalence (1 + T)ψ_{0} : C^{2n+1−∗}⟶C induces the Poincaré duality isomorphisms

[M] ∩ − : H^{2n+1−∗}(C) = K^{2n+1−∗}(M) ⟶ H_{∗}(C) = K_{∗}(M) .

Remark 6.4 The (2n + 1)-complex (C, ψ) of 6.3 can also be obtained by working inside M, assuming that X has a single (2n + 1)-cell

X = X_{0} ∪ D^{2n+1} ,

so that there is defined a degree 1 map

collapse : X ⟶ X/X_{0} = S^{2n+1} ,

the Poincaré Gauss map. Let U ⊂ M^{2n+1} be the disjoint union of the m embeddings S^{n}×D^{n+1} ⊂ M with null-homotopies in X, so that (f, b) has a Heegard decomposition as a union of normal maps

\begin{matrix}(f, b) = (e, a) ∪ (f_{0}, b_{0}) : \\ M = (U, ∂U) ∪ (M_{0}, ∂M_{0}) ⟶ X = (D^{2n+1}, S^{2n}) ∪ (X_{0}, ∂X_{0}) \end{matrix}

with the inclusion (6.2) of the lagrangian

( \begin{matrix}ψ_{0} \\ d^{∗} \end{matrix} ) : C^{n} ⟶ C_{n+1} ⊕ C^{n+1}

in the hyperbolic (−)^{n}-quadratic form H_{(−)^{n}}(C_{n+1}) given by

inclusion_{∗} : K_{n+1}(M_{0}, ∂U) ⟶ K_{n}(∂U) = K_{n+1}(U, ∂U) ⊕ K_{n}(U) .

Wall obtained the surgery obstruction of (f, b) using an extension (cf. 5.7) of this inclusion to an automorphism

α : H_{(−)^{n}}(C_{n+1}) ⟶ H_{(−)^{n}}(C_{n+1}) ,

which will be discussed further in §10 below. The presentation of (f, b) used to obtain (C, ψ) in 6.3 is the trace of the m type (n, n) surgeries on U ⊂ M

\begin{matrix}(g, c) = (e_{1}, a_{1}) ∪ (f_{0}, b_{0}) × id : \\ (W; M, M′) = (V; U, U′) ∪ M_{0} × ([0, 1]; \{0\}, \{1\})⟶X × ([0, 1]; \{0\}, \{1\}) \end{matrix}

with

(V; U, U′) = ∪_{m}(D^{n+1} × D^{n+1}; S^{n} × D^{n+1}, D^{n+1} × S^{n}) .

Example 6.5 There is also a relative version of 6.3. A presentation of an (n − 1)- connected (2n + 1)-dimensional normal map (M, ∂M)⟶(X, ∂X) with ∂M⟶∂X a homotopy equivalence is a normal map of triads

(W^{2n+2}; M^{2n+1}, M^{′2n+1}; ∂M ×[0, 1]) ⟶ (X ×[0, 1]; X ×{0}, X ×{1}; ∂X ×[0, 1])

such that W ⟶X × [0, 1] is n-connected. Again, the presentation determines a (2n + 1)-complex (C, ψ) over ℤ[π_{1}(X)] with

C_{n} = K_{n+1}(W, ∂W) , C_{n+1} = K_{n+1}(W, M′) , H_{∗}(C) = K_{∗}(M) .

Remark 6.6 (Realization of complexes) For every 2n-dimensional manifold X^{2n} with n ≥ 2 and every (2n+1)-complex (C, ψ) over ℤ[π_{1}(X)] there exists an (n − 1)-connected (2n + 1)-dimensional normal map

(M^{2n+1}; ∂_{0}M, ∂_{1}M) ⟶ X^{2n} × ([0, 1]; {0}, {1})

with ∂_{0}M = X⟶X the identity and ∂_{1}M⟶X a homotopy equivalence, which admits a presentation with respect to which it has (2n + 1)-complex (C, ψ). Such a normal map is constructed from the identity X⟶X in two stages. First, choose a base {b_{1}, b_{2}, …, b_{m}} for C_{n+1}, and perform type (n − 1, n) surgeries on m disjoint trivial embeddings S^{n−1} × D^{n+1} ⊂ X^{2n} with trace

(U; X, ∂_{1}U) = (X × [0, 1] ∪ ∪_{m}D^{n} × D^{n+1}; X × {0}, X##_{m}S^{n} × S^{n})

⟶ X × ([0, 1/2]; {0}, {1/2}) .

The (n − 1)-connected 2n-dimensional normal map ∂_{1}U⟶X × {1/2} has kernel (−)^{n}-quadratic form

(K_{n}(∂_{1}U), λ, μ) = H_{(−)^{n}}(ℤ[π_{1}(X)]^{m}) = H_{(−)^{n}}(C_{n+1}) .

Second, choose a base {c_{1}, c_{2}, …, c_{m}} for C^{n} and realize the inclusion of the lagrangian in H_{(−)^{n}}(C_{n+1}) by type (n, n−1) surgeries on m disjoint framed embeddings S^{n} × D^{n} ⊂ ∂_{1}U with trace

(M_{0}; ∂_{1}U, ∂_{1}M) ⟶ X × ([1/2, 1]; {1/2}, {1})

such that

( \begin{matrix}ψ_{0} \\ d^{∗} \end{matrix} ) = ∂ : C^{n} = K_{n+1}(M_{0}, ∂_{1}U)

⟶ C_{n+1} ⊕ C^{n+1} = K_{n+1}(U, ∂_{1}U) ⊕ K_{n}(U) = K_{n}(∂_{1}U) .

The required (2n + 1)-dimensional normal map realizing (C, ψ) is the union

(M; ∂_{0}M, ∂_{1}M) = (U; X, ∂_{1}U) ∪ (M_{0}; ∂_{1}U, ∂_{1}M) ⟶ X × ([0, 1]; {0}, {1}) .

The corresponding presentation is the trace of surgeries on m disjoint framed embeddings S^{n} × D^{n+1} ⊂ U ⊂ M^{2n+1}. This is the terminology (and result) of Chapter 6 of Wall’s book.

The choice of presentation (6.3) for an (n−1)-connected (2n+1)-dimensional normal map (f, b) : M^{2n+1}⟶X does not change the “homotopy type” of the associated (2n + 1)-complex (C, ψ), in the following sense.

Definition 6.7 i) A map of (2n + 1)-complexes over A

f : (C, ψ) ⟶ (C′, ψ′)

is a chain map f : C⟶C′ such that there exist A-module morphisms

χ_{0} : C^{′n+1} ⟶ C′_{n+1} , χ_{1} : C^{′n} ⟶ C^{′n}

with

\begin{matrix}fψ_{0}f^{∗} − ψ′_{0} = (χ_{0} + (−)^{n+1}χ_{0}^{∗})d^{′∗} : C^{′n} ⟶ C′_{n+1} , \\ fψ_{1}f^{∗} − ψ′_{1} = −d′χ_{0}d^{′∗} + χ_{1} + (−)^{n}χ_{1}^{∗} : C^{′n} ⟶ C′_{n} . \end{matrix}

ii) A homotopy equivalence of (2n + 1)-complexes is a map with f : C⟶C′ a chain equivalence. iii) An isomorphism of (2n + 1)-complexes is a map with f : C⟶C′ an isomorphism of chain complexes.

Proposition 6.8 Homotopy equivalence is an equivalence relation on (2n + 1)- complexes. Proof : For m ≥0 let E(m) be the contractible f.g. free A-module chain complex defined by

d_{E(m)} = 1 : E(m)_{n+1} = A^{m} ⟶ E(m)_{n} = A^{m} ,
E(m)_{r} = 0 for r ≠ n, n + 1 .

A map f : (C, ψ)⟶(C′, ψ′) is a homotopy equivalence if and only if for some m, m′ ≥ 0 there exists an isomorphism

f′ : (C, ψ) ⊕ (E(m), 0) ⟶ (C′, ψ′) ⊕ (E(m′), 0)

such that the underlying chain map f′ is chain homotopic to

f ⊕ 0 : C ⊕ E(m) ⟶ C′ ⊕ E(m′) .

It is clear that isomorphism is an equivalence relation on (2n + 1)-complexes, and hence that so is homotopy equivalence.

Example 6.9 The (2n + 1)-complexes (C, ψ), (C′, ψ′) associated by 6.3 to any two presentations

(W; M, M̂)⟶X × ([0, 1]; {0}, {1}) , (W′; M, M̂′)⟶X × ([0, 1]; {0}, {1})

of an (n − 1)-connected normal map M^{2n+1}⟶X are homotopy equivalent. Without loss of generality it may be assumed that W and W′ are the traces of type (n, n) surgeries on disjoint framed embeddings

g^{i} : S^{n} × D^{n+1} ⟶ M , g^{′j} : S^{n} × D^{n+1}⟶M ,

corresponding to two sets of ℤ[π_{1}(X)]-module generators of K_{n}(M). Define a presentation of M⟶X

(W′′; M, M′′) = (W; M, M̂) ∪ (V; M̂, M′′) = (W′; M, M̂′) ∪ (V′; M̂′, M′′)

with (V; M̂, M′′) the presentation of M̂⟶X defined by the trace of the surgeries on the copies ĝ^{′j} : S^{n}⟶M̂ of g^{′j} : S^{n}⟶M, and (V′; M̂′, M′′) the presentation of M̂′⟶X defined by the trace of the surgeries on the copies ĝ^{i} : S^{n}⟶M̂′ of g^{i} : S^{n}⟶M. The projections

C′′ ⟶ C , C′′ ⟶ C′

define homotopy equivalences of (2n + 1)-complexes

(C′′, ψ′′) ⟶ (C, ψ) , (C′′, ψ′′) ⟶ (C′, ψ′) .

Definition 6.10 A (2n + 1)-complex (C, ψ) over A is contractible if it is homotopy equivalent to the zero complex (0, 0), or equivalently if d : C_{n+1}⟶C_{n} is an A-module isomorphism.

Example 6.11 A (2n+1)-complex (C, ψ) associated to an (n−1)-connected (2n+1)- dimensional normal map (f, b) : M^{2n+1}⟶X is contractible if (and for n ≥ 2 only if) f is a homotopy equivalence, by the theorem of J.H.C. Whitehead. The (2n + 1)- complexes (C, ψ) associated to the various presentations of a homotopy equivalence (f, b) : M^{2n+1}⟶X are contractible, by 6.9. The zero complex (0, 0) is associated to the presentation

(f, b) × id. : M × ([0, 1]; {0}, {1}) ⟶ X × ([0, 1]; {0}, {1}) .

§7. Complex cobordism

The cobordism of (2n + 1)-complexes is the equivalence relation which corresponds to the normal bordism of (n − 1)-connected (2n + 1)-dimensional normal maps. The (2n + 1)-dimensional surgery obstruction groups L_{2n+1}(A) will be defined in §8 below to be the cobordism groups of (2n + 1)-complexes over A.

Definition 7.1 A cobordism of (2n + 1)-complexes (C, ψ), (C′, ψ′)

((j j′) : C ⊕ C′⟶D, (δψ, ψ ⊕ −ψ′))

is a f.g. free A-module chain complex of the type

D : … ⟶ 0 ⟶ D_{n+1} ⟶ 0 ⟶ …

together with A-module morphisms

j : C_{n+1} ⟶ D_{n+1} , j′ : C′_{n+1} ⟶ D_{n+1} ,
δψ_{0} : D^{n+1} = (D_{n+1})^{∗} ⟶ D_{n+1}

such that the duality A-module chain map

(1 + T)(δψ_{0}, ψ_{0} ⊕ −ψ′_{0}) : C(j′)^{2n+2−∗} ⟶ C(j)

defined by

\begin{matrix}(1 + T)(δψ_{0}, ψ_{0} ⊕ −ψ′_{0}) = ( \begin{matrix}δψ_{0} + (−)^{n+1}δψ_{0}^{∗} & j′ψ′_{0} \\ ψ_{0}^{∗}j^{∗} & 0 \end{matrix} ) \\ : C(j′)^{2n+1} = D^{n+1} ⊕ C^{′n} ⟶ C(j)_{n+1} = D_{n+1} ⊕ C_{n} \end{matrix}

is a chain equivalence, with C(j), C(j′) the algebraic mapping cones of the chain maps j : C⟶D, j′ : C′⟶D.

The duality chain map C(j′)^{2n+2−∗}⟶C(j) is given by

\begin{matrix}C(j′)^{2n+2−∗} : & … ⟶0⟶0⟶D^{n+1} ⊕ C^{′n}⟶C^{′n+1}⟶0⟶ … \\ C(j) : & … ⟶0⟶C_{n+1}⟶D_{n+1} ⊕ C_{n}⟶0⟶0⟶ …. \end{matrix}

The condition for it to be a chain equivalence is just that the A-module morphism

\begin{aligned}( & \begin{matrix}d & 0 & ψ_{0}^{∗}j^{∗} \\ 0 & d^{′∗} & j^{′∗} \\ (−)^{n+1}j & j′ψ′_{0} & δψ_{0} + (−)^{n+1}δψ_{0}^{∗} \end{matrix} & ) \\ & : C_{n+1} ⊕ C^{′n} ⊕ D^{n+1} ⟶ C_{n} ⊕ C^{′n+1} ⊕ D_{n+1} \end{aligned}

be an isomorphism.

Example 7.2 Suppose given two (n−1)-connected (2n+1)-dimensional normal maps M^{2n+1}⟶X, M^{′2n+1}⟶X with presentations (6.3)

(W^{2n+2}; M^{2n+1}, M̂^{2n+1}) ⟶ X × ([0, 1]; {0}, {1}) ,

(W^{′2n+2}; M^{′2n+1}, M̂^{′2n+1}) ⟶ X × ([0, 1]; {0}, {1})

and corresponding (2n + 1)-complexes (C, ψ), (C′, ψ′). An n-connected normal bordism

(V^{2n+2}; M^{2n+1}, M^{′2n+1}) ⟶ X × ([0, 1]; {0}, {1})

determines a cobordism ((j j′) : C ⊕ C′⟶D, (δψ, ψ ⊕ −ψ′)) (again, up to some choices) from (C, ψ) to (C′, ψ′). Define an n-connected normal bordism (V′; M̂, M̂′) = (W; M̂, M)∪(V; M, M′)∪(W′; M′, M̂′) ⟶ X ×([0, 1]; {0}, {1}) . The exact sequence of stably f.g. free ℤ[π_{1}(X)]-modules 0 ⟶ K_{n+1}(V) ⟶ K_{n+1}(V′, ∂V′) ⟶ K_{n+1}(W, ∂W) ⊕ K_{n+1}(W′, ∂W′) ⟶ 0 splits. Choosing any splitting K_{n+1}(V′, ∂V′)⟶K_{n+1}(V) define j, j′ by

(j j′) : C_{n+1} ⊕ C′_{n+1} = K_{n+1}(W, M̂) ⊕ K_{n+1}(W′, M̂′) ⟶^{incl_{∗}⊕incl_{∗}}

K_{n+1}(V′, ∂V′) ⟶ K_{n+1}(V) = D_{n+1} .

Geometric intersection numbers provide a (−)^{n+1}-quadratic form (D^{n+1}, δψ_{0}) over ℤ[π_{1}(X)] such that the duality chain map C(j′)^{2n+2−∗}⟶C(j) is a chain equivalence inducing the Poincaré duality isomorphisms

[V] ∩ − : H^{2n+2−∗}(j′) = K^{2n+2−∗}(V, M′) ⟶ H_{∗}(j) = K_{∗}(V, M) .

Definition 7.3 A null − cobordism of a (2n + 1)-complex (C, ψ) is a cobordism (j : C⟶D, (δψ, ψ)) to (0, 0).

Example 7.4 Let (W^{2n+2}; M^{′2n+1}, M^{2n+1}) ⟶ X × ([0, 1]; {0}, {1}) be a presentation of an (n − 1)-connected (2n + 1)-dimensional normal map M⟶X, with (2n + 1)-complex (C, ψ). For n ≥ 2 there is a one-one correspondence between nconnected normal bordisms of M⟶X

(V^{2n+2}; M^{2n+1}, N^{2n+1}) ⟶ X × ([0, 1]; {0}, {1})

to homotopy equivalences N⟶X and null-cobordisms (j : C⟶D, (δψ, ψ)). (Every normal bordism of (n − 1)-connected (2n + 1)-dimensional normal maps can be made n-connected by surgery below the middle dimension on the interior). Any such (V; M, N)⟶X × ([0, 1]; {0}, {1}) determines by 7.2 a null-cobordism (j : C⟶D,- (δψ, ψ)) of (C, ψ).

Cobordisms of (2n + 1)-complexes arise in the following way:

Construction 7.5 An isomorphism of hyperbolic split (−)^{n}-quadratic forms over A

(( \begin{matrix}γ & γ̃ \\ μ & μ̃ \end{matrix} ) , ( \begin{matrix}θ & 0 \\ γ̃^{∗}μ & θ̃ \end{matrix})) : H_{(−)^{n}}(G) ⟶ H_{(−)^{n}}(F)

with F, G f.g. free determines a cobordism of (2n + 1)-complexes ((j j′) : C ⊕ C′⟶D, (δψ, ψ ⊕ −ψ′)) by

d = μ^{∗} : C_{n+1} = F ⟶ C_{n} = G^{∗} ,
ψ_{0} = γ : C^{n} = G ⟶ C_{n+1} = F ,
ψ_{1} = −θ : C^{n} = G ⟶ C_{n} = G^{∗} ,
j = μ̃^{∗} : C_{n+1} = F ⟶ D_{n+1} = G ,
d′ = γ^{∗} : C′_{n+1} = F^{∗} ⟶ C_{n} = G^{∗} ,
ψ′_{0} = μ : C^{′n} = G ⟶ C′_{n+1} = F^{∗} ,
ψ′_{1} = −θ : C^{′n} = G ⟶ C′_{n} = G^{∗} ,
j′ = γ̃^{∗} : C′_{n+1} = F^{∗} ⟶ D_{n+1} = G ,
δψ_{0} = 0 : D^{n+1} = G^{∗} ⟶ D_{n+1} = G .

It can be shown that every cobordism of (2n+1)-complexes is homotopy equivalent to one constructed as in 7.5.

Example 7.6 An (n − 1)-connected (2n + 2)-dimensional normal bordism

((g, c); (f, b), (f′, b′)) : (W^{2n+2}; M^{2n+1}, M^{′2n+1}) ⟶ X × ([0, 1]; {0}, {1})

with g : W ⟶X × [0, 1] n-connected can be regarded both as a presentation of (f, b) and as a presentation of (f′, b′). The cobordism of (2n + 1)-complexes ((j j′) : C ⊕ C′⟶D, (δψ, ψ ⊕ −ψ′)) obtained in 7.2 with W = V = W′, M̂ = M′, M̂′ = M′ is the construction of 7.5 for an extension of the inclusion of the lagrangian (6.2)

(( \begin{matrix}γ \\ μ \end{matrix} ) , θ) = (( \begin{matrix}ψ_{0} \\ d^{∗} \end{matrix} ) , −ψ_{1}) : (C^{n}, 0) ⟶ H_{(−)^{n}}(C_{n+1})

to an isomorphism of hyperbolic split (−)^{n}-quadratic forms

(( \begin{matrix}γ & γ̃ \\ μ & μ̃ \end{matrix} ) , ( \begin{matrix}θ & 0 \\ γ̃^{∗}μ & θ̃ \end{matrix})) : H_{(−)^{n}}(C^{n}) ⟶ H_{(−)^{n}}(C_{n+1}) ,

with

\begin{matrix}j = μ̃^{∗} : C_{n+1} = K_{n+1}(W, M′) ⟶ D_{n+1} = K_{n+1}(W) , \\ j′ = γ̃^{∗} : C′_{n+1} = K_{n+1}(W, M) ⟶ D_{n+1} = K_{n+1}(W) \end{matrix}

etc.

Remark 7.7 Fix a (2n+1)-dimensional geometric Poincaré complex X with a stable vector bundle ν_{X} : X⟶BO in the Spivak normal class, and consider the set of (n−1)-connected normal maps (f : M^{2n+1}⟶X, b : ν_{M}⟶ν_{X}). The relation defined on this set by

(M⟶X) ∼ (M′⟶X) if there exists an n−connected normal bordism

(W; M, M′) ⟶ X × ([0, 1]; {0}, {1})

is an equivalence relation. Symmetry and transitivity are verified in the same way as for any geometric cobordism relation. For reflexitivity form the cartesian product of an (n − 1)-connected normal map M^{2n+1}⟶X with ([0, 1]; {0}, {1}), as usual. The product is an (n − 1)-connected normal bordism

M × ([0, 1]; {0}, {1}) ⟶ X × ([0, 1]; {0}, {1})

which can be made n-connected by surgery killing the n-dimensional kernel K_{n}(M × [0, 1]) = K_{n}(M). The following verification that the cobordism of (2n + 1)-complexes is an equivalence relation uses algebraic surgery in exactly the same way.

Proposition 7.8 Cobordism is an equivalence relation on (2n + 1)-complexes (C, ψ) over A, such that (C, ψ)⊕(C, −ψ) is null-cobordant. Homotopy equivalent complexes are cobordant. Proof : Symmetry is easy: if ((j j′) : C ⊕ C′⟶D, (δψ, ψ ⊕ −ψ′)) is a cobordism from (C, ψ) to (C′, ψ′) then ((j′ j) : C′⊕C⟶D′, (−δψ, ψ′⊕−ψ)) is a cobordism from (C′, ψ′) to (C, ψ). For transitivity, suppose given adjoining cobordisms of (2n + 1)- complexes

\begin{matrix}((j j′) : C ⊕ C′⟶D, (δψ, ψ ⊕ −ψ′)) , \\ ((j̃′ j′′) : C′ ⊕ C′′⟶D′, (δψ′, ψ′ ⊕ −ψ′′)) . \end{matrix}

Define the union cobordism between (C, ψ) and (C′′, ψ′′)

((j̃ j̃′′) : C ⊕ C′′ ⟶ D′′, (δψ′′, ψ ⊕ −ψ′′))

by

D′′_{n+1} = coker(i = ( j′ ; d′ ; j̃′ ) : C′_{n+1}⟶D_{n+1} ⊕ C′_{n} ⊕ D′_{n+1}) ,

j̃ = [j ⊕ 0 ⊕ 0] : C_{n+1} ⟶ D′′_{n+1},

j̃′′ = [0 ⊕ 0 ⊕ j′′] : C′′_{n+1} ⟶ D′′_{n+1} ,

δψ′′_{0} = [ \begin{matrix}δψ_{0} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & δψ′_{0} \end{matrix} ] : D^{′′n+1} ⟶ D′′_{n+1} .

The A-module morphism i : C′_{n+1}⟶D_{n+1} ⊕ C′_{n} ⊕ D′_{n+1} is a split injection since the dual A-module morphism i^{∗} is a surjection, as is clear from the Mayer-Vietoris exact sequence

H^{n+2}(D, C′) ⊕ H^{n+2}(D′, C′) = 0 ⊕ 0 ⟶ H^{n+2}(D′′) ⟶ H^{n+2}(C′) = 0 .

Given any (2n + 1)-complex (C, ψ) let (C′, ψ′) be the (2n + 1)-complex defined by

d′ = (−)^{n}ψ_{0}^{∗} : C′_{n+1} = C^{n+1} ⟶ C′_{n} = C_{n} ,
ψ′_{0} = d^{∗} : C^{′n} = C^{n} ⟶ C′_{n+1} = C^{n+1} ,
ψ′_{1} = −ψ_{1} : C^{′n} = C^{n} ⟶ C′_{n} = C_{n} .

Apply 5.7 to extend the inclusion of the lagrangian in H_{(−)^{n}}(C_{n+1})

( \begin{matrix}ψ_{0} \\ d^{∗} \end{matrix} ) : C^{n} ⟶ C_{n+1} ⊕ C^{n+1}

to an isomorphism of (−)^{n}-quadratic forms

( \begin{matrix}ψ_{0} & ψ̃_{0} \\ d^{∗} & d̃^{∗} \end{matrix} ) : H_{(−)^{n}}(C^{n}) ⟶ H_{(−)^{n}}(C_{n+1})

with ψ̃_{0} ∈ Hom_{A}(C_{n}, C_{n+1}), d̃∈ Hom_{A}(C_{n+1}, C^{n}). Now apply 7.5 to construct from any such extension a cobordism ((j j′) : C ⊕ C′⟶D, (δψ, ψ ⊕ −ψ′)) with

j = d̃ : C_{n+1} ⟶ D_{n+1} = C^{n} ,
j′ = ψ̃_{0}^{∗} : C′_{n+1} = C^{n+1} ⟶ D_{n+1} = C^{n} ,
d′ = ψ_{0}^{∗} : C′_{n+1} = C^{n+1} ⟶ C′_{n} = C_{n} ,
ψ′_{0} = d^{∗} : C^{′n} = C^{n} ⟶ C^{′n+1} = C^{n+1} ,
δψ_{0} = 0 : D^{n+1} = C_{n} ⟶ D_{n+1} = C^{n} .

(This is the algebraic analogue of the construction of a presentation (6.3)

(W^{2n+2}; M^{2n+1}, M^{′2n+1}) ⟶ X × ([0, 1]; {0}, {1})

of an (n − 1)-connected (2n + 1)-dimensional normal map M^{2n+1}⟶X by surgery on a finite set of ℤ[π_{1}(X)]-module generators of K_{n}(M)). The union of the cobordisms

((j j′) : C ⊕ C′⟶D, (δψ, ψ ⊕ −ψ′)) , ((j′ j) : C′ ⊕ C⟶D, (−δψ, ψ′ ⊕ −ψ))

is a cobordism

((j̃ j̃′) : C ⊕ C⟶D′, (δψ′, ψ ⊕ −ψ))

with an A-module isomorphism

{[ \begin{matrix}1 & 0 & −1 \\ ψ_{0} & (−)^{n}ψ̃_{0} & 0 \end{matrix} ] : \atop D′_{n+1} = coker(( ψ̃_{0}^{∗} ; ψ_{0}^{∗} ; ψ̃_{0}^{∗} ) : C^{n+1}⟶C^{n} ⊕ C_{n} ⊕ C^{n}) ⟶ C^{n} ⊕ C_{n+1} .}

This verifies that cobordism is reflexive, and also that (C, ψ) ⊕ (C, −ψ) is nullcobordant.

Suppose given a homotopy equivalence of (2n + 1)-complexes

f : (C, ψ) ⟶ (C′, ψ′) ,

with χ_{0} : C^{′n+1}⟶C′_{n+1} as in 6.6. By reflexitivity there exists a cobordism ((j′′ j′) : C′ ⊕ C′⟶D, (δψ′, ψ′ ⊕ −ψ′)) from (C′, ψ′) to itself. Define a cobordism ((j j′) : C ⊕ C′⟶D, (δψ, ψ ⊕ −ψ′)) from (C, ψ) to (C′, ψ′) by

{j = j′′f : C_{n+1} ⟶^{f} C′_{n+1} ⟶^{j′′} D_{n+1} , \atop δψ_{0} = δψ′_{0} + j′′χ_{0}j^{′′∗} : D^{n+1} ⟶ D_{n+1} .}

Definition 7.9 i) A weak map of (2n + 1)-complexes over A

f : (C, ψ) ⟶ (C′, ψ′)

is a chain map f : C⟶C′ such that there exist A-module morphisms

χ_{0} : C^{′n+1} ⟶ C′_{n+1} , χ_{1} : C^{′n} ⟶ C^{′n}

with

fψ_{0}f^{∗} − ψ′_{0} = (χ_{0} + (−)^{n+1}χ_{0}^{∗})d^{′∗} : C^{′n} ⟶ C′_{n+1} .

ii) A weak equivalence of (2n + 1)-complexes is a weak map with f : C⟶C′ a chain equivalence. iii) A weak isomorphism of (2n + 1)-complexes is a weak map with f : C⟶C′ an isomorphism of chain complexes.

Proposition 7.10 Weakly equivalent (2n + 1)-complexes are cobordant. Proof: The proof in 7.8 that homotopy equivalent (2n+1)-complexes are cobordant works just as well for weakly equivalent ones.

Given a (2n + 1)-complex (C, ψ) let

(( \begin{matrix}ψ_{0} \\ d^{∗} \end{matrix} ) , −ψ_{1}) : (C^{n}, 0) ⟶ H_{(−)^{n}}(C_{n+1})

be the inclusion of a lagrangian in a hyperbolic split (−)^{n}-quadratic form given by 6.2. The result of 7.10 is that the cobordism class of (C, ψ) is independent of the hessian (−)^{n+1}-quadratic form (C^{n}, −ψ_{1}).

§8. The odd-dimensional L-groups

We define the odd-dimensional surgery obstruction groups L_{2n+1}(A) of a ring with involution A to be the cobordism groups of (2n + 1)-complexes over A.

Definition 8.1 Let L_{2n+1}(A) be the abelian group of cobordism classes of (2n + 1)- complexes over A, with addition and inverses by

(C, ψ) + (C′, ψ′) = (C ⊕ C′, ψ ⊕ ψ′) ∈ L_{2n+1}(A) ,

−(C, ψ) = (C, −ψ) ∈ L_{2n+1}(A) .

It should be clear that L_{2n+1}(A) depends only on the residue n(mod 2), so that only two L-groups have actually been defined, L_{1}(A) and L_{3}(A). Note that 8.1 uses 7.8 to justify (C, ψ) ⊕ (C, −ψ) = 0 ∈ L_{2n+1}(A).

Example 8.2 The odd-dimensional L-groups of A = ℤ are trivial

L_{2n+1}(ℤ) = 0 .

8.2 was implicit in the work of Kervaire and Milnor (1963) on the simply-connected surgery technique used to classify exotic 2n-dimensional spheres.

Example 8.3 The surgery obstruction of an (n − 1)-connected (2n + 1)-dimensional normal map (f, b) : M^{2n+1}⟶X is the cobordism class

σ_{∗}(f, b) = (C, ψ) ∈ L_{2n+1}(ℤ[π_{1}(X)])

of the (2n + 1)-complex (C, ψ) associated in 6.3 to any choice of presentation

(W; M, M′) ⟶ X × ([0, 1]; {0}, {1}) .

The surgery obstruction vanishes σ_{∗}(f, b) = 0 if (and for n ≥ 2 only if) (f, b) is normal bordant to a homotopy equivalence.

Definition 8.4 A surgery (j : C⟶D, (δψ, ψ)) on a (2n + 1)-complex (C, ψ) is an A-module chain map j : C⟶D with D_{r} = 0 for r ≠ n + 1 and D_{n+1} a f.g. free A-module, together with an A-module morphism

δψ_{0} : D^{n+1} = (D_{n+1})^{∗} ⟶ D_{n+1} ,

such that the A-module morphism

( d ψ_{0}^{∗}j^{∗}) : C_{n+1} ⊕ D^{n+1} ⟶ C_{n}

is onto. The effect of the surgery is the (2n + 1)-complex (C′, ψ′) defined by

d′ = ( \begin{matrix}d & ψ_{0}^{∗}j^{∗} \\ (−)^{n+1}j & δψ_{0} + (−)^{n+1}δψ_{0}^{∗} \end{matrix} )

: C′_{n+1} = C_{n+1} ⊕ D^{n+1} ⟶ C′_{n} = C_{n} ⊕ D_{n+1} ,

ψ′_{0} = ( \begin{matrix}ψ_{0} & 0 \\ 0 & 1 \end{matrix} )

: C^{′n} = C^{n} ⊕ D^{n+1} ⟶ C′_{n+1} = C_{n+1} ⊕ D^{n+1} ,

ψ′_{1} = ( \begin{matrix}ψ_{1} & −ψ_{0}^{∗}j^{∗} \\ 0 & −δψ_{0} \end{matrix} )

: C^{′n} = C^{n} ⊕ D^{n+1} ⟶ C′_{n} = C_{n} ⊕ D_{n+1} .

Choose a splitting of (d ψ_{0}^{∗}j^{∗}) : C_{n+1} ⊕ D^{n+1}⟶C_{n}, and let

\begin{aligned}j′′ = (j′ k) : & C′_{n+1} = C_{n+1} ⊕ D^{n+1} \\ & ⟶ D′_{n+1} = ker(( d ψ_{0}j^{∗}) : C_{n+1} ⊕ D^{n+1}⟶C_{n}) \end{aligned}

be the resulting split surjection. The trace of the surgery is the cobordism of (2n+1)- complexes ((j′ j′′) : C ⊕ C′⟶D′, (δψ′, ψ ⊕ −ψ′)) with

δψ′_{0} = βjα^{∗} : D^{′n+1} ⟶ D′_{n+1} .

Example 8.5 Let

((e, a); (f, b), (f′, b′)) : (V^{2n+2}; M^{2n+1}, M^{′2n+1}) ⟶ X × ([0, 1]; {0}, {1})

be the trace of a sequence of m type (n, n) surgeries on an (n − 1)-connected (2n + 1)-dimensional normal map (f, b) : M⟶X, with e n-connected and f′ (n − 1)- connected. V has a handle decomposition on M of the type

V = M × I ∪ ∪_{m}(n + 1) − handles D^{n+1} × D^{n+1} ,

and also a handle decomposition on M′ of the same type

V = M′ × I ∪ ∪_{m}(n + 1) − handles D^{n+1} × D^{n+1} .

A presentation of (f, b)

((g, c); (f̂, b̂), (f, b)) : (W^{2n+2}; M̂^{2n+1}, M^{2n+1}) ⟶ X × ([0, 1]; {0}, {1})

with (2n + 1)-complex (C, ψ) determines a presentation of (f′, b′)

{((g′, c′); (f̂, b̂), (f′, b′)) = ((g, c); (f̂, b̂), (f, b)) ∪ ((e, a); (f, b), (f′, b′)) \atop : (W′; M̂, M′) = (W; M̂, M) ∪ (V; M, M′) ⟶ X × ([0, 1]; \{0\}, \{1\})}

such that the (2n + 1)-complex (C′, ψ′) is the effect of a surgery (j : C⟶D, (δψ, ψ)) on (C, ψ) with

D_{n+1} = K_{n+1}(V, M′) = ℤ[π_{1}(X)]^{m} ,
C′_{n+1} = K_{n+1}(W′, M̂) = K_{n+1}(W, M̂) ⊕ K_{n+1}(V, M) = C_{n+1} ⊕ D^{n+1} ,
C′_{n} = K_{n+1}(W′, ∂W′) = K_{n+1}(W, ∂W) ⊕ K_{n+1}(V, M′) = C_{n} ⊕ D_{n+1} .

Also, the geometric trace determines the algebraic trace, with

D′_{n+1} = K_{n+1}(V) .

It can be shown that (2n + 1)-complexes (C, ψ), (C′, ψ′) are cobordant if and only if (C′, ψ′) is homotopy equivalent to the effect of a surgery on (C, ψ). We shall only need this in the special case (C′, ψ′) = (0, 0):

Proposition 8.6 A (2n + 1)-complex (C, ψ) represents 0 in L_{2n+1}(A) if and only if there exists a surgery (j : C⟶D, (δψ, ψ)) with contractible effect. Proof: The effect of a surgery is contractible if and only if it is a null-cobordism.

Given an (n − 1)-connected (2n + 1)-dimensional normal map (f, b) : M^{2n+1}⟶X it is possible to kill every element x ∈ K_{n}(M) by a type (n, n)-surgery on a framed embedding S^{n} × D^{n+1} ⊂ M to obtain a bordant normal map

(f′, b′) : M^{′2n+1} = (M − S^{n} × D^{n+1}) ∪ D^{n+1} × S^{n} ⟶X .

There are many ways of carrying out the surgery, which are quantified by the surgeries on the kernel (2n + 1)-complex (C, ψ). In general, K_{n}(M′) need not be smaller than K_{n}(M).

Example 8.7 The kernel (2n + 1)-complex (C, ψ) over ℤof the identity normal map

(f, b) = id. : M^{2n+1} = S^{2n+1} ⟶ S^{2n+1}

is (0, 0). Given any commutative diagram

S^{n}  w ω u  SO(n + 1) u

Ω

D^{n+1}  wSO

define a null-homotopic framed embedding of S^{n} in M

e_{ω} : S^{n} × D^{n+1} ⟶ M ; (x, y) ⟶ (x, ω(x)(y))/|(x, ω(x)(y))| .

Use Ω to kill 0 ∈ K_{n}(M) by a type (n, n) surgery on (f, b), with effect a normal bordant (n − 1)-connected (2n + 1)-dimensional normal map

(f_{Ω}, b_{Ω}) : M_{Ω}^{2n+1} = (M − e_{ω}(S^{n} × D^{n+1})) ∪ D^{n+1} × S^{n} ⟶ S^{2n+1} .

The homotopy type of M_{Ω} is determined by the relative homotopy class

Ω ∈ π_{n+1}(SO, SO(n + 1)) = Q_{(−)^{n+1}}(ℤ) ,

and the kernel complex (C′, ψ′) of (f_{Ω}, b_{Ω}) is such that

d′ = (1 + T_{(−)^{n+1}})(Ω) : C′_{n+1} = ℤ ⟶ C′_{n} = ℤ .

In particular, for Ω = 0, 1 this gives the (2n + 1)-manifolds

M′ = M_{0} = S^{n} × S^{n+1} ,
M′′ = M_{1} = the tangent S^{n} − bundle of S^{n+1} = SO(n + 2)/SO(n)

= the Stiefel manifold V_{2,n} of orthonormal 2-frames in ℝ^{n+2}

( = SO(3) = ℝℙ^{3} for n = 1) ,

corresponding to the algebraic surgeries (0 : 0⟶D, (δψ′, 0)), (0 : 0⟶D, (δψ′′, 0)) on (0, 0) with

D_{n+1} = ℤ , δψ′_{0} = 0 , δψ′′_{0} = 1 .

§9. Formations

As before, let A be a ring with involution, and let ϵ = ±1.

Definition 9.1 An ϵ-quadratic formation over A (Q, ϕ; F, G) is a non-singular ϵquadratic form (Q, ϕ) together with an ordered pair of lagrangians F,G.

Formations with ϵ = (−)^{n} are essentially the (2n + 1)-complexes of §6 expressed in the language of forms and lagrangians of §4. In the general theory it is possible to consider formations (Q, ϕ; F, G) with Q, F, G f.g. projective, but in view of the more immediate topological applications we only consider the f.g. free case here. Strictly speaking, 9.1 defines a “nonsingular formation”. In the general theory a formation (Q, ϕ; F, G) is a nonsingular form (Q, ϕ) together with a lagrangian F and a sublagrangian G.

In dealing with formations we assume that A is such that the rank of f.g. free A-modules is well-defined (e.g. A = ℤ[π]). The rank of a f.g. free A-module M is such that

rank_{A}(M) = m ∈ ℤ^{+}

if and only if M is isomorphic to A^{m}. Also, since A^{m} ≅ (A^{m})^{∗}

rank_{A}(M) = rank_{A}(M^{∗}) ∈ ℤ^{+} .

Definition 9.2 An isomorphism of ϵ-quadratic formations over A

f : (Q, ϕ; F, G) ⟶ (Q′, ϕ′; F′, G′)

is an isomorphism of forms f : (Q, ϕ)⟶(Q′, ϕ′) such that

f(F) = F′ , f(G) = G′ .

Proposition 9.3 i) Every ϵ-quadratic formation (Q, ϕ; F, G) is isomorphic to one of the type (H_{ϵ}(F); F, G). ii) Every ϵ-quadratic formation (Q, ϕ; F, G) is isomorphic to one of the type (H_{ϵ}(F);- F, α(F)) for some automorphism α : H_{ϵ}(F)⟶H_{ϵ}(F). Proof: i) By Theorem 5.7 the inclusion of the lagrangian F ⟶Q extends to an isomorphism of forms

f : H_{ϵ}(F) ⟶ (Q, ϕ) ,

defining an isomorphism of formations

f : (H_{ϵ}(F); F, f^{−1}(G)) ⟶ (Q, ϕ; F, G) .

ii) As in i) extend the inclusions of the lagrangians to isomorphisms of forms

f : H_{ϵ}(F) ⟶ (Q, ϕ) , g : H_{ϵ}(G) ⟶ (Q, ϕ) .

Then

rank_{A}(F) = rank_{A}(M)/2 = rank_{A}(G) ∈ ℤ^{+} ,

so that F is isomorphic to G. Choosing an A-module isomorphism β : G⟶F there is defined an automorphism of H_{ϵ}(F)

α : H_{ϵ}(F)\underrightarrow{f}(Q, ϕ)\underrightarrow{g^{−1}}H_{ϵ}(G)\underrightarrow{( \begin{matrix}β & 0 \\ 0 & β^{∗−1} \end{matrix} )}H_{ϵ}(F)

such that there is defined an isomorphism of formations

f : (H_{ϵ}(F); F, α(F)) ⟶ (Q, ϕ; F, G) .

Proposition 9.4 The weak isomorphism classes of (2n + 1)-complexes (C, ψ) over A are in natural one-one correspondence with the isomorphism classes of (−)^{n}-quadratic formations (Q, ϕ; F, G) over A, with

H_{n}(C) = Q/(F + G) , H_{n+1}(C) = F ∩ G .

Moreover, if the complex (C, ψ) corresponds to the formation (Q, ϕ; F, G) then (C,- −ψ) corresponds to (Q, −ϕ; F, G). Proof: We shall only describe how to pass from complexes to formations and then back again, leaving to the reader the verification that the equivalence classes are preserved. Given a (2n + 1)-complex (C, ψ) define a (−)^{n}-quadratic formation

(Q, ϕ; F, G) = (H_{(−)^{n}}(C_{n+1}); C_{n+1}, im(( \begin{matrix}ψ_{0} \\ d^{∗} \end{matrix} ) : C^{n}⟶C_{n+1} ⊕ C^{n+1})) .

The formation associated in this way to the (2n + 1)-complex (C, −ψ) is isomorphic to (Q, −ϕ; F, G), by the isomorphism

{( \begin{matrix}−1 & 0 \\ 0 & 1 \end{matrix} ) : \atop (Q, −ϕ; F, G) ⟶ (H_{(−)^{n}}(C_{n+1}); C_{n+1}, im(( \begin{matrix}−ψ_{0} \\ d^{∗} \end{matrix} ) : C^{n}⟶C_{n+1} ⊕ C^{n+1})) .}

Conversely, suppose given an (−)^{n}-quadratic formation (Q, ϕ; F, G). By 9.3 i) this can be replaced by an isomorphic formation with (Q, ϕ) = H_{(−)^{n}}(F). Let γ ∈ Hom_{A}(G, F), μ ∈ Hom_{A}(G, F^{∗}) be the components of the inclusion

i = ( \begin{matrix}γ \\ μ \end{matrix} ) : G ⟶ Q = F ⊕ F^{∗} .

Choose any θ ∈ Hom_{A}(G, G^{∗}) such that

γ^{∗}μ = θ + (−)^{n+1}θ^{∗} ∈ Hom_{A}(G, G^{∗}) .

Define a (2n + 1)-complex (C, ψ) by

d = μ^{∗} : C_{n+1} = F ⟶ C_{n} = G^{∗} ,
ψ_{0} = γ : C^{n} = G ⟶ C_{n+1} = F ,
ψ_{1} = (−)^{n}θ : C^{n} = G ⟶ C_{n} = G^{∗} .

The exact sequence

0 ⟶ G ⟶^{i} Q⟶^{i^{∗}(ϕ+(−)^{n}ϕ^{∗})}G^{∗} ⟶ 0

is the algebraic mapping cone

0 ⟶ G\underrightarrow{( \begin{matrix}γ \\ μ \end{matrix} )}F ⊕ F^{∗} \underrightarrow{( μ^{∗} (−)^{n}γ^{∗})}G^{∗} ⟶ 0

of the chain equivalence (1 + T)ψ_{0} : C^{2n+1−∗}⟶C.

Example 9.5 An (n−1)-connected (2n+1)-dimensional normal map M^{2n+1}⟶X together with a choice of presentation (W; M, M′)⟶X×([0, 1]; {0}, {1}) determines by 9.3 a (2n+1)-complex (C, ψ), and hence by 9.4 a (−)^{n}-quadratic formation (Q, ϕ; F, G) over ℤ[π_{1}(X)] such that

Q/(F + G) = H_{n}(C) = K_{n}(M) , F ∩ G = H_{n+1}(C) = K_{n+1}(M) .

Next, we introduce the equivalence relation on formations which corresponds to the weak equivalence (7.9) of (2n + 1)-complexes.

Definition 9.6 i) An ϵ-quadratic formation (Q, ϕ; F, G) is trivial if it is isomorphic to (H_{ϵ}(P); P, P^{∗}) for some f.g. free A-module P. ii) A stable isomorphism of ϵ-quadratic formations

[f] : (Q, ϕ; F, G) ⟶ (Q′, ϕ′; F′, G′)

is an isomorphism of ϵ-quadratic formations of the type

f : (Q, ϕ; F, G) ⊕ (trivial) ⟶ (Q′, ϕ′; F′, G′) ⊕ (trivial′) .

Example 9.7 The (−)^{n}-quadratic formations associated in 9.5 to all the presentations of an (n−1)-connected (2n+1)-dimensional normal map M^{2n+1}⟶X define a stable isomorphism class.

Proposition 9.8 The weak equivalence classes of (2n + 1)-complexes over A are in natural one-one correspondence with the stable isomorphism classes of (−)^{n}-quadratic formations over A. Proof: The (2n + 1)-complex (C, ψ) associated (up to weak equivalence) to a (−)^{n}- quadratic formation (Q, ϕ; F, G) in 9.4 is contractible if and only if the formation is trivial.

Next, we describe the formations which correspond to the null-cobordant complexes.

Definition 9.9 The boundary of a (−ϵ) quadratic form (P, θ) is the ϵ-quadratic formation

∂(P, θ) = (H_{ϵ}(P); P, Γ_{(P,θ)})

with Γ_{(P,θ)} the graph lagrangian

Γ_{(P,θ)} = {(x, (θ − ϵθ^{∗})(x)) ∈ P ⊕ P^{∗}|x ∈ P } .

Note that the form (P, θ) may be singular, that is the A-module morphism θ −ϵθ^{∗} : P ⟶P^{∗} need not be an isomorphism. The graphs Γ_{(P,θ)} of (−ϵ)-quadratic forms (P, θ) are precisely the lagrangians of H_{ϵ}(P) which are direct complements of P^{∗}.

Proposition 9.10 A (−)^{n}-quadratic formation (Q, ϕ; F, G) is stably isomorphic to a boundary ∂(P, θ) if and only if the corresponding (2n + 1)-complex (C, ψ) is nullcobordant. Proof: Given a (−)^{n+1}-quadratic form (P, θ) let (C, ψ) be the (2n + 1)-complex associated by 9.4 to the boundary formation ∂(P, θ), so that

d = θ + (−)^{n+1}θ^{∗} : C_{n+1} = P ⟶ C_{n} = P^{∗} ,
ψ_{0} = 1 : C^{n} = P ⟶ C_{n+1} = P ,
ψ_{1} = −θ : C^{n} = P ⟶ C_{n} = P^{∗} .

Then (C, ψ) is null-cobordant, with a null-cobordism (j : C⟶D, (δψ, ψ)) defined by

j = 1 : C_{n+1} = P ⟶ D_{n+1} = P ,
δψ_{0} = 0 : D^{n+1} = P^{∗} ⟶ D_{n+1} = P .

Conversely, suppose given a (2n + 1)-complex (C, ψ) with a null-cobordism (j : C⟶D, (δψ, ψ)) as in 8.1. The (2n + 1)-complex (E, θ) defined by

d = ( \begin{matrix}ψ_{1} + (−)^{n+1}ψ_{1}^{∗} & d & ψ_{0}^{∗}j^{∗} \\ (−)^{n+1}d^{∗} & 0 & −j^{∗} \\ (−)^{n+1}jψ_{0}^{∗} & (−)^{n}j & δψ_{0} + (−)^{n+1}δψ_{0}^{∗} \end{matrix} ) :

E_{n+1} = C^{n} ⊕ C_{n+1} ⊕ D^{n+1} ⟶ E_{n} = C_{n} ⊕ C^{n+1} ⊕ D_{n+1} ,

θ_{0} = 1 : E^{n} = C^{n} ⊕ C_{n+1} ⊕ D^{n+1} ⟶ E_{n+1} = C^{n} ⊕ C_{n+1} ⊕ D^{n+1} ,

θ_{1} = ( \begin{matrix}−ψ_{1} & −d & −ψ_{0}^{∗}j^{∗} \\ 0 & 0 & j^{∗} \\ 0 & 0 & −δψ_{0} \end{matrix} ) :

E^{n} = C^{n} ⊕ C_{n+1} ⊕ D^{n+1} ⟶ E_{n} = C_{n} ⊕ C^{n+1} ⊕ D_{n+1} corresponds to the boundary (−)^{n}-quadratic formation ∂(E^{n}, θ_{1}) of the (−)^{n+1}- quadratic form (E^{n}, θ_{1}), and there is defined a homotopy equivalence

f : (E, θ) ⟶ (C, ψ)

with

f_{n} = (1 ψ_{0}^{∗} 0) : E_{n} = C_{n} ⊕ C^{n+1} ⊕ D_{n+1} ⟶ C_{n} ,
f_{n+1} = (0 1 0) : E_{n+1} = C^{n} ⊕ C_{n+1} ⊕ D^{n+1} ⟶ C_{n+1} .

Proposition 9.11 The cobordism group L_{2n+1}(A) of (2n+1)-complexes is naturally isomorphic to the abelian group of equivalence classes of (−)^{n}-quadratic formations over A, subject to the equivalence relation

(Q, ϕ; F, G) ∼ (Q′, ϕ′; F′, G′) if there exists a stable isomorphism
[f] : (Q, ϕ; F, G) ⊕ (Q′, −ϕ′; F′, G′) ⟶ ∂(P, θ)

for some (−)^{n+1}-quadratic form (P, θ) over A, with addition and inverses by

(Q, ϕ; F, G) + (Q′, ϕ′; F′, G′) = (Q ⊕ Q′, ϕ ⊕ ϕ′; F ⊕ F′, G ⊕ G′) ,
−(Q, ϕ; F, G) = (Q, −ϕ; F, G) .

Proof: This is just the translation of the definition (8.1) of L_{2n+1}(A) into the language of (−)^{n}-quadratic formations, using 9.4, 9.8 and 9.10.

It is much easier to prove (7.8) that the cobordism of (2n + 1)-complexes is an equivalence relation than the relation defined in 9.11 on (−)^{n}-quadratic formations. We use 9.11 as an identification of L_{2n+1}(A) with the group of equivalence classes of (−)^{n}-quadratic formations over A.

Corollary 9.12 A (−)^{n}-quadratic formation (Q, ϕ; F, G) over A is such that

(Q, ϕ; F, G) = 0 ∈ L_{2n+1}(A)

if and only if it is stably isomorphic to the boundary ∂(P, θ) of a (−)^{n+1}-quadratic form (P, θ) on a f.g. free A-module P. Proof: Immediate from 9.10.

Next, we need to establish the relation

(Q, ϕ; F, G) ⊕ (Q, ϕ; G, H) = (Q, ϕ; F, H) ∈ L_{2n+1}(A) .

This is the key step in the identification in §10 below of L_{2n+1}(A) with a stable unitary group.

Lemma 9.13 i) An ϵ-quadratic formation (Q, ϕ; F, G) is trivial if and only if the lagrangians F and G are direct complements in Q. ii) An ϵ-quadratic formation (Q, ϕ; F, G) is isomorphic to a boundary if and only if (Q, ϕ) has a lagrangian H which is a direct complement of both the lagrangians F,G. Proof: i) If F and G are direct complements in Q express any representative ϕ ∈ Hom_{A}(Q, Q^{∗}) of ϕ ∈ Q_{ϵ}(Q) as

ϕ = ( \begin{matrix}λ − ϵλ^{∗} & γ \\ δ & μ − ϵμ^{∗} \end{matrix} ) : Q = F ⊕ G ⟶ Q^{∗} = F^{∗} ⊕ G^{∗} .

Then γ + ϵδ^{∗} ∈ Hom_{A}(G, F^{∗}) is an A-module isomorphism, and there is defined an isomorphism of ϵ-quadratic formations

( \begin{matrix}1 & 0 \\ 0 & (γ + ϵδ^{∗})^{−1} \end{matrix} ) : (H_{ϵ}(F); F, F^{∗}) ⟶ (Q, ϕ; F, G)

so that (Q, ϕ; F, G) is trivial. The converse is obvious. ii) For the boundary ∂(P, θ) of a (−ϵ)-quadratic form (P, θ) the lagrangian P^{∗} of H_{ϵ}(P) is a direct complement of both the lagrangians P,Γ_{(P,θ)}. Conversely, suppose that (Q, ϕ; F, G) is such that there exists a lagrangian H in (Q, ϕ) which is a direct complement to both F and G. By the proof of i) there exists an isomorphism of formations

f : (H_{ϵ}(F); F, F^{∗}) ⟶ (Q, ϕ; F, H)

which is the identity on F. Now f^{−1}(G) is a lagrangian of H_{ϵ}(F) which is a direct complement of F^{∗}, so that it is the graph Γ_{(F,θ)} of a (−ϵ)-quadratic form (F, θ), and f defines an isomorphism of ϵ-quadratic formations

f : ∂(F, θ) = (H_{ϵ}(F); F, Γ_{(F,θ)}) ⟶ (Q, ϕ; F, G) .

Proposition 9.14 For any lagrangians F, G, H in a nonsingular (−)^{n}-quadratic form (Q, ϕ) over A

(Q, ϕ; F, G) ⊕ (Q, ϕ; G, H) = (Q, ϕ; F, H) ∈ L_{2n+1}(A) .

Proof: Choose lagrangians F^{∗}, G^{∗}, H^{∗} in (Q, ϕ) complementary to F, G, H respectively. The (−)^{n}-quadratic formations (Q_{i}, ϕ_{i}; F_{i}, G_{i}) (1 ≤ i ≤ 4) defined by

(Q_{1}, ϕ_{1}; F_{1}, G_{1}) = (Q, −ϕ; G^{∗}, G^{∗}),

(Q_{2}, ϕ_{2}; F_{2}, G_{2}) = (Q⊕Q, ϕ⊕−ϕ; F ⊕F^{∗}, H ⊕G^{∗})⊕(Q⊕Q, −ϕ⊕ϕ; Δ_{Q}, H^{∗} ⊕G), (Q_{3}, ϕ_{3}; F_{3}, G_{3}) = (Q ⊕ Q, ϕ ⊕ −ϕ, F ⊕ F^{∗}, G ⊕ G^{∗}),

(Q_{4}, ϕ_{4}; F_{4}, G_{4}) = (Q⊕Q, ϕ⊕−ϕ; G⊕G^{∗}, H ⊕G^{∗})⊕(Q⊕Q, −ϕ⊕ϕ; Δ_{Q}, H^{∗} ⊕G)

are such that

(Q, ϕ; F, G) ⊕ (Q, ϕ; G, H) ⊕ (Q_{1}, ϕ_{1}; F_{1}, G_{1}) ⊕ (Q_{2}, ϕ_{2}; F_{2}, G_{2})

= (Q, ϕ; F, H) ⊕ (Q_{3}, ψ_{3}; F_{3}, G_{3}) ⊕ (Q_{4}, ϕ_{4}; F_{4}, G_{4}) .

Each of (Q_{i}, ϕ_{i}; F_{i}, G_{i}) (1 ≤ i ≤ 4) is isomorphic to a boundary, since there exists a lagrangian H_{i} in (Q_{i}, ϕ_{i}) complementary to both F_{i} and G_{i}, so that 9.12 ii) applies and (Q_{i}, ϕ_{i}; F_{i}, G_{i}) represents 0 in L_{2n+1}(A). Explicitly, we can take

H_{1} = G ⊂ Q_{1} = Q ,
H_{2} = Δ_{Q⊕Q} ⊂ Q_{2} = (Q ⊕ Q) ⊕ (Q ⊕ Q) ,
H_{3} = Δ_{Q} ⊂ Q_{3} = Q ⊕ Q ,
H_{4} = Δ_{Q⊕Q} ⊂ Q_{4} = (Q ⊕ Q) ⊕ (Q ⊕ Q) .

Remark 9.15 It is also possible to express L_{2n+1}(A) as the abelian group of equivalence classes of (−)^{n}-quadratic formations over A subject to the equivalence relation generated by

i) (Q, ϕ; F, G) ∼ (Q′, ϕ′; F′, G′)
if (Q, ϕ; F, G) is stably isomorphic to (Q′, ϕ′; F′, G′),
ii) (Q, ϕ; F, G) ⊕ (Q, ϕ; G, H) ∼ (Q, ϕ; F, H) ,

with addition and inverses by

(Q, ϕ; F, G) + (Q′, ϕ′; F′, G′) = (Q ⊕ Q′, ϕ ⊕ ϕ′; F ⊕ F′, G ⊕ G′) ∈ L_{2n+1}(A) ,

−(Q, ϕ; F, G) = (Q, ϕ; G, F) ∈ L_{2n+1}(A) .

This is immediate from 9.13 and the observation that for any (−)^{n+1}-quadratic form (P, θ) on a f.g. free A-module P the lagrangian P^{∗} in H_{(−)^{n}}(P) is a complement to both P and the graph Γ_{(P,θ)}, so that

∂(P, θ) ∼ (H_{(−)^{n}}(P); P, Γ_{(P,θ)}) ⊕ (H_{(−)^{n}}(P); Γ_{(P,θ)}, P^{∗}) ∼ (H_{(−)^{n}}(P); P, P^{∗}) ∼ 0 .

§10. Automorphisms

The (2n+1)-dimensional L-group L_{2n+1}(A) of a ring with involution A is identified with a quotient of the stable automorphism group of hyperbolic (−)^{n}-quadratic forms over A, as in the original definition of Wall.

Given an A-module M let Aut_{A}(M) be the group of automorphisms M⟶M, with the composition as group law.

Example 10.1 The automorphism group of the f.g. free A-module A^{m} is the general linear group GL_{m}(A) of invertible m × m matrices with entries in A

Aut_{A}(A^{m}) = GL_{m}(A) ,

with the multiplication of matrices as group law (cf. Remark 1.10).

In general, Aut_{A}(M) is non-abelian.

Example 10.2 GL_{2}(ℤ) is not abelian, since

( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) ≠ ( \begin{matrix}1 & 0 \\ 1 & 1 \end{matrix} ) ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

Definition 10.3 For any ϵ-quadratic form (M, ψ) on a f.g. projective A-module M let Aut_{A}(M, ψ) be the subgroup of Aut_{A}(M) consisting of the automorphisms f : (M, ψ)⟶(M, ψ), i.e. the automorphisms f ∈ Aut_{A}(M) such that

f^{∗}ψf = ψ ∈ Q_{ϵ}(M) .

Definition 10.4 The (ϵ, m)-unitary group of A is defined for ϵ = ±1, m ≥ 0 to be the automorphism group

U_{ϵ,m}(A) = Aut_{A}(H_{ϵ}(A^{m}))

of the ϵ-quadratic hyperbolic form H_{ϵ}(A^{m}).

Proposition 10.5 U_{ϵ,m}(A) is the group of invertible 2m × 2m matrices

( \begin{matrix}α & β \\ γ & δ \end{matrix} ) ∈ GL_{2m}(A)

such that

α^{∗}δ + ϵγ^{∗}β = 1 ∈ M_{m,m}(A) , α^{∗}γ = β^{∗}δ = 0 ∈ Q_{ϵ}(A^{m}) .

Proof: This is just the decoding of the condition

( \begin{matrix}α^{∗} & γ^{∗} \\ β^{∗} & δ^{∗} \end{matrix} ) ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) ∈ Q_{ϵ}(A^{m} ⊕ (A^{m})^{∗})

for ( \begin{matrix}α & β \\ γ & δ \end{matrix} ) to define an automorphism of the hyperbolic ϵ-quadratic form

H_{ϵ}(A^{m}) = (A^{m} ⊕ (A^{m})^{∗}, ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} )) .

We use 10.5 to express the automorphisms of H_{ϵ}(A^{m}) as matrices.

Example 10.6 U_{ϵ,1}(A) is the subgroup of GL_{2}(A) consisting of the 2 × 2 matrices

( \begin{matrix}a & b \\ c & d \end{matrix} ) such that

dā + ϵbc̄ = 1 ∈ A , cā = db̄ = 0 ∈ Q_{ϵ}(A) .

Definition 10.7 The elementary (ϵ, m)-quadratic unitary group of A is the normal subgroup

EU_{ϵ,m}(A) ⊆ U_{ϵ,m}(A)

of the full (ϵ, m)-quadratic unitary group generated by the elements of the following two types:

i) ( \begin{matrix}α & 0 \\ 0 & α^{∗−1} \end{matrix} ) for any automorphism α ∈ GL_{m}(A) ,

ii) ( \begin{matrix}1 & 0 \\ θ − ϵθ^{∗} & 1 \end{matrix} ) for any (−ϵ)- quadratic form (A^{m}, θ ∈ Q_{−ϵ}(A^{m})).

Lemma 10.8 For any (−ϵ)-quadratic form (A^{m}, θ ∈ Q_{−ϵ}(A^{m}))

( \begin{matrix}1 & θ − ϵθ^{∗} \\ 0 & 1 \end{matrix} ) ∈ EU_{ϵ,m}(A) .

Proof: This is immediate from the identity

( \begin{matrix}1 & θ − ϵθ^{∗} \\ 0 & 1 \end{matrix} ) = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )^{−1} ( \begin{matrix}1 & 0 \\ θ − ϵθ^{∗} & 1 \end{matrix} ) ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

Use the identifications

A^{m+1} = A^{m} ⊕ A , H_{ϵ}(A^{m+1}) = H_{ϵ}(A^{m}) ⊕ H_{ϵ}(A)

to define injections of groups

U_{ϵ,m}(A) ⟶ U_{ϵ,m+1}(A) ; f ⟶ f ⊕ 1,

such that EU_{ϵ,m}(A) is sent into EU_{ϵ,m+1}(A).

Definition 10.9 i) The stable ϵ-quadratic unitary group of A is the union

U_{ϵ}(A) = ⋃_{m=1}^{∞} U_{ϵ,m}(A) .

ii) The elementary stable ϵ-quadratic unitary group of A is the union

EU_{ϵ}(A) = ⋃_{m=1}^{∞} EU_{ϵ,m}(A) ,

a normal subgroup of U_{ϵ}(A). iii) The ϵ-quadratic M-group of A is the quotient

M_{ϵ}(A) = U_{ϵ}(A)/{EU_{ϵ}(A), σ_{ϵ}}

with

σ_{ϵ} = ( \begin{matrix}0 & 1 \\ ϵ & 0 \end{matrix} ) ∈ U_{ϵ,1}(A) ⊆ U_{ϵ}(A) .

M_{ϵ}(A) is the original definition due to Wall of the odd-dimensional L-group L_{2n+1}(A), with ϵ = (−)^{n}. Wall verified that M_{ϵ}(A) is abelian using a somewhat complicated matrix identity, corresponding to the formation identity 9.14. We now use formations to identify M_{(−)^{n}}(A) with the a priori abelian L-group L_{2n+1}(A) defined in §8.

Given an automorphism of a hyperbolic (−)^{n}-quadratic form

α = ( \begin{matrix}γ & γ̃ \\ μ & μ̃ \end{matrix} ) : H_{(−)^{n}}(A^{m}) ⟶ H_{(−)^{n}}(A^{m})

define a (2n + 1)-complex (C, ψ) by

d = μ^{∗} : C_{n+1} = A^{m} ⟶ C_{n} = A^{m} ,
ψ_{0} = γ : C^{n} = A^{m} ⟶ C_{n+1} = A^{m} ,

corresponding to the (−)^{n}-quadratic formation

Φ_{m}(α) = (H_{(−)^{n}}(A^{m}); A^{m}, im(( \begin{matrix}γ \\ μ \end{matrix} ) : A^{m}⟶A^{m} ⊕ (A^{m})^{∗})) .

Lemma 10.10 The formations Φ_{m}(α_{1}), Φ_{m}(α_{2}) associated to two automorphisms

α_{i} = ( \begin{matrix}γ_{i} & γ̃_{i} \\ μ_{i} & μ̃_{i} \end{matrix} ) : H_{(−)^{n}}(A^{m}) ⟶ H_{(−)^{n}}(A^{m}) (i = 1, 2)

are isomorphic if and only if there exist β_{i} ∈ GL_{m}(A), θ_{i} ∈ Q_{(−)^{n+1}}(A^{m}) such that

( \begin{matrix}β_{1} & 0 \\ 0 & β_{1}^{∗−1} \end{matrix} ) ( \begin{matrix}1 & θ_{1} + (−)^{n+1}θ_{1}^{∗} \\ 0 & 1 \end{matrix} ) ( \begin{matrix}γ_{1} & γ̃_{1} \\ μ_{1} & μ̃_{1} \end{matrix} )

= ( \begin{matrix}γ_{2} & γ̃_{2} \\ μ_{2} & μ̃_{2} \end{matrix} ) ( \begin{matrix}β_{2} & 0 \\ 0 & β_{2}^{∗−1} \end{matrix} ) ( \begin{matrix}1 & θ_{2} + (−)^{n+1}θ_{2}^{∗} \\ 0 & 1 \end{matrix} )

: H_{(−)^{n}}(A^{m}) ⟶ H_{(−)^{n}}(A^{m}) .

Proof: An automorphism α : H_{(−)^{n}}(A^{m})⟶H_{(−)^{n}}(A^{m}) of the hyperbolic (−)^{n}- quadratic form H_{(−)^{n}}(A^{m}) preserves the lagrangian A^{m} ⊂ A^{m} ⊕ (A^{m})^{∗} if and only if there exist β ∈ GL_{m}(A), θ ∈ Q_{(−)^{n+1}}(A^{m}) such that

α = ( \begin{matrix}β & 0 \\ 0 & β^{∗−1} \end{matrix} ) ( \begin{matrix}1 & θ + (−)^{n+1}θ^{∗} \\ 0 & 1 \end{matrix} ) : H_{(−)^{n}}(A^{m}) ⟶ H_{(−)^{n}}(A^{m}) .

Proposition 10.11 The function

Φ : M_{(−)^{n}}(A) ⟶ L_{2n+1}(A) ; α ⟶ Φ_{m}(α) (α ∈ U_{(−)^{n},m}(A))

is an isomorphism of groups. Proof: The function

Φ_{m} : U_{(−)^{n},m}(A) ⟶ L_{2n+1}(A) ; α ⟶ Φ_{m}(α)

is a group morphism by 9.14. Each of the generators (10.7) of the elementary subgroup EU_{(−)^{n},m}(A) is sent to 0 with

i) Φ_{m} ( \begin{matrix}β & 0 \\ 0 & β^{∗−1} \end{matrix} ) = (H_{(−)^{n}}(A^{m}); A^{m}, A^{m}) = ∂(A^{m}, 0) = 0 ∈ L_{2n+1}(A) ,

ii) Φ_{m} ( \begin{matrix}1 & 0 \\ θ + (−)^{n+1}θ^{∗} & 1 \end{matrix} ) = ∂(A^{m}, θ) = 0 ∈ L_{2n+1}(A) .

Also, abbreviating σ_{(−)^{n}} to σ

Φ_{1}(σ) = (H_{(−)^{n}}(A); A, A^{∗}) = 0 ∈ L_{2n+1}(A) ,
Φ_{m+1}(α ⊕ 1) = Φ_{m}(α) ⊕ (H_{(−)^{n}}(A); A, A) = Φ_{m}(α) ∈ L_{2n+1}(A) .

Thus the morphisms Φ_{m} (m ≥ 0) fit together to define a group morphism

Φ : M_{(−)^{n}}(A) ⟶ L_{2n+1}(A) ; α ⟶ Φ_{m}(α) if α ∈ U_{(−)^{n},m}(A)

such that

Φ(α_{1}α_{2}) = Φ(α_{1} ⊕ α_{2}) = Φ(α_{1}) ⊕ Φ(α_{2}) ∈ L_{2n+1}(A) .

Φ is onto by 9.3 ii). It remains to prove that Φ is one-one.

For any α_{i} ∈ U_{(−)^{n},m_{i}}(A) (i = 1, 2) we have

α_{1} ⊕ α_{2} = α_{2} ⊕ α_{1} ∈ M_{(−)^{n}}(A),

since

\begin{aligned}( \begin{matrix}α_{1} & 0 \\ 0 & α_{2} \end{matrix} ) & = ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )^{−1} ( \begin{matrix}α_{2} & 0 \\ 0 & α_{1} \end{matrix} ) ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) \\ & : H_{(−)^{n}}(A^{m_{1}+m_{2}}) ⟶ H_{(−)^{n}}(A^{m_{1}+m_{2}}) . \end{aligned}

Now σ = 1 ∈ M_{(−)^{n}}(A) (by construction), so that for any α ∈ U_{(−)^{n},m}(A)

α ⊕ σ = σ ⊕ α = (σ ⊕ 1)(1 ⊕ α) = α ∈ M_{(−)^{n}}(A) .

It follows that for every m ≥ 1

σ ⊕ σ ⊕ … ⊕ σ = 1 ∈ M_{(−)^{n}}(A) (m−fold sum) .

If α ∈ U_{(−)^{n},m}(A) is such that Φ(α) = 0 ∈ L_{2n+1}(A) then by 9.12 the (−)^{n}- quadratic formation Φ_{m}(α) is stably isomorphic to the boundary ∂(A^{m′}, θ) of a (−)^{n+1}-quadratic form (A^{m′}, θ ∈ Q_{(−)^{n+1}}(A^{m′})), which can be expressed as

∂(A^{m′}, θ) = Φ_{m′} ( \begin{matrix}1 & 0 \\ θ + (−)^{n+1}θ^{∗} & 1 \end{matrix} ) .

Thus for a sufficiently large m′′ ≥ 0 there exist by 10.10 β_{i} ∈ GL_{m′′}(A), θ_{i} ∈ Q_{(−)^{n+1}}(A^{m′′}) (i = 1, 2) such that

( \begin{matrix}β_{1} & 0 \\ 0 & β_{1}^{∗−1} \end{matrix} ) ( \begin{matrix}1 & θ_{1} + (−)^{n+1}θ_{1}^{∗} \\ 0 & 1 \end{matrix} ) (α ⊕ σ ⊕ … ⊕ σ)

= (( \begin{matrix}1 & 0 \\ θ + (−)^{n+1}θ^{∗} & 1 \end{matrix} ) ⊕ σ ⊕ σ … ⊕ σ) ( \begin{matrix}β_{2} & 0 \\ 0 & β_{2}^{∗−1} \end{matrix} ) ( \begin{matrix}1 & θ_{2} + (−)^{n+1}θ_{2}^{∗} \\ 0 & 1 \end{matrix} )

: H_{(−)^{n}}(A^{m′′}) ⟶ H_{(−)^{n}}(A^{m′′})

so that by another application of 10.8

α = ( \begin{matrix}1 & 0 \\ θ + (−)^{n+1}θ^{∗} & 1 \end{matrix} ) = 1 ∈ M_{(−)^{n}}(A) ,

verifying that Φ is one-one.