ALGEBRAIC POINCARÉ COBORDISM

by Andrew Ranicki

Introduction

1. Rings with involution

2. Chain complexes

3. Symmetric, quadratic and hyperquadratic structures 4. Wu classes

5. Algebraic Poincaré complexes

6. Chain bundles

7. Normal complexes

8. Normal cobordism

9. Normal Wu classes

10. Forms

11. An example.

Introduction

The Wall surgery obstruction σ_{∗}(f, b) ∈ L_{n}(π_{1}(X)) of an n-dimensional normal map (f, b) : M⟶X was expressed in Ranicki [ ] as the cobordism class (C, ψ) of a ℤ[π_{1}(X)]-module chain complex C such that

H_{∗}(C) = ker(f_{∗} : H_{∗}(M̃)⟶H_{∗}(X̃))

together with an n-dimensional quadratic Poincaré duality

ψ : H^{n−∗}(C) ≅ H_{∗}(C) .

The passage from the bundle map b : ν_{M}⟶ν_{X} to ψ used an equivariant chain level version of the relationship established by Thom between the Wu classes of the normal bundle ν_{M} of a manifold M and the action of the Steenrod algebra on the Thom class of ν_{M}. The theory of algebraic Poincaré complexes was developed further by Weiss [ ], with generalized L-groups L^{n}(B, γ) defined for any space B with a (k −1)-spherical fibration γ : B⟶BG(k). Given an n-dimensional geometric Poincaré complex X and a map g : X⟶B such that g^{∗}γ = ν_{X} is the Spivak normal fibration there is defined an invariant

σ^{∗}(X, g) ∈ L^{n}(B, γ)

generalizing the signature. For a normal map (f, b) : M⟶X and g : X⟶B inducing an isomorphism of fundamental groups π_{1}(X)⟶π_{1}(B) the image of the surgery obstruction σ_{∗}(f, b) under a canonical map L_{n}(π_{1}(X))⟶L^{n}(B, γ) is the difference of the generalized signature invariants

[σ_{∗}(f, b)] = σ^{∗}(M, gf) − σ^{∗}(X, g) ∈ L^{n}(B, γ) .

In this paper we study the generalized signature invariants in greater detail.

§1. Rings with involution

In §1 we show how an involution a⟶\overline{a} on a ring A determines a duality involution functor

(f.g. projective left A-modules) ⟶ (f.g. projective left A-modules) .

More generally, duality can be defined using an antistructure on A in the sense of Wall [ ], and the L-theory results described in this paper all have versions for rings with antistructure.

Let A be an associative ring with 1, together with an involution

: A ⟶ A ; a ⟶ \overline{a} ,

that is a function satisfying

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

In the topological applications A = ℤ[π] is a group ring, for some group π equipped with a morphism w : π⟶ℤ_{2} = {+1, −1}, and the involution is defined by

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

A-modules are understood to be left A-modules unless a right A-action is expressly specified. Given an A-module M there is defined a right A-module M^{t} with the same additive group and

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

The dual of an A-module M is the A-module with additive group

M^{∗} = Hom_{A}(M, A)

and A acting by

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

The dual of an A-module morphism f ∈ Hom_{A}(M, N) is the A-module morphism

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

For a f.g. (finitely generated) projective A-module M the natural A-module isomorphism

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

will be used to identify

M^{∗∗} = M .

§2. Chain complexes

In order to adequately deal with the quadratic nature of the function C⟶C^{t} ⊗_{A} C sending an A-module chain complex C to the ℤ-module chain complex C^{t} ⊗_{A} C it is necessary to use the Kan-Dold equivalence between positive ℤ-module chain complexes and simplicial ℤ-modules. This is now recalled, along with some other properties of chain complexes that we shall require.

An A-module chain complex

C : … ⟶ C_{r+1} ⟶^{d} C_{r} ⟶^{d} C_{r−1} ⟶ … (r ∈ ℤ)

is n-dimensional if each C_{r} (0 ≤ r ≤ n) is a f.g. projective A-module and C_{r} = 0 for r < 0 and r > n. By an abuse of terminology chain complexes of the chain homotopy type of an n-dimensional chain complex will also be called n-dimensional.

The suspension of an A-module chain complex C is the A-module chain complex defined by

d_{SC} = d_{C} : SC_{r} = C_{r−1} ⟶ SC_{r−1} = C_{r−2} .

If C is n-dimensional then SC is (n + 1)-dimensional.

Given an A-module chain complex C let

C^{r} = (C_{r})^{∗} (r ∈ ℤ) .

The dual A-module chain complex C^{−∗} is defined by

d_{C^{−∗}} = (d_{C})^{∗} : (C^{−∗})_{r} = C^{−r} ⟶ (C^{−∗})_{r−1} = C^{−r+1} .

The n-dual A-module chain complex C^{n−∗} is defined by

d_{C^{n−∗}} = (−)^{r}(d_{C})^{∗} : (C^{n−∗})_{r} = C^{n−r} ⟶ (C^{n−∗})_{r−1} = C^{n−r+1} .

The n-fold suspension of the dual S^{n}C^{−∗} is related to the n-dual C^{n−∗} by the isomorphism

S^{n}C^{∗} ⟶ C^{n−∗} ; x ⟶ (−)^{r(r−1)/2}x (x ∈ C^{n−r}) .

A chain map up to sign between A-module chain complexes

f : C ⟶ D

is a collection of A-module morphisms

{f ∈ Hom_{A}(C_{r}, D_{r}) | r ∈ ℤ}

such that

d_{D}f = ±fd_{C} : C_{r} ⟶ D_{r−1} (r ∈ ℤ) .

If the sign is always + this is just a chain map f : C⟶D, as usual.

Given A-module chain complexes C, D let C^{t} ⊗_{A} D, Hom_{A}(C, D) be the ℤ-module chain complexes defined by

(C^{t} ⊗_{A} D)_{n} = ∑_{p+q=n} C_{p} ⊗_{A} D_{q} ,

d_{C^{t}}⊗_{A}D(x ⊗ y) = x ⊗ d_{D}(y) + (−)^{q}d_{C}(x) ⊗ y ,

Hom_{A}(C, D)_{n} = ∑_{q−p=n} Hom_{A}(C_{p}, D_{q}) ,

d_{Hom_{A}(C,D)}(f)(x) = d_{D}(f(x)) + (−)^{q}f(d_{C}(x)) .

A cycle f ∈ Hom_{A}(C, D)_{n} is a chain map up to sign f : S^{n}C⟶D, and

H_{n}(Hom_{A}(C, D)) = H_{0}(Hom_{A}(S^{n}C, D))

is the ℤ-module of chain homotopy classes of chain maps S^{n}C⟶D.

For finite-dimensional C the slant isomorphism of ℤ-module chain complexes

C^{t} ⊗_{A} D ⟶ Hom_{A}(C^{∗}, D) ; x ⊗ y ⟶ (f ⟶ \overline{f(x)} . y)

will be used to identify

C^{t} ⊗_{A} D = Hom_{A}(C^{∗}, D) .

A cycle

f ∈ (C^{t} ⊗_{A} D)_{n} = Hom_{A}(C^{∗}, D)_{n}

is a chain map f : C^{n−∗}⟶D. Thus

H_{n}(C^{t} ⊗_{A} D) = H_{n}(Hom_{A}(C^{∗}, D)) = H_{0}(Hom_{A}(C^{n−∗}, D))

is the ℤ-module of chain homotopy classes of chain maps C^{n−∗}⟶D.

The algebraic mapping cone C(f) of an A-module chain map f : C⟶D is the A-module chain complex defined as usual by

d_{C(f)} = ( \begin{matrix}d_{D} & (−)^{r−1}f \\ 0 & d_{C} \end{matrix} ) : C(f)_{r} = D_{r} ⊕ C_{r−1} ⟶ C(f)_{r−1} = D_{r−1} ⊕ C_{r−2} .

The relative homology A-modules

H_{n}(f) = H_{n}(C(f))

are such that there is defined an exact sequence

… ⟶ H_{n}(C) ⟶^{f_{∗}} H_{n}(D) ⟶ H_{n}(f) ⟶ H_{n−1}(C) ⟶ ….

Let C(Δ^{n}) denote the cellular chain complex of the standard n-simplex Δ^{n} with

the standard cell structure consisting of \binom{n + 1}{r + 1} r-cells (0 ≤ r ≤ n).

Given a ℤ-module chain complex C let K(C) denote the simplicial ℤ-module defined by the Kan-Dold construction, with one n-simplex for each chain map C(Δ^{n})⟶C and the evident face and degeneracy maps d_{i}, s_{i}, such that

π_{n}(K(C)) = H_{n}(C) (n ≥ 0) .

Given a chain y ∈ C_{n} and cycles x_{i} ∈ C_{n−1} (0 ≤ i ≤ n) such that

dy = ∑_{i=0}^{n}(−)^{i}x_{i} ∈ C_{n−1}

let (y; x_{0}, …, x_{n}) denote the n-simplex of K(C) defined by the chain map

f : C(Δ^{n}) ⟶ C

with

\begin{matrix}f : C(Δ^{n})_{n} = ℤ ⟶ C_{n} ; 1 ⟶ y , \\ d_{i}f : C(Δ^{n−1})_{n−1} = ℤ ⟶ C_{n−1} ; 1 ⟶ x_{i} (0 ≤ i ≤ n) . \end{matrix}

The chain x ∈ C_{n} is identified with the n-simplex (x; dx, 0, …, 0) ∈ K(C)^{(n)}.

Given ℤ-module chain complexes C, D and a simplicial map

f : K(C) ⟶ K(D)

(which need not preserve the ℤ-module structure) define functions

f : C_{n} ⟶ D_{n} ; x ⟶ f(x) = f(x; dx, 0, …, 0)

and

[ , ] : ker(d : C_{n}⟶C_{n−1}) × ker(d : C_{n}⟶C_{n−1})⟶D_{n+1} ;
(y, y′) ⟶ [y, y′] = f(0; y + y′, y, −y′, 0, …, 0) − f(0; y, y, 0, …, 0)
− f(0; y′, 0, −y′, 0, …, 0)

such that

df(x) = f(dx) ∈ D_{n−1} ,
f(y + y′) − f(y) − f(y′) = d[y, y′] ∈ D_{n}
(x ∈ C_{n} , y, y′ ∈ ker(d : C_{n}⟶C_{n−1})) .

There is defined an exact sequence of abelian groups

… ⟶ H_{n+1}(f) ⟶ H_{n}(C) ⟶^{f_{∗}} H_{n}(D) ⟶ H_{n}(f) ⟶ H_{n−1}(C) ⟶ …,

with the relative group H_{n}(f) (= π_{n}(f)) the set of equivalence classes of pairs (x, y) ∈ C_{n} × D_{n+1} such that

dx = 0 ∈ C_{n−1} , f(x) = dy ∈ D_{n} ,

subject to the equivalence relation

(x, y) ∼ (x′, y′) if there exist (u, v) ∈ C_{n+1} × D_{n+2} such that

x − x′ = du ∈ C_{n}, y − y′ = f(u; x, x′, 0, …, 0) + dv ∈ D_{n+1} ,

and addition by

(x, y) + (x′, y′) = (x + x′, y + y′ + [x, x′]) ∈ H_{n}(f) .

If f : K(C)⟶K(D) does preserve the ℤ-module structure (so that [ , ] = 0) then it is defined by a chain map f : C⟶D, and the relative homology groups H_{∗}(f) are just the homology groups H_{∗}(C(f)) of the algebraic mapping cone C(f).

§3. Symmetric, quadratic and hyperquadratic structures

An n-dimensional \{ \begin{matrix}symmetric \\ quadratic \\ hyperquadratic \end{matrix} structure on a finite-dimensional A-module

chain complex C is a cycle representing an element of the \{ \begin{matrix}ℤ_{2}-hypercohomology \\ ℤ_{2}-hyperhomology \\ Tate ℤ_{2}-hypercohomology \end{matrix}

group \{ \begin{matrix}H^{n}(ℤ_{2}; C^{t} ⊗_{A} C) \\ H_{n}(ℤ_{2}; C^{t} ⊗_{A} C) \\ Ĥ^{n}(ℤ_{2}; C^{t} ⊗_{A} C) \end{matrix} in the sense of Cartan and Eilenberg [ ], with the generator

T ∈ ℤ_{2} acting on C^{t} ⊗_{A} C by x ⊗ y⟶ ± y ⊗ x. Given an A-module chain complex C let the generator T ∈ ℤ_{2} act on C^{t} ⊗_{A} C by the transposition involution

T : C^{t} ⊗_{A} C ⟶ C^{t} ⊗_{A} C ; x ⊗ y ⟶ (−)^{pq}y ⊗ x (x ∈ C_{p}, y ∈ C_{q}) .

For finite-dimensional C use the slant isomorphism to identify

C^{t} ⊗_{A} C = Hom_{A}(C^{∗}, C) .

Under this identification the transposition involution corresponds to the duality involution on Hom_{A}(C^{∗}, C)

T : Hom_{A}(C^{∗}, C) ⟶ Hom_{A}(C^{∗}, C) ; ϕ⟶(−)^{pq}ϕ^{∗} (ϕ ∈ Hom_{A}(C^{p}, C_{q})) .

A cycle ϕ ∈ Hom_{A}(C^{∗}, C)_{n} = (C^{t} ⊗_{A} C)_{n} is a chain map ϕ : C^{n−∗}⟶C, and H_{n}(Hom_{A}(C^{∗}, C)) is the ℤ-module of chain homotopy classes of A-module chain maps C^{n−∗}⟶C. Let W be the standard free ℤ[ℤ_{2}]-module resolution of ℤ

W : … ⟶ ℤ[ℤ_{2}] ⟶^{1−T} ℤ[ℤ_{2}] ⟶^{1+T} ℤ[ℤ_{2}] ⟶^{1−T} ℤ[ℤ_{2}] ⟶ 0

and let Ŵ be the complete resolution

Ŵ : … ⟶ ℤ[ℤ_{2}] ⟶^{1−T} ℤ[ℤ_{2}] ⟶^{1+T} ℤ[ℤ_{2}] ⟶^{1−T} ℤ[ℤ_{2}] ⟶ ….

The \{ \begin{matrix}ℤ_{2}-hypercohomology \\ ℤ_{2}-hyperhomology \\ Tate ℤ_{2}-hypercohomology \end{matrix} groups of a ℤ[ℤ_{2}]-module chain complex C are defined by

\{ \begin{matrix}H^{n}(ℤ_{2}; C) = H_{n}(Hom_{ℤ[ℤ_{2}]}(W, C)) \\ H_{n}(ℤ_{2}; C) = H_{n}(W ⊗_{ℤ[ℤ_{2}]} C) \\ Ĥ^{n}(ℤ_{2}; C) = H_{n}(Hom_{ℤ[ℤ_{2}]}(Ŵ, C)) . \end{matrix}

The evident short exact sequence of ℤ[ℤ_{2}]-module chain complexes

0 ⟶ SW^{−∗} ⟶ Ŵ ⟶ W ⟶ 0

induces a long exact sequence of abelian groups

… ⟶ H_{n}(ℤ_{2}; C) ⟶^{1+T} H^{n}(ℤ_{2}; C)⟶^{J} Ĥ^{n}(ℤ_{2}; C)⟶^{H} H_{n−1}(ℤ_{2}; C) ⟶ ….

An element \{ \begin{matrix}ϕ ∈ H^{n}(ℤ_{2}; C) \\ ψ ∈ H_{n}(ℤ_{2}; C) \\ θ ∈ Ĥ^{n}(ℤ_{2}; C) \end{matrix} is represented by an n-cycle of \{ \begin{matrix}Hom_{ℤ[ℤ_{2}]}(W, C) \\ W ⊗_{ℤ[ℤ_{2}]} C \\ Hom_{ℤ[ℤ_{2}]}(Ŵ, C) \end{matrix} which

is just a collection of chains of C \{ \begin{matrix}\{ϕ_{s} ∈ C_{n+s} | s ≥ 0\} \\ \{ψ_{s} ∈ C_{n−s} | s ≥ 0\} \\ \{θ_{s} ∈ C_{n+s} | s ∈ ℤ\} \end{matrix} such that

\{ \begin{matrix}d_{C}(ϕ_{s}) + (−)^{n+s−1}(ϕ_{s−1} + (−)^{s}T ϕ_{s−1}) = 0 ∈ C_{n+s−1} (s ≥ 0, ϕ_{−1} = 0) \\ d_{C}(ψ_{s}) + (−)^{n−s−1}(ψ_{s+1} + (−)^{s+1}T ψ_{s+1}) = 0 ∈ C_{n−s−1} (s ≥ 0) \\ d_{C}(θ_{s}) + (−)^{n+s−1}(θ_{s−1} + (−)^{s}T θ_{s−1}) = 0 ∈ C_{n+s−1} (s ∈ ℤ) \end{matrix}

with

1 + T : H_{n}(ℤ_{2}; C) ⟶ H^{n}(ℤ_{2}; C) ;

ψ = \{ψ_{s} | s ≥ 0\} ⟶ \{(1 + T)ψ_{s} = \{ \begin{matrix}(1 + T)ψ_{0} & if s = 0 \\ 0 & if s ≥ 1 \end{matrix}\} ,

J : H^{n}(ℤ_{2}; C) ⟶ Ĥ^{n}(ℤ_{2}; C) ;

ϕ = \{ϕ_{s} | s ≥ 0\} ⟶ \{Jϕ_{s} = \{ \begin{matrix}ϕ_{s} & if s ≥ 0 \\ 0 & if s ≤ −1 \end{matrix}\}

H : Ĥ^{n}(ℤ_{2}; C) ⟶ H_{n−1}(ℤ_{2}; C) ;

θ = {θ_{s} | s ∈ ℤ} ⟶ Hθ = {Hθ_{s} = θ_{−s−1} | s ≥ 0} .

Given an A-module chain complex C use the action of T ∈ ℤ_{2} on C^{t} ⊗_{A} C by the transposition involution to define the ℤ-module chain complex

W^{%}C = Hom_{ℤ[ℤ_{2}]}(W, C^{t} ⊗_{A} C)
W_{%}C = W ⊗_{ℤ[ℤ_{2}]} (C^{t} ⊗_{A} C)
Ŵ^{%}C = Hom_{ℤ[ℤ_{2}]}(Ŵ, C^{t} ⊗_{A} C) .

We shall be mainly concerned with finite-dimensional C, using the slant isomorphism to identify

C^{t} ⊗_{A} C = Hom_{A}(C^{∗}, C)

and

W^{%}C = Hom_{ℤ[ℤ_{2}]}(W, Hom_{A}(C^{∗}, C))
W_{%}C = W ⊗_{ℤ[ℤ_{2}]} Hom_{A}(C^{∗}, C)
Ŵ^{%}C = Hom_{ℤ[ℤ_{2}]}(Ŵ, Hom_{A}(C^{∗}, C)) .

An n-dimensional \{ \begin{matrix}symmetric \\ quadratic \\ hyperquadratic \end{matrix} structure on a finite-dimensional A-module chain

complex C is a cycle \{ \begin{matrix}ϕ ∈ (W^{%}C)_{n} \\ ψ ∈ (W_{%}C)_{n} \\ θ ∈ (Ŵ^{%}C)_{n} , \end{matrix} which is just a collection of A-module mor-

phisms

{ϕ_{s} ∈ Hom_{A}(C^{n−r+s}, C_{r}) | r ∈ ℤ, s ≥ 0}
{ψ_{s} ∈ Hom_{A}(C^{n−r−s}, C_{r}) | r ∈ ℤ, s ≥ 0}
{θ_{s} ∈ Hom_{A}(C^{n−r+s}, C_{r}) | r ∈ ℤ, s ∈ ℤ} .

such that

dϕ_{s} + (−)^{r}ϕ_{s}d^{∗} + (−)^{n+s−1}(ϕ_{s−1} + (−)^{s}T ϕ_{s−1}) = 0
: C^{n−r+s−1} ⟶ C_{r} (s ≥ 0, ϕ_{−1} = 0)
dψ_{s} + (−)^{r}ψ_{s}d^{∗} + (−)^{n−s−1}(ψ_{s+1} + (−)^{s+1}T ψ_{s+1}) = 0
: C^{n−r−s−1} ⟶ C_{r} (s ≥ 0)
dθ_{s} + (−)^{r}θ_{s}d^{∗} + (−)^{n+s−1}(θ_{s−1} + (−)^{s}T θ_{s−1}) = 0
: C^{n−r+s−1} ⟶ C_{r} (s ∈ ℤ) .

An equivalence of n-dimensional \{ \begin{matrix}symmetric \\ quadratic \\ hyperquadratic \end{matrix} structures on C \{ \begin{matrix}ξ : ϕ⟶ϕ′ \\ χ : ψ⟶ψ′ \\ ν : θ⟶θ′ \end{matrix}

is a chain \{ \begin{matrix}ξ ∈ (W^{%}C)_{n+1} \\ χ ∈ (W_{%}C)_{n+1} \\ ν ∈ (Ŵ^{%}C)_{n+1} \end{matrix} such that

\{ \begin{matrix}ϕ′ − ϕ = d(ξ) ∈ (W^{%}C)_{n} \\ ψ′ − ψ = d(χ) ∈ (W_{%}C)_{n} \\ θ′ − θ = d(ν) ∈ (Ŵ^{%}C)_{n} . \end{matrix}

The n-dimensional \{ \begin{matrix}symmetric \\ quadratic \\ hyperquadratic \end{matrix} structure group \{ \begin{matrix}Q^{n}(C) \\ Q_{n}(C) \\ Q̂^{n}(C) \end{matrix} of a chain complex C

is the abelian group of equivalence classes of n-dimensional \{ \begin{matrix}symmetric \\ quadratic \\ hyperquadratic \end{matrix} struc-

tures on C, that is

\{ \begin{matrix}Q^{n}(C) = H^{n}(ℤ_{2}; C^{t} ⊗_{A} C) = H_{n}(W^{%}C) \\ Q_{n}(C) = H_{n}(ℤ_{2}; C^{t} ⊗_{A} C) = H_{n}(W_{%}C) \\ Q̂^{n}(C) = Ĥ^{n}(ℤ_{2}; C^{t} ⊗_{A} C) = H_{n}(Ŵ^{%}C) . \end{matrix}

The Q-groups are related by a long exact sequence

… ⟶ Q_{n}(C) ⟶^{1+T} Q^{n}(C) ⟶^{J} Q̂^{n}(C) ⟶^{H} Q_{n−1}(C) ⟶ …

involving the morphisms induced in homology by the ℤ-module chain maps

1 + T : W_{%}C ⟶ W^{%}C , J : W^{%}C ⟶ Ŵ^{%}C , H : Ŵ^{%}C ⟶ S(W_{%}C) defined by

1 + T : (W_{%}C)_{n} ⟶ (W^{%}C)_{n} ;

\{ψ_{s} ∈ (C^{t} ⊗_{A} C)_{n−s} | s ≥ 0\} ⟶ \{((1 + T)ψ)_{s} = \{ \begin{matrix}(1 + T)ψ_{0} & if s = 0 \\ 0 & if s ≥ 1 \end{matrix}\} ,

J : (W^{%}C)_{n} ⟶ (Ŵ^{%}C)_{n} ;

\{ϕ_{s} ∈ (C^{t} ⊗_{A} C)_{n+s} | s ≥ 0\} ⟶ \{(Jϕ)_{s} = \{ \begin{matrix}ϕ_{s} & if s ≥ 0 \\ 0 & if s ≤ −1 \end{matrix}\} ,

H : (Ŵ^{%}C)_{n} ⟶ (W_{%}C)_{n−1} ;

{θ_{s} ∈ (C^{t} ⊗_{A} C)_{n+s} | ∈ ℤ} ⟶ {(Hθ)_{s} = θ_{−s−1} | s ≥ 0} . An n-dimensional symmetric structure ϕ ∈ (W^{%}C)_{n} is equivalent to the symmetrization (1 + T)ψ of an n-dimensional quadratic structure ψ ∈ (W_{%}C)_{n} if and only if the n-dimensional hyperquadratic structure J(ϕ) ∈ (Ŵ^{%}C)_{n} is equivalent to 0. An A-module chain map f : C⟶D induces a ℤ[ℤ_{2}]-module chain map

f^{t} ⊗_{A} f : C^{t} ⊗_{A} C ⟶ D^{t} ⊗_{A} D ; x ⊗ y ⟶ f(x) ⊗ f(y)

and hence ℤ-module chain maps

f^{%} : W^{%}C ⟶ W^{%}D ,
f_{%} : W_{%}C ⟶ W_{%}D ,
f̂^{%} : Ŵ^{%}C ⟶ Ŵ^{%}D .

An A-module chain homotopy

g : f ≃ f′ : C ⟶ D

determines ℤ-module chain homotopies

(g; f, f′)^{%} : f^{%} ≃ f^{′%} : W^{%}C ⟶ W^{%}D ,
(g; f, f′)_{%} : f_{%} ≃ f′_{%} : W_{%}C ⟶ W_{%}D ,
ˆ (g; f, f′)^{%} : f̂^{%} ≃ f̂^{′%} : Ŵ^{%}C ⟶ Ŵ^{%}D

with

(g; f, f′)^{%} : (W^{%}C)_{n} = ∑_{s=0}^{∞}(C^{t} ⊗_{A} C)_{n+s}

⟶ (W^{%}D)_{n+1} = ∑_{s=0}^{∞} ∑_{q} D_{n−q+s+1}^{t} ⊗_{A} D_{q} ;

∑_{s=0}^{∞} ϕ_{s} ⟶ ∑_{s=0}^{∞}((f^{t} ⊗_{A} g + g^{t} ⊗_{A} f′)(ϕ_{s}) + (−)^{q+s−1}(g^{t} ⊗_{A} g)(T ϕ_{s−1}))

and similarly for (g; f, f′)_{%}, ˆ (g; f, f′)^{%}. Thus the induced morphisms in the Q-groups

f^{%} : Q^{n}(C) ⟶ Q^{n}(D)
f_{%} : Q_{n}(C) ⟶ Q_{n}(D)
f̂^{%} : Q̂^{n}(C) ⟶ Q̂^{n}(D)

depend only on the chain homotopy class of f, and are isomorphisms if f is a chain equivalence. For finite-dimensional C, D the slant isomorphisms are used to identify f^{t} ⊗_{A} f : C^{t} ⊗_{A} C⟶D^{t} ⊗_{A} D with

Hom_{A}(f^{∗}, f) : Hom_{A}(C^{∗}, C) ⟶ Hom_{A}(D^{∗}, D) ; θ ⟶ fθf^{∗} ,

and similarly for f^{%}, f_{%}, f̂^{%} and (g; f, f′)^{%}, (g; f, f′)_{%}, ˆ (g; f, f′)^{%}.

Although all the Q-groups are chain homotopy invariant, only the hyperquadratic Q-groups Q̂^{∗}(C) are additive. The sum of A-module chain maps f, g : C⟶D is an A-module chain map f + g : C⟶D such that

(f + g)^{%} − f^{%} − g^{%} : Q^{n}(C) ⟶ H_{n}(C^{t} ⊗_{A} C) ⟶^{f^{t}⊗_{A}g} H_{n}(D^{t} ⊗_{A} D) ⟶ Q^{n}(D) ,

(f + g)_{%} − f_{%} − g_{%} : Q_{n}(C) ⟶ H_{n}(C^{t} ⊗_{A} C) ⟶^{f^{t}⊗_{A}g} H_{n}(D^{t} ⊗_{A} D) ⟶ Q_{n}(D) ,

ˆ (f + g)^{%} − f̂^{%} − g^{%} = 0 : Q̂^{n}(C) ⟶ Q̂^{n}(D)

with

Q^{n}(C) ⟶ H_{n}(C^{t} ⊗_{A} C) ; ϕ = {ϕ_{s} | s ≥ 0} ⟶ ϕ_{0} ,

Q_{n}(C) ⟶ H_{n}(C^{t} ⊗_{A} C) ; ψ = {ψ_{s} | s ≥ 0} ⟶ (1 + T)ψ_{0} ,

H_{n}(D^{t} ⊗_{A} D) ⟶ Q^{n}(D) ; θ ⟶ \{ϕ_{s} = \{ \begin{matrix}(1 + T)θ & if s = 0 \\ 0 & if s ≥ 1 \end{matrix}\} ,

H_{n}(D^{t} ⊗_{A} D) ⟶ Q_{n}(D) ; θ ⟶ \{ψ_{s} = \{ \begin{matrix}θ & if s = 0 \\ 0 & if s ≥ 1 \end{matrix} \} .

Given a finite-dimensional A-module chain complex C and n ≥ 0 define the n-fold suspension chain isomorphism

S^{n} : S^{n}(Ŵ^{%}C) ⟶ Ŵ^{%}(S^{n}C) ;
θ = {θ_{s} ∈ Hom_{A}(C^{r}, C_{m−r+s}) | s ∈ ℤ}
⟶ S^{n}θ = {(S^{n}θ)_{s} = θ_{s−n} ∈ Hom_{A}(C^{r}, C_{m−n+r+s}) | s ∈ ℤ} .

The methods of Weiss [ ] show that for any (finite-dimensional) A-module chain complexes C, D there is defined a simplicial map

I : K(Hom_{A}(C, D)) ⟶ K(Hom_{ℤ}(Ŵ^{%}C, Ŵ^{%}D))

sending a cycle f ∈ Hom_{A}(C, D)_{n} (= a chain map up to sign f : S^{n}C⟶D) to the ℤ-module chain map up to sign

I(f) = f̂^{%}S^{n} : S^{n}(Ŵ^{%}C) ⟶^{S^{n}} Ŵ^{%}S^{n}C \underrightarrow{_{f^{%}}} W^{%}D .

An n-simplex (g; f, f′, 0, …, 0) ∈ K(Hom_{A}(C, D))^{(n)} (= an A-module chain homotopy up to sign g : f ≃ f′ : S^{n}C⟶D) is sent to the ℤ-module chain homotopy up to sign

I(g; f, f′) = (g; f, f′)^{%}S^{n} : I(f) ≃ I(f′) : S^{n}(Ŵ^{%}C) ⟶ Ŵ^{%}D .

The failure of I to be linear on chain maps up to sign f : S^{n}C⟶D is given by the chain homotopy up to sign

[f, f′] : ˆ (f + f′)^{%} ≃ f̂^{%} + f̂^{′%} : Ŵ^{%}(S^{n}C) ⟶ Ŵ^{%}D

defined by

[f, f′] : (S^{n}Ŵ^{%}C)_{m} ⟶ (Ŵ^{%}D)_{m+n+1} ;
θ = {θ_{s} ∈ Hom_{A}(C^{r}, C_{m−r+s}) | s ∈ ℤ}
⟶ [f, f′]θ = {T^{n+1}θ_{s−n+1}f^{′∗} ∈ Hom_{A}(D^{r}, D_{m−n+r+s+1}) | s ∈ ℤ} .

§4. Wu classes

The Wu classes are the fundamental invariants of a structure on a chain complex C, which are obtained by an algebraic analogue of the Steenrod squares in the cohomology groups of a topological space. In the topological applications the algebraic Wu classes are closely related to the Steenrod squares and the topological Wu classes, as explained in Ranicki [ ].

Let S^{r}A (r ∈ ℤ) denote the A-module chain complex

S^{r}A : … ⟶ 0 ⟶ A ⟶ 0 ⟶ …

concentrated in degree r. For any A-module chain complex C there are defined natural isomorphisms

H_{0}(Hom_{A}(C, S^{r}A)) ⟶ H^{r}(C) ; (f : C_{r}⟶A) ⟶ f^{∗}(1) .

An element f ∈ H^{r}(C) is just a chain homotopy class of chain maps f : C⟶S^{r}A. The Wu classes of a quadratic structure on C are the invariants of the equivalence class defined by sending an element f ∈ H^{r}(C) to the induced equivalence class of quadratic structures on S^{r}A. The quadratic structure groups of the elementary complexes S^{r}A are identified with subquotients of the ground ring A.

An A-group M is an abelian group together with an A-action

A × M ⟶ M ; (a, x) ⟶ ax

such that

a(x + y) = ax + ay , a(bx) = (ab)x , 1x = x (x, y ∈ M, a, b ∈ M) .

An A-module is an A-group M such that also

(a + b)x = ax + bx ∈ M .

An A-morphism of A-groups is a morphism of abelian groups

f : M ⟶ N

such that

f(ax) = af(x) ∈ N (x ∈ M, a ∈ A) .

The set of A-group morphisms f : M⟶N defines an abelian group Hom_{A}(M, N), with addition by

(f + g)(x) = f(x) + g(x) ∈ N .

For A-modules M, N the A-morphisms f : M⟶N coincide with A-module morphisms.

For ϵ = ±1 let the generator T ∈ ℤ_{2} act on A by the ϵ-involution

T_{ϵ} : A ⟶ A ; a ⟶ ϵ\overline{a} .

Define the \{ \begin{matrix}ℤ_{2} − cohomology \\ ℤ_{2} − homology \\ T ateℤ_{2} − cohomology \end{matrix} A-groups \{ \begin{matrix}H^{∗}(ℤ_{2}; A, ϵ) \\ H_{∗}(ℤ_{2}; A, ϵ) \\ Ĥ^{∗}(ℤ_{2}; A, ϵ) \end{matrix} by

H^{r}(ℤ_{2}; A, ϵ) = \{ \begin{matrix}ker(1 − T_{ϵ} : A⟶A) & if r = 0 \\ Ĥ^{r}(ℤ_{2}; A, ϵ) & if r ≥ 1 \\ 0 & if r < 0 \end{matrix}

H_{r}(ℤ_{2}; A, ϵ) = \{ \begin{matrix}coker(1 − T_{ϵ} : A⟶A) & if r = 0 \\ Ĥ^{r+1}(ℤ_{2}; A, ϵ) & if r ≥ 1 \\ 0 & if r < 0, \end{matrix}

Ĥ^{r}(ℤ_{2}; A, ϵ) = ker(1 − (−)^{r}T_{ϵ} : A⟶A)/im(1 + (−)^{r}T_{ϵ} : A⟶A) (r ∈ ℤ) . The A-action

A × Ĥ^{r}(ℤ_{2}; A, ϵ) ⟶ Ĥ^{r}(ℤ_{2}; A, ϵ) ; (a, x) ⟶ ax\overline{a}

defines an A-module structure Ĥ^{r}(ℤ_{2}; A, ϵ). The A-actions

\begin{matrix}A × H^{0}(ℤ_{2}; A, ϵ) ⟶ H^{0}(ℤ_{2}; A, ϵ) ; (a, x) ⟶ ax\overline{a} \\ A × H_{0}(ℤ_{2}; A, ϵ) ⟶ H_{0}(ℤ_{2}; A, ϵ) ; (a, x) ⟶ ax\overline{a} \end{matrix}

are not linear in A, and so do not define A-module structures.

For ϵ = +1 the groups \{ \begin{matrix}H^{∗}(ℤ_{2}; A, ϵ) \\ H_{∗}(ℤ_{2}; A, ϵ) \\ Ĥ^{∗}(ℤ_{2}; A, ϵ) \end{matrix} are denoted by \{ \begin{matrix}H^{∗}(ℤ_{2}; A) \\ H_{∗}(ℤ_{2}; A) \\ Ĥ^{∗}(ℤ_{2}; A). \end{matrix}

The natural ℤ-module isomorphisms

Q^{n}(S^{r}A) \underrightarrow{≃} H^{2r−n}(ℤ_{2}; A, (−)^{r}) ; ϕ ⟶ ϕ_{2r−n}(1)(1)
Q_{n}(S^{r}A) \underrightarrow{≃} H_{n−2r}(ℤ_{2}; A, (−)^{r}) ; ψ ⟶ ψ_{n−2r}(1)(1)
Q̂^{n}(S^{r}A) \underrightarrow{≃} Ĥ^{r−n}(ℤ_{2}; A, (−)^{r}) ; θ ⟶ θ_{2r−n}(1)(1)

will be used as identifications.

The Wu classes of an n-dimensional \{ \begin{matrix}symmetric \\ quadratic \\ hyperquadratic \end{matrix} structure \{ \begin{matrix}ϕ ∈ (W^{%}C)_{n} \\ ψ ∈ (W_{%}C)_{n} \\ θ ∈ (Ŵ^{%}C)_{n} \end{matrix}

are the invariants of the equivalence class of structures defined by the A-morphisms

v_{r}(ϕ) : H^{n−r}(C) = H_{0}(Hom_{A}(C, S^{n−r}A))

⟶ Q^{n}(S^{n−r}A) = H^{n−2r}(ℤ_{2}; A, (−)^{n−r}) ; f ⟶ (f ⊗ f)(ϕ_{n−2r}) ,

v^{r}(ψ) : H^{n−r}(C) = H_{0}(Hom_{A}(C, S^{n−r}A))

⟶ Q_{n}(S^{n−r}A) = H_{2r−n}(ℤ_{2}; A, (−)^{n−r}) ; f ⟶ (f ⊗ f)(ψ_{2r−n})

v̂_{r}(θ) : H^{n−r}(C) = H_{0}(Hom_{A}(C, S^{n−r}A)) ⟶ Q̂^{n}(S^{n−r}A) = Ĥ^{r}(ℤ_{2}; A) ; f ⟶ (f ⊗ f)(θ_{n−2r}) .

§5. Algebraic Poincaré complexes

An algebraic Poincaré complex is a chain complex with Poincaré duality, such as arises from a compact n-manifold or a normal map.

An n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} (Poincaré) complex over A \{ \begin{matrix}(C, ϕ) \\ (C, ψ) \end{matrix} is an n-dimen-

sional A-module chain complex C together with an n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} struc-

ture \{ \begin{matrix}ϕ ∈ (W^{%}C) \\ ψ ∈ (W_{%}C)_{n} \end{matrix} (such that \{ \begin{matrix}ϕ_{0} \\ (1 + T)ψ_{0} \end{matrix} : C^{n−∗}⟶C is a chain equivalence.)

An (n + 1)-dimensional \{ \begin{matrix}symmetric \\ quadratic \\ (symmetric, quadratic) \end{matrix} (Poincaré) pair over A

( f : C ⟶ D , \{ \begin{matrix}(δϕ, ϕ) \\ (δψ, ψ) \\ (δϕ, ψ) \end{matrix} )

consists of an n-dimensional A-module chain complex C, an (n + 1)-dimensional Amodule chain complex D, a chain map f : C⟶D and a cycle

\{ \begin{matrix}(δϕ, ϕ) ∈ C(f^{%} : W^{%}C⟶W^{%}D)_{n+1} = (W^{%}D)_{n+1} ⊕ (W^{%}C)_{n} \\ (δψ, ψ) ∈ C(f_{%} : W_{%}C⟶W_{%}D)_{n+1} = (W_{%}D)_{n+1} ⊕ (W_{%}C)_{n} \\ (δϕ, ψ) ∈ C((1 + T)f_{%} : W_{%}C⟶W^{%}D)_{n+1} = (W^{%}D)_{n+1} ⊕ (W_{%}C)_{n} \end{matrix}

(such that the A-module chain map D^{n+1−∗}⟶C(f) defined by

\{ \begin{matrix}(δϕ, ϕ)_{0} = ( \begin{matrix}δϕ_{0} \\ ϕ_{0}f^{∗} \end{matrix} ) \\ (1 + T)(δψ, ψ)_{0} = ( \begin{matrix}(1 + T)δψ_{0} \\ (1 + T)ψ_{0}f^{∗} \end{matrix} ) \\ (δϕ, (1 + T)ψ)_{0} = ( \begin{matrix}δϕ_{0} \\ (1 + T)ψ_{0}f^{∗} \end{matrix} ) \end{matrix} : D^{n+1−r} ⟶ C(f)_{r} = D_{r} ⊕ C_{r−1}

is a chain equivalence.) The boundary of the pair is the n-dimensional \{ \begin{matrix}symmetric \\ quadratic \\ quadratic \end{matrix}

(Poincaré) complex \{ \begin{matrix}(C, ϕ) \\ (C, ψ) \\ (C, ψ). \end{matrix}

A homotopy equivalence of n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} complexes

\{ \begin{matrix}(f, χ) : (C, ϕ)⟶(C′, ϕ′) \\ (f, ξ) : (C, ψ)⟶(C′, ψ′) \end{matrix}

is a chain equivalence f : C⟶C′ together with an equivalence of \{ \begin{matrix}symmetric \\ quadratic \end{matrix} structures on C′

\{ \begin{matrix}χ : f^{%}(ϕ) ⟶ ϕ′ \\ ξ : f_{%}(ψ) ⟶ ψ′ . \end{matrix}

There is a similar notion of homotopy equivalence for pairs.

An n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} complex \{ \begin{matrix}(C, ϕ) \\ (C, ψ) \end{matrix} is connected if

\{ \begin{matrix}H_{0}(ϕ_{0} : C^{n−∗}⟶C) = 0 \\ H_{0}((1 + T)ψ_{0} : C^{n−∗}⟶C) = 0 . \end{matrix}

It was shown in Ranicki [ ] that there is a natural one-one correspondence between the homotopy equivalence classes of connected n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} complexes

over A and the homotopy equivalence classes of n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} Poincaré

pairs over A. A connected n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} complex \{ \begin{matrix}(C, ϕ) \\ (C, ψ) \end{matrix} determines the n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} Poincaré pair

\{ \begin{matrix}(i_{C} : ∂C⟶C^{n−∗}, (0, ∂ϕ)) \\ (i_{C} : ∂C⟶C^{n−∗}, (0, ∂ψ)) \end{matrix}

defined by i_{C} = (01) : ∂C_{r} = C_{r+1} ⊕ C^{n−r}⟶C^{n−r},

d_{∂C} = \{ \begin{matrix}( & \begin{matrix}d_{C} & (−)^{r}ϕ_{0} \\ 0 & (−)^{r}d_{C}^{∗}) \end{matrix} & ) \\ ( & \begin{matrix}d_{C} & (−)^{r}(1 + T)ψ_{0} \\ 0 & (−)^{r}d_{C}^{∗}) \end{matrix} & ) \end{matrix} :

∂C_{r} = C_{r+1} ⊕ C^{n−r} ⟶ ∂C_{r−1} = C_{r} ⊕ C^{n−r+1} ,

\{ \begin{matrix}∂ϕ_{0} = \binom{(−)^{n−r−1}T ϕ_{1}(−)^{r(n−r−1)}}{1 0} \\ ∂ψ_{0} = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) \end{matrix} :

∂C^{n−r−1} = C^{n−r} ⊕ C_{r+1} ⟶ ∂C_{r} = C_{r+1} ⊕ C^{n−r} ,

\{ \begin{matrix}∂ϕ_{s} = ( \begin{matrix}(−)^{n−r+s−1}T ϕ_{s+1} & 0 \\ 0 & 0 \end{matrix} ) : \\ ∂C^{n−r+s−1} = C^{n−r+s} ⊕ C_{r−s+1} ⟶ ∂C_{r} = C_{r+1} ⊕ C^{n−r} (s ≥ 1) \\ ∂ψ_{s} = ( \begin{matrix}(−)^{n−r−s}T ψ_{s−1} & 0 \\ 0 & 0 \end{matrix} ) \end{matrix} :

∂C^{n−r−s−1} = C^{n−r−s} ⊕ C_{r+s+1} ⟶ ∂C_{r} = C_{r+1} ⊕ C^{n−r} (s ≥ 1) .

The (n − 1)-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} Poincaré complex

\{ \begin{matrix}∂(C, ϕ) = (∂C, ∂ϕ) \\ ∂(C, ψ) = (∂C, ∂ψ) \end{matrix}

is the boundary of the connected n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} complex \{ \begin{matrix}(C, ϕ) \\ (C, ψ). \end{matrix} The connected complex \{ \begin{matrix}(C, ϕ) \\ (C, ψ) \end{matrix} is a Poincaré complex if and only if the boundary \{ \begin{matrix}∂(C, ϕ) \\ (C, ψ) \end{matrix} is contractible (= homotopy equivalent to 0). A Poincaré complex \{ \begin{matrix}(C, ϕ) \\ (C, ψ) \end{matrix} is the boundary of an (n+1)-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} Poincaré pair \{ \begin{matrix}(f : C⟶D, (δϕ, ϕ)) \\ (f : C⟶D, (δψ, ψ)) \end{matrix} if and only if it is homotopy equivalent to the boundary of a connected (n + 1)-

dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} complex.

The n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} Poincaré complexes \{ \begin{matrix}(C, ϕ) \\ (C, ψ) \end{matrix}, \{ \begin{matrix}(C′, ϕ′) \\ (C′, ψ′) \end{matrix} are cobor-

dant if \{ \begin{matrix}(C, ϕ) ⊕ (C′, −ϕ′) \\ (C, ψ) ⊕ (C′, −ψ′) \end{matrix} is the boundary of an (n + 1)-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix}

Poincaré pair \{ \begin{matrix}((f f′) : C ⊕ C′⟶D, (δϕ, ϕ ⊕ −ϕ′)) \\ ((f f′) : C ⊕ C′⟶D, (δψ, ψ ⊕ −ψ′)) . \end{matrix} Homotopy equivalent Poincaré

complexes are cobordant.

The \{ \begin{matrix}symmetric \\ quadratic \end{matrix} L-groups \{ \begin{matrix}L^{n}(A) \\ L_{n}(A) \end{matrix} (n ≥ 0) are the cobordism groups of n-

dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} Poincaré complexes over A. The quadratic L-groups L_{∗}(A)

are 4-periodic, with isomorphisms

L_{n}(A) ⟶ L_{n+4}(A) ; (C, ψ) ⟶ (S^{2}C, ψ) (n ≥ 0) ,

and are just the surgery obstruction groups of Wall [ ]. The symmetric L-groups L^{∗}(A) were introduced by Mishchenko [ ]. The corresponding maps in the symmetric L-groups

L^{n}(A) ⟶ L^{n+4}(A) ; (C, ϕ) ⟶ (S^{2}C, ϕ) (n ≥ 0)

are not isomorphisms in general. The symmetric and quadratic L-groups are related by an exact sequence

… ⟶ L_{n}(A) ⟶^{1+T} L^{n}(A) ⟶ L̂^{n}(A) ⟶ L_{n−1}(A) ⟶ …

with

1 + T : L_{n}(A) ⟶ L^{n}(A) ; (C, ψ) ⟶ (C, (1 + T)ψ)

and L̂^{n}(A) the relative cobordism group of n-dimensional (symmetric,quadratic) Poincaré pairs over A. The relative L-groups L̂^{∗}(A) are 8-torsion, so that the symmetrization maps 1 + T : L_{n}(A)⟶L^{n}(A) are isomorphisms modulo 8-torsion. If Ĥ^{∗}(ℤ_{2}; A) = 0 (e.g. if 1/2 ∈ A) then L̂^{∗}(A) = 0 and the symmetrization maps are isomorphisms.

§6. Chain bundles

A bundle over a finite-dimensional A-module chain complex C is a 0-dimensional hyperquadratic structure on C^{0−∗}, that is a cycle

γ ∈ (Ŵ^{%}C^{0−∗})_{0} ,

as defined by a collection of A-module morphisms

{γ_{s} ∈ Hom_{A}(C_{r−s}, C^{−r}) | r, s ∈ ℤ}

such that

(−)^{r+1}d_{C}^{∗}γ_{s} + (−)^{s}γ_{s}d_{C} + (−)^{s−1}(γ_{s−1} + (−)^{s}T γ_{s−1}) = 0 : C_{r−s+1} ⟶ C^{−r} .

An equivalence of bundles over C

χ : γ ⟶ γ′

is an equivalence of hyperquadratic structures, as defined by a collection of A-module morphisms

{χ_{s} ∈ Hom_{A}(C_{r−s−1}, C^{−r}) | r, s ∈ ℤ}

such that

γ′_{s} − γ_{s} = (−)^{r+1}d_{C}^{∗}χ_{s} + (−)^{s}χ_{s}d_{C} + (−)^{s}(χ_{s−1} + (−)^{s}T χ_{s−1}) : C_{r−s} ⟶ C^{−r} .

Thus

Q̂^{0}(C^{0−∗}) = H_{0}(Ŵ^{%}C^{0−∗})

is the abelian group of equivalence classes of bundles over C.

A chain bundle over A (C, γ) is a finite-dimensional A-module chain complex C together with a bundle γ ∈ (Ŵ^{%}C^{0−∗})_{0}.

Given a chain bundle (C, γ) over A and an A-module chain map f : B⟶C define the pullback chain bundle (B, f^{∗}γ) using the image of γ under the ℤ-module chain map

f̂^{∗} : Ŵ^{%}C^{0−∗} ⟶ Ŵ^{%}B^{0−∗}

induced by the dual A-module chain map f^{∗} : C^{0−∗}⟶B^{0−∗}. The equivalence class of the pullback bundle f^{∗}γ depends only on the chain homotopy class of the chain map f, by the chain homotopy invariance of the Q-groups.

A map of chain bundles over A

(f, χ) : (C, γ) ⟶ (C′, γ′)

is a chain map f : C⟶C′ together with an equivalence of bundles over C

χ : γ ⟶ f^{∗}γ′ .

The composite of chain bundle maps

(f, χ) : (C, γ) ⟶ (C′, γ′) , (f′, χ′) : (C′, γ′) ⟶ (C′′, γ′′)

is the chain bundle map

(f′, χ′)(f, χ) = (f′f, χ + f̂^{∗}^{%}(χ′)) : (C, γ) ⟶ (C′′, γ′′) .

A homotopy of chain bundle maps

(g, η) : (f, χ) ≃ (f′, χ′) : (C, γ) ⟶ (C′, γ′)

is a chain homotopy

g : f ≃ f′ : C ⟶ C′

together with an equivalence of 1-dimensional hyperquadratic structures on C^{0−∗}

η : χ − χ′ + (g^{∗}; f^{∗}, f^{′∗})^{%}(γ′) ⟶ 0 .

A map of chain bundles (f, χ) : (C, γ)⟶(C′, χ′) is an equivalence if there exists a homotopy inverse, which happens precisely when f : C⟶C′ is a chain equivalence, with any chain homotopy inverse

f′ = f^{−1} : C′ ⟶ C

such that there is defined an inverse equivalence

(f′, χ′) : (C′, γ′) ⟶ (C, γ) .

Given a chain bundle (B, β) over A and a finite-dimensional A-module chain complex C use the pullback construction to define abelian group morphisms

I_{β} : H_{n}(B^{t} ⊗_{A} C) ⟶ Q̂^{n}(C) ; f ⟶ f̂^{%}(S^{n}β) ,

using the slant isomorphism

B^{t} ⊗_{A} C \underrightarrow{≃} Hom_{A}(B^{∗}, C)

to identify a cycle f ∈ (B^{t} ⊗_{A} C)_{n} with a chain map f : B^{n−∗}⟶C. Weiss [ ] developed an algebraic analogue of the representation theorem of Brown [ ] to obtain for any ring with involution A the existence of a directed system {(B(r), β(r)) | r ≥ 0} of chain bundles over A and chain bundle maps

(B(r), β(r)) ⟶ (B(r + 1), β(r + 1))

such that the abelian group morphisms

\begin{matrix}lim \\ ⟶ \\ r \end{matrix} I_{β_{r}} : \begin{matrix}lim \\ ⟶ \\ r \end{matrix} H_{n}(B(r)^{t} ⊗_{A} C) ⟶ Q̂^{n}(C)

are isomorphisms for any finite-dimensional A-module chain complex C. In general, the direct limit A-module chain complex

B(∞) = \begin{matrix}lim \\ ⟶ \\ r \end{matrix} B(r)

is not finite-dimensional. As in Weiss [ ] we shall ignore this inconvenience and treat B(∞) as if it were finite-dimensional, so that there is defined the universal chain bundle over A

(B(∞), β(∞)) = \begin{matrix}lim \\ ⟶ \\ r \end{matrix} (B(r), β(r)) ,

with the universal property that for any finite-dimensional A-module chain complex C the abelian group morphisms

I_{β(∞)} : H_{n}(B(∞)^{t} ⊗_{A} C) ⟶ Q̂^{n}(C)

are isomorphisms. In particular, there are defined isomorphisms

H_{0}(Hom_{A}(C, B(∞))) \underrightarrow{≃} Q̂^{0}(C^{0−∗}) ; f ⟶ f^{∗}(β(∞))

for any finite-dimensional C. Thus every chain bundle (C, γ) has a classifying map

(f, χ) : (C, γ) ⟶ (B(∞), β(∞))

and the equivalence classes of bundles γ ∈ (Ŵ^{%}C^{0−∗})_{0} over C are in one-one correspondence with the chain homotopy classes of chain maps f : C⟶B(∞).

The Wu classes of a chain bundle (C, γ) are the Wu classes of γ, the A-module morphisms

v̂_{r}(γ) : H_{r}(C) ⟶ Ĥ^{r}(ℤ_{2}; A) ; x ⟶ ⟨γ_{−2r}, x ⊗ x⟩ .

The universal chain bundle (B(∞), β(∞)) is characterized by the property that the Wu classes define A-module isomorphisms

v̂_{r}(γ) : H_{r}(C) \underrightarrow{≃} Ĥ^{r}(ℤ_{2}; A) (r ≥ 0) .

For example, if A = ℤ the chain bundle (B(∞), β(∞)) defined by

d_{B(∞)} = \{ \begin{matrix}2 & if r is odd \\ 0 & if r is even \end{matrix} : B(∞)_{r} = ℤ ⟶ B(∞)_{r−1} = ℤ ,

β(∞)_{s} = \{ \begin{matrix}1 & if 2r = s \\ 0 & otherwise \end{matrix} : B(∞)_{r−s} = ℤ ⟶ B(∞)^{−r} = ℤ

is universal, with the Wu classes defining isomorphisms

v̂_{r}(β(∞)) : H_{r}(B(∞)) \underrightarrow{≃} Ĥ^{r}(ℤ_{2}; ℤ) = \{ \begin{matrix}ℤ_{2} & if r is even \\ 0 & if r is odd \end{matrix} .

§7. Normal complexes

An n-dimensional normal space (X, ν_{X}, ρ_{X}) (Quinn [ ]) is a finite n-dimensional CW complex X together with a (k − 1)-spherical fibration ν_{X} : X⟶BG(k) and a map ρ_{X} : S^{n+k}⟶T (ν_{X}) to the Thom space of ν_{X}. An n-dimensional geometric Poincaré complex X has a unique equivalence class of normal structures (ν_{X}, ρ_{X}), with ν_{X} the Spivak normal fibration and ρ_{X} representing the fundamental class [X] ∈ H_{n}(X). A normal complex is the algebraic analogue of a normal space, consisting of a symmetric complex with normal chain bundle.

A normal structure (γ, θ) on an n-dimensional symmetric complex (C, ϕ) is a bundle γ ∈ (Ŵ^{%}C^{0−∗})_{0} together with an equivalence of n-dimensional hyperquadratic structures on C

θ : J(ϕ) ⟶ (ϕ̂_{0})^{%}(S^{n}γ) ,

as defined by a chain θ ∈ (Ŵ^{%}C)_{n+1} such that

J(ϕ) − (ϕ̂_{0})^{%}(S^{n}γ) = dθ ∈ (Ŵ^{%}C)_{n} .

The Wu classes of ϕ and γ are then related by a commutative diagram

\begin{matrix}H^{n−r}(C)  w ϕ_{0} u v_{r}(ϕ) \\ H_{r}(C) uv̂_{r}(γ) \end{matrix} ; \begin{matrix}H^{n−2r}(ℤ_{2}; A, (−)^{n−r})  w J \\ Ĥ^{r}(ℤ_{2}; A) . \end{matrix}

An equivalence of n-dimensional normal structures on (C, ϕ)

(χ, η) : (γ, θ) ⟶ (γ′, θ′)

is an equivalence of bundles χ : γ⟶γ′ together with an equivalence of (n + 1)- dimensional hyperquadratic structures on C

η : θ − θ′ + (ϕ̂_{0})^{%}(S^{n}χ) ⟶ 0 .

An n-dimensional symmetric Poincaré complex (C, ϕ ∈ (W^{%}C)_{n}) has a unique equivalence class of normal structures (γ, θ), with the equivalence class of bundles [γ] ∈ Q̂^{0}(C^{0−∗}) the image of the equivalence class of symmetric structures [ϕ] ∈ Q^{n}(C) under the composite

Q^{n}(C) ⟶^{J} Q̂^{n}(C) ⟶^{((ϕ_{0})^{%})^{−1}} Q̂^{n}(C^{n−∗}) ⟶^{(S^{n})^{−1}} Q̂^{0}(C^{0−∗}).

If (γ, θ), (γ′, θ′) are two such normal structures on (C, ϕ) there exists an equivalence of bundles χ : γ⟶γ′. As ϕ_{0} : C^{n−∗}⟶C is a chain equivalence the cycle

θ − θ′ + (ϕ_{0})^{%}(S^{n}χ) ∈ (Ŵ^{%}C)_{n+1}

is such that there exist a cycle λ ∈ Ŵ^{%}C^{n−∗})_{n+1} and a chain μ ∈ (Ŵ^{%}C)_{n+2} such that

θ − θ′ + (ϕ_{0})^{%}(S^{n}χ) = (ϕ_{0})^{%}(λ) + dμ ∈ (Ŵ^{%}C)_{n+1} .

There is now defined an equivalence of normal structures on (C, ϕ)

(χ − (S^{n})^{−1}(λ), μ) : (γ, θ) ⟶ (γ′, θ′) .

An n-dimensional normal (Poincaré) complex over A (C, ϕ, γ, θ) is an n-dimensional symmetric (Poincaré) complex (C, ϕ) together with a normal structure (γ, θ). Symmetric Poincaré complexes are regarded as normal Poincaré complexes by choosing a normal structure in the unique equivalence class.

An n-dimensional normal complex (C, ϕ, γ, θ) is connected if the n-dimensional symmetric complex (C, ϕ) is connected, that is

H_{0}(ϕ_{0} : C^{n−∗}⟶C) = 0 .

The correspondence described in §5 between the homotopy equivalence classes of

connected n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} complexes and those of n-dimensional

\{ \begin{matrix}symmetric \\ quadratic \end{matrix} Poincaré pairs has the following generalization to connected normal

complexes and (symmetric, quadratic) Poincaré pairs.

A connected n-dimensional normal complex (C, ϕ, γ, θ) determines the n-dimensional (symmetric, quadratic) Poincaré pair

(i_{C}: ∂C⟶C^{n−∗}, (δϕ, ψ))

defined by

i_{C} = (0 1) : ∂C_{r} = C_{r+1} ⊕ C^{n−r} ⟶ C^{n−r} ,

d_{∂C} = ( \begin{matrix}d_{C} & (−)^{r}ϕ_{0} \\ 0 & (−)^{r}d_{C}^{∗} \end{matrix} ) :

∂C_{r} = C_{r+1} ⊕ C^{n−r} ⟶ ∂C_{r−1} = C_{r} ⊕ C^{n−r+1} ,

ψ_{0} = ( \begin{matrix}χ_{0} & 0 \\ 1 + γ_{−n}ϕ_{0}^{∗} & γ_{−n−1}^{∗} \end{matrix} ) :

∂C^{r} = C^{r+1} ⊕ C_{n−r} ⟶ ∂C_{n−r−1} = C_{n−r} ⊕ C^{r+1} ,

ψ_{s} = ( \begin{matrix}χ_{−s} & 0 \\ γ_{−n−s}ϕ_{0}^{∗} & γ_{−n−s−1}^{∗} \end{matrix} ) :

∂C^{r} = C^{r+1} ⊕ C_{n−r} ⟶ ∂C_{n−r−s−1} = C_{n−r−s} ⊕ C^{r+s+1} (s ≥ 1) ,

δϕ_{s} = γ_{−n−s} : C_{r} ⟶ C^{n−r+s} (s ≥ 0) .

The (n − 1)-dimensional quadratic Poincaré complex

∂(C, ϕ, γ, θ) = (∂C, ψ)

is the quadratic boundary of the connected n-dimensional normal complex (C, ϕ, γ, θ).

(Compare with the definition in §6 of the boundary (n − 1)-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix}

Poincaré complex

\{ \begin{matrix}∂(C, ϕ) \\ ∂(C, ψ) \end{matrix} of a connected n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} complex \{ \begin{matrix}(C, ϕ) \\ (C, ψ) \end{matrix}.) Conversely,

given an n-dimensional (symmetric, quadratic) Poincaré pair (f : C⟶D, (δϕ, ψ)) there is defined a connected n-dimensional normal complex (C(f), ϕ, γ, θ) with the symmetric structure

\begin{aligned}ϕ_{s} = & \{ \begin{matrix}( \begin{matrix}δϕ_{0} & 0 \\ (1 + T)ψ_{0}f^{∗} & 0 \end{matrix} ) & if s = 0 \\ ( \begin{matrix}δϕ_{1} & 0 \\ 0 & (1 + T)ψ_{0} \end{matrix} ) & if s = 1 \\ ( \begin{matrix}δϕ_{s} & 0 \\ 0 & 0 \end{matrix} ) & if s ≥ 2 \end{matrix} \\ & : C(f)^{r} = D^{r} ⊕ C^{r−1} ⟶ C(f)_{n−r+s} = D_{n−r+s} ⊕ C_{n−r+s−1} . \end{aligned}

The normal structure (γ, χ) is determined up to equivalence by the Poincaré duality, with γ ∈ Q̂^{0}(D^{−∗}) the image of (δϕ/(1 + T)ψ) ∈ Q^{n}(C(f)) under the composite

Q^{n}(C(f)) ⟶^{((δϕ_{0},(1+T)ψ_{0})^{%})^{−1}} Q^{n}(D^{n−∗}) ⟶^{J} Q̂^{n}(D^{n−∗}) ⟶^{S^{−n}} Q̂^{0}(D^{−∗}) .

The composite isomorphism

Q̂^{0}(C(f)^{0−∗}) ⟶^{S^{n}} Q̂^{n}(C(f)^{n−∗}) ⟶^{(δϕ,(1+T)ψ)^{%}} Q̂^{n}(D)

sends the equivalence class [γ] ∈ Q̂^{0}(C(f)^{0−∗}) to the element α ∈ Q̂^{n}(D) represented by

α_{s} = \{ \begin{matrix}δϕ_{s} & if s ≥ 0 \\ fψ_{−s−1}f^{∗} & if s ≤ −1 \end{matrix} : D^{r} ⟶ D_{n−r+s} .

There is thus a natural one-one correspondence between the homotopy equivalence classes of connected n-dimensional normal complexes over A and the homotopy equivalence classes of n-dimensional (symmetric, quadratic) Poincaré pairs over A. In §8 below this correspondence will be used to identify the cobordism group L̂^{n}(A) of ndimensional (symmetric, quadratic) Poincaré pairs over A with the cobordism group of n-dimensional normal complexes over A.

Let (B, β) be a chain bundle over A. A normal (B, β)-structure (γ, θ, f, χ) on an n-dimensional symmetric complex (C, ϕ) over A is a normal structure (γ, θ) on (C, ϕ) together with a chain bundle map

(f, χ) : (C, γ) ⟶ (B, β) .

There are also the corresponding relative notions of normal (B, β)-structure on symmetric and (symmetric, quadratic) pairs. For the universal chain bundle (B(∞), β(∞)) over A a normal (B(∞), β(∞))-structure (γ, θ, f, χ) on a symmetric complex (C, ϕ) is to all intents and purposes the same as a normal structure (γ, θ).

A normal (0,0)-structure (γ, θ, 0, χ) on an n-dimensional symmetric complex (C, ϕ) determines an equivalence to 0 of the hyperquadratic structure J(ϕ) ∈ (Ŵ^{%}C)_{n}

ξ = θ + ϕ_{0}^{%}(S^{n}χ) : J(ϕ) ⟶ 0 .

Such an equivalence ξ : J(ϕ)⟶0 consists of a quadratic structure ψ ∈ (W_{%}C)_{n} and an equivalence of symmetric structures

η : (1 + T)ψ ⟶ ϕ ,

with

\begin{matrix}ψ_{s} = ξ_{−s−1} ∈ Hom_{A}(C^{∗}, C)_{n−s} (s ≥ 0) , \\ η_{s} = ξ_{s} ∈ Hom_{A}(C^{∗}, C)_{n+s+1} (s ≥ 0) . \end{matrix}

Thus a normal (0,0)-structure on a symmetric complex (C, ϕ) is to all intents and purposes an equivalence of the symmetric structure ϕ to (1 + T)ψ for some quadratic structure ψ on C.

An n-dimensional (B, β)-normal (Poincaré) complex (C, ϕ, γ, θ, f, χ) is an n-dimen-sional symmetric (Poincaré) complex (C, ϕ) together with a normal (B, β)-structure (γ, θ, ϕ, χ).

In §8 below the cobordism group L̂⟨B, β⟩^{n}(A) of n-dimensional (B, β)-normal complexes over A will be identified with the twisted quadratic group Q_{n}(B, β) (introduced by Weiss [ ]) of equivalence classes of pairs (ϕ, θ) such that (B, ϕ, β, θ, 1, 0) is an ndimensional (B, β)-normal complex.

An n-dimensional symmetric structure (ϕ, θ) on a chain bundle (C, γ ∈ (Ŵ^{%}C^{0−∗})_{0}) is an n-dimensional symmetric structure ϕ ∈ (W^{%}C)_{n} together with an equivalence of n-dimensional hyperquadratic structures on C

θ : J(ϕ) ⟶ (ϕ_{0})^{%}(S^{n}γ) ,

as defined by a chain θ ∈ (Ŵ^{%}C)_{n+1} such that

J(ϕ) − (ϕ_{0})^{%}(S^{n}γ) = d(θ) ∈ (Ŵ^{%}C)_{n} .

Thus (C, ϕ) is an n-dimensional symmetric complex with normal structure (γ, θ).

An equivalence of n-dimensional symmetric structures on (C, γ)

(ξ, η) : (ϕ, θ) ⟶ (ϕ′, θ′)

is defined by an equivalence of symmetric structures ξ : ϕ⟶ϕ′ together with an equivalence of hyperquadratic structures on C

η : θ − θ′ + J(ξ) + (ξ_{0}; ϕ_{0}, ϕ′_{0})^{%}(S^{n}γ) ⟶ 0 ,

as defined by chains ξ ∈ (W^{%}C)_{n+1}, η ∈ (Ŵ^{%}C)_{n+2} such that

\begin{matrix}ϕ′ − ϕ = d(ξ) ∈ (W^{%}C)_{n} , \\ θ′ − θ + J(ξ) + (ξ_{0}; ϕ_{0}, ϕ′_{0})^{%}(S^{n}γ) = d(η) ∈ (Ŵ^{%}C)_{n+1} . \end{matrix}

The twisted quadratic Q-group Q_{n}(C, γ) is the abelian group of equivalence classes of n-dimensional symmetric structures on a chain bundle (C, γ), with addition by

(ϕ, θ) + (ϕ′, θ′) = (ϕ + ϕ′, θ + θ′ + [ϕ_{0}, ϕ′_{0}](S^{n}γ)) ∈ Q_{n}(C, γ) .

The twisted quadratic Q-groups Q_{∗}(C, γ) fit into an exact sequence of abelian groups

… ⟶ Q̂^{n+1}(C) \underrightarrow{H_{γ}} Q_{n}(C, γ) \underrightarrow{N_{γ}} Q^{n}(C) \underrightarrow{J_{γ}} Q̂^{n}(C) ⟶ …

with the morphisms

H_{γ} : Q̂^{n+1}(C) ⟶ Q_{n}(C, γ) ; θ ⟶ (0, θ) ,
N_{γ} : Q_{n}(C, γ) ⟶ Q^{n}(C) ; (ϕ, θ) ⟶ ϕ ,
J_{γ} : Q^{n}(C) ⟶ Q̂^{n}(C) ; ϕ ⟶ J(ϕ) − (ϕ_{0})^{%}(S^{n}γ)

induced by simplicial maps. In the untwisted case γ = 0 there is defined an isomorphism of exact sequences

…  wQ̂^{n+1}(C)  w H Q_{n}(C)  w 1 + T u≅  Q^{n}(C)  w J Q̂^{n}(C)  w…

…  wQ̂^{n+1}(C)  w H_{0} Q_{n}(C, 0)  w N_{0} Q^{n}(C)  w J_{0}  Q̂^{n}(C)  w…

with

\begin{matrix}Q_{n}(C) \underrightarrow{≃} Q_{n}(C, 0) ; ψ ⟶ ((1 + T)ψ, θ) , \\ θ_{s} = \{ \begin{matrix}ψ_{−s−1} & if s ≤ −1 \\ 0 & if s ≥ 0. \end{matrix} \end{matrix}

The twisted quadratic groups Q_{∗}(C, γ) are covariant in (C, γ). Given a map of chain bundles (f, χ) : (C, γ)⟶(C′, γ′) and an n-dimensional symmetric structure (ϕ, θ) on (C, γ) define an n-dimensional symmetric structure on (C′, γ′)

(f, χ)_{%}(ϕ, θ) = (f^{%}(ϕ), f̂^{%}(θ) + (fϕ_{0})^{%}(S^{n}χ)) .

The resulting morphisms of the twisted quadratic Q-groups

(f, χ)_{%} : Q_{n}(C, γ) ⟶ Q_{n}(C′, γ′)

depend only on the homotopy class of (f, χ). There is defined a morphism of exact sequences

…  wQ̂^{n+1}(C)  w H_{γ} u f̂^{%}  Q_{n}(C, γ)  w N_{γ} u (f, χ)_{%}  Q^{n}(C)  w J_{γ} u f^{%}  Q̂^{n}(C)  w u f̂^{%}  …

…  wQ̂^{n+1}(C′)  w H_{γ′} Q_{n}(C′, γ′)  w N_{γ′} Q^{n}(C′)  w J_{γ′}  Q̂^{n}(C′)  w…

which is an isomorphism if (f, χ) is an equivalence.

The characteristic element of an n-dimensional (B, β)-normal complex (C, ϕ, γ, θ, f, χ) is defined by

(f, χ)_{%}(ϕ, θ) ∈ Q_{n}(B, β) .

In §8 the cobordism class of a (B, β)-normal complex will be identified with the characteristic element.

A map of n-dimensional \{ \begin{matrix}normal \\ (B, β)−normal \end{matrix} complexes

\{ \begin{matrix}(f, ξ, χ, η) : (C, ϕ, γ, θ) ⟶ (C′, ϕ′, γ′, θ′) \\ (f, ξ, χ, η, h, μ) : (C, ϕ, γ, θ, g, λ) ⟶ (C′, ϕ′, γ′, θ′, g′, λ′) \end{matrix}

consists of

(i) a chain map f : C⟶C′, (ii) an equivalence ξ : f^{%}(ϕ)⟶ϕ′ of n-dimensional symmetric structures on C′, (iii) an equivalence χ : γ⟶f^{∗}γ′ of bundles on C, (iv) an equivalence of (n + 1)-dimensional hyperquadratic structures on C′

η : J(ξ) + θ′ − f̂^{%}(θ) + (ξ_{0}; f^{%}ϕ_{0}, ϕ′_{0})^{%}(S^{n}γ′) + (fϕ_{0})^{%}(S^{n}χ) ⟶ 0 , and in the (B, β)-normal case also

(v) a homotopy of bundle maps (h, μ) : (g, λ) ≃ (g′, λ′)(f, χ) : (C, γ) ⟶ (B, β) .

Note that (C, ϕ, γ, θ, g, λ) and (C′, ϕ′, γ′, θ′, g′, λ′) have the same characteristic element (g, λ)_{%}(ϕ, θ) = (g′, λ′)_{%}(ϕ′, θ′) ∈ Q_{n}(B, β) .

It is convenient for computational purposes to describe the behaviour of the twisted quadratic groups under direct sum. The direct sum of chain bundles (C, γ), (C′, γ′) is the chain bundle

(C, γ) ⊕ (C′, γ′) = (C ⊕ C′, γ ⊕ γ′) .

Let

i = ( \begin{matrix}1 \\ 0 \end{matrix} ) : C ⟶ C ⊕ C′ ,

i′ = ( \begin{matrix}0 \\ 1 \end{matrix} ) : C′ ⟶ C ⊕ C′ ,

j = ( 1 0 ) : C ⊕ C′ ⟶ C ,

j′ = ( 0 1 ) : C ⊕ C′ ⟶ C′ .

The twisted quadratic groups of the direct sum are such that there is defined a long exact sequence

\begin{matrix}… ⟶ Q_{n}(C, γ) ⊕ Q_{n}(C′, γ′) ⟶^{i_{∗}} Q_{n}(C ⊕ C′, γ ⊕ γ′) ⟶^{j_{∗}} H_{n}(C^{t} ⊗_{A} C′) \\ ⟶^{k_{∗}} Q_{n−1}(C, γ) ⊕ Q_{n−1}(C′, γ′) ⟶^{i_{∗}} Q_{n−1}(C ⊕ C′, γ ⊕ γ′) ⟶ … \end{matrix}

with

i_{∗} = ( i_{%} i′_{%} ) : Q_{n}(C, γ) ⊕ Q_{n}(C′, γ′) ⟶ Q_{n}(C ⊕ C′, γ ⊕ γ′) ,
j_{∗} : Q_{n}(C ⊕ C′, γ ⊕ γ′) ⟶ H_{n}(C^{t} ⊗_{A} C′) ; (ϕ, θ) ⟶ (j ⊗ j′)ϕ_{0} ,
k_{∗} : H_{n}(C^{t} ⊗_{A} C′) ⟶ Q_{n−1}(C, γ) ⊕ Q_{n−1}(C′, γ′) ;
(f : C^{n−∗}⟶C′) ⟶ ((0, f̂^{%}(S^{n}γ′)), (0, −f ∗^{%} (S^{n}γ)) .

For γ = 0 and γ′ = 0 the long exact sequence collapses into split exact sequences of the untwisted quadratic Q-groups

0 ⟶ Q_{n}(C) ⊕ Q_{n}(C′) ⟶ Q_{n}(C ⊕ C′) ⟶ H_{n}(C^{t} ⊗_{A} C′) ⟶ 0 .

The relative \{ \begin{matrix}symmetric \\ quadratic \\ hyperquadratic \end{matrix} Q-groups of an A-module chain map f : C⟶D

\{ \begin{matrix}Q^{n}(f) = H_{n}(f^{%} : W^{%}C⟶W^{%}D) \\ Q_{n}(f) = H_{n}(f_{%} : W_{%}C⟶W_{%}D) \\ Q̂^{n}(f) = H_{n}(f^{%} : Ŵ^{%}C⟶Ŵ^{%}D) \end{matrix}

fit into the exact sequences

\{ \begin{matrix}… ⟶ Q^{n}(C) ⟶^{f^{%}} Q^{n}(D) ⟶ Q^{n}(f) ⟶ Q^{n−1}(C) ⟶ … \\ … ⟶ Q_{n}(C) ⟶^{f_{%}} Q_{n}(D) ⟶ Q_{n}(f) ⟶ Q_{n−1}(C) ⟶ … \\ … ⟶ Q̂^{n}(C) ⟶^{f̂^{%}} Q̂^{n}(D) ⟶ Q̂^{n}(f) ⟶ Q̂^{n−1}(C) ⟶ …. \end{matrix}

The natural maps Q̂^{n}(f)⟶Q̂^{n}(C(f)) are isomorphisms. Let g : D⟶C(f) be the inclusion. The natural maps \{ \begin{matrix}Q^{n}(f)⟶Q^{n}(C(f)) \\ Q_{n}(f)⟶Q_{n}(C(f)) \end{matrix} fit into commutative braids of exact sequences

[ [ [ [ [ ’ ’ ’ ’ ’)  [ [ [ [ [ ’ ’ ’ ’ ’) f^{%}

H_{n}(C(f)^{t} ⊗_{A} C)  Q^{n−1}(C)  Q^{n−1}(D)

Q^{n}(f)^{64 4 4 4}  ^{h h hh j}  Q^{n}(g)^{64 4 4 4}  ^{h h hh j}

Q^{n}(D)^{h h hh j}  Q^{n}(C(f))^{64 4 4 4}  ^{h h hh j}  H_{n−1}(C(f)^{t} ⊗_{A} C)^{64 4 4 4}

N N N N N^{‚ ‚ ‚ ‚‚ ›} g^{%}  ^{N N N N N}^{‚ ‚ ‚ ‚‚ ›}
[ [ [ [ [ ’ ’ ’ ’ ’)  [ [ [ [ [ ’ ’ ’ ’ ’) f_{%}

H_{n}(C(f)^{t} ⊗_{A} C)  Q_{n−1}(C)  Q_{n−1}(D)

Q_{n}(f)^{64 4 4 4}  ^{h h hh j}  Q_{n}(g)^{64 4 4 4}  ^{h h hh j}

Q_{n}(D)^{h h hh j}  Q_{n}(C(f))^{64 4 4 4}  ^{h h hh j}  H_{n−1}(C(f)^{t} ⊗_{A} C)^{64 4 4 4}

N N N N N^{‚ ‚ ‚ ‚‚ ›} g_{%}  ^{N N N N N}^{‚ ‚ ‚ ‚‚ ›}

Given a chain bundle γ ∈ Q̂^{0}(C(f)^{−∗}) over C(f) let δ = g^{∗}γ ∈ Q̂^{0}(D^{0−∗}) be the pullback chain bundle over D, such that

f^{∗}δ = 0 ∈ Q̂^{0}(C^{0−∗}) .

The relative twisted quadratic groups Q_{∗}(f, γ) fit into a commutative diagram of exact sequences

⋮ u  ⋮ u  ⋮ u  ⋮ u

…  wQ_{n}(C)  w u  Q_{n}(D, δ)  w u  Q_{n}(f, γ)  w u  Q_{n−1}(C)  w u  …

…  wQ^{n}(C)  w u  Q^{n}(D)  w u  Q^{n}(f)  w u  Q^{n−1}(C)  w u  …

…  wQ̂^{n}(C)  w u  Q̂^{n}(D)  w u  Q̂^{n}(f)  w u  Q̂^{n−1}(C)  w u  …

…  w Q_{n−1}(C) w u Q_{n−1}(D, δ) w u Q_{n−1}(f, γ) w u Q_{n−2}(C) w u …

⋮  ⋮  ⋮  ⋮

The natural maps Q_{n}(f, γ)⟶Q_{n}(C(f), γ) fit into a commutative braid of exact sequences

[ [ [ [ [ ’ ’ ’ ’ ’)  [ [ [ [ [ ’ ’ ’ ’ ’)

H_{n}(C(f)^{t} ⊗_{A} C)  Q^{n}(f)  Q̂^{n}(f)

Q_{n}(f, γ)^{64 4 4 4}  ^{h h hh j}  Q^{n}(C(f))^{64 4 4 4}  ^{h h hh j}

Q̂^{n+1}(f)^{h h hh j}  Q_{n}(C(f), γ)^{64 4 4 4}  ^{h h hh j}  H_{n−1}(C(f)^{t} ⊗_{A} C)^{64 4 4 4}

N N N N N^{‚ ‚ ‚ ‚‚ ›}  ^{N N N N N}^{‚ ‚ ‚ ‚‚ ›}

§8. Normal cobordism

Given a k-plane vector bundle ν : X⟶BO(k) over a space X let Ω_{n}(X, ν) (n ≥ 0) denote the bordism groups of bundle maps

(f, b) : (M^{n}, ν_{M}) ⟶ (X, ν)

with M^{n} a smooth closed n-manifold and ν_{M} : M⟶BO(k) the normal bundle of an embedding M^{n} ⊂ S^{n+k}. The Thom space of ν_{M} is given by

T (ν_{M}) = E(ν_{M})/∂E(ν_{M})

with E(ν_{M}) the tubular neighbourhood of M^{n} in S^{n+k}, so that there is defined a collapse map

ρ_{M} : S^{n+k} ⟶ S^{n+k}/(S^{n+k}\E(ν_{M})) = E(ν_{M})/∂E(ν_{M}) = T (ν_{M}) .

The Pontrjagin-Thom isomorphism

\begin{matrix}Ω_{n}(X, ν) \underrightarrow{≃} π_{n+k}(T (ν)) ; \\ (f : M^{n}⟶X, b : ν_{M}⟶ν) ⟶ (T (b)(ρ_{M}) : S^{n+k} ⟶^{ρ_{M}} T (ν_{M}) ⟶^{T (b)} T (ν)) \end{matrix}

has inverse

\begin{matrix}π_{n+k}(T (ν)) \underrightarrow{≃} Ω_{n}(X, ν) ; \\ (ρ : S^{n+k}⟶T (ν)) ⟶ (f = ρ| : M^{n} = ρ^{−1}(X)⟶X, b : ν_{M}⟶ν) , \end{matrix}

using smooth transversality to choose a representative ρ transverse regular at the zero section X ⊂ T (ν).

Given a (k − 1)-spherical fibration ν : X⟶BG(k) over a space X let Ω_{n}^{N}(X, ν) (resp. Ω_{n}^{P} (X, ν)) denote the bordism group of fibration maps

(f, b) : (M^{n}, ν_{M}) ⟶ (X, ν)

with (M^{n}, ν_{M} : M⟶BG(k), ρ_{M} : S^{n+k}⟶T (ν_{M})) an n-dimensional normal space (resp. geometric Poincaré complex with Spivak normal structure). According to the theory of Quinn [ ] there is a geometric theory of transversality for normal spaces, so that by analogy with the Pontrjagin-Thom isomorphism for smooth bordism there is defined an isomorphism

\begin{matrix}Ω_{n}^{N}(X, ν) \underrightarrow{≃} π_{n+k}(T (ν)) ; \\ (f : M^{n}⟶X, b : ν_{M}⟶ν) ⟶ (T (b)(ρ_{M}) : S^{n+k} ⟶^{ρ_{M}} T (ν_{M}) ⟶^{T (b)} T (ν)) , \end{matrix}

with inverse

\begin{matrix}π_{n+k}(T (ν)) \underrightarrow{≃} Ω_{n}^{N}(X, ν) ; \\ (ρ : S^{n+k}⟶T (ν))⟶(f = ρ| : M^{n} = ρ^{−1}(X)⟶X, b : ν_{M}⟶ν) . \end{matrix}

The geometric Poincaré and normal bordism groups for n ≥ 5 are related by the Levitt-Jones-Quinn an exact sequence

… ⟶ L_{n}(ℤ[π_{1}(X)]) ⟶ Ω_{n}^{P} (X, ν) ⟶ Ω_{n}^{N}(X, ν) ⟶ L_{n−1}(ℤ[π_{1}(X)]) ⟶ ….

A map of n-dimensional normal spaces

(f, b, c) : (M^{n}, ν_{M}, ρ_{M})⟶(X^{n}, ν_{X}, ρ_{X})

is defined by a map of fibrations (f, b) : (M, ν_{M})⟶(X, ν_{X}) together with a homotopy

c : T (b)ρ_{M} ≃ ρ_{X} : S^{n+k} ⟶ T (ν_{X}) .

The mapping cylinder of f

M(f) = M × [0, 1] ∪ X/{(m, 1) = f(m)|m ∈ M}

defines a cobordism (M(f); M, X) of normal spaces, identifying M = M × {0} ⊂ M(f). If M^{n} and X^{n} are Poincaré complexes the corresponding element of the relative bordism group is just the surgery obstruction

(M(f); M ∪ −X) = σ_{∗}(f, b) ∈ Ω_{n+1}^{N,P}(X, ν_{X}) = L_{n}(ℤ[π_{1}(X)]) .

Ignoring questions of finite-dimensionality (or assuming that X is a finite n-dimensional CW complex) it is therefore possible to define the inverse isomorphism to Ω_{n}^{N}(X, ν)

\underrightarrow{≃}π_{n+k}(T (ν)) by

π_{n+k}(T (ν)) \underrightarrow{≃} Ω_{n}^{N}(X, ν) ; ρ ⟶ (X, ν, ρ) ,

without an appeal to the transversality of normal spaces. The group π_{n+k}(T (ν)) consists of the equivalence classes of normal structures (ν_{X} : X⟶BG(k), ρ_{X} : S^{n+k}⟶T (ν_{X})) on X with ν_{X} = ν.

Following Weiss [ ] we shall now identify the algebraic normal bordism groups L̂⟨B, β⟩^{n}(A) with the twisted quadratic groups Q_{n}(B, β), the algebraic analogues of the homotopy groups of the Thom space π_{n+k}(T (ν)).

A cobordism of n-dimensional normal complexes (C, ϕ, γ, θ), (C′, ϕ′, γ′, θ′) is defined by an (n + 1)-dimensional symmetric pair

((f f′) : C ⊕ C′⟶D, (δϕ, ϕ ⊕ −ϕ′))

together with bundle maps

\begin{matrix}(f, ζ) : (C, γ) ⟶ (D, δγ) , \\ (f′, ζ′) : (C′, γ′) ⟶ (D, δγ) \end{matrix}

and an equivalence of hyperquadratic structures on D

δθ : J(δϕ) − (δϕ_{0}; fϕ_{0}f^{∗}, f′ϕ′_{0}f^{′∗})^{%}(S^{n}δγ) + f; (ϕ′_{0})^{%}(S^{n}ζ′)) ⟶ 0 . Similarly for the cobordism of (B, β)-normal complexes.

The symmetric (B, β)-structure L-groups of A L⟨B, β⟩^{n}(A) (n ≥ 0) of Weiss [ ] are the cobordism groups of n-dimensional (B, β)-normal Poincaré complexes over A

(C, ϕ, γ, θ, f, χ). For the \{ \begin{matrix}universal \\ zero \end{matrix} chain bundle \{ \begin{matrix}(B(∞), β(∞)) \\ (0, 0) \end{matrix} over A these are

just the \{ \begin{matrix}symmetric \\ quadratic \end{matrix} L-groups

\{ \begin{matrix}L⟨B(∞), β(∞)⟩^{n}(A) = L^{n}(A) \\ L⟨0, 0⟩^{n}(A) = L_{n}(A) . \end{matrix}

The symmetric (B, β)-structure L̂-groups L̂⟨B, β⟩^{n}(A) (n ≥ 0) are the cobordism

groups of n-dimensional (B, β)-normal complexes over A. For the \{ \begin{matrix}universal \\ zero \end{matrix} chain

bundle \{ \begin{matrix}(B(∞), β(∞)) \\ (0, 0) \end{matrix} over A these are just the

\{ \begin{matrix}L̂⟨B(∞), β(∞)⟩^{n}(A) = L̂^{n}(A) \\ L̂⟨0, 0⟩^{n}(A) = 0 . \end{matrix}

Algebraic surgery was used in Ranicki [ ] to prove that every n-dimensional quadratic Poincaré complex (C, ψ) is cobordant to a highly-connected complex (C′, ψ′), with

H_{r}(C′) = 0 (2r ≤ n − 2) .

The boundary of an n-dimensional normal complex (C, ϕ, γ, θ) is an (n−1)-dimensional quadratic Poincaré complex (∂C, ψ). Glueing on to (C, ϕ, γ, θ) the trace of the surgery making (∂C, ψ) highly-connected there is obtained an n-dimensional normal complex (C′, ϕ′, γ′, θ′) which is cobordant to (C, ϕ, γ, θ) and which has a highly-connected boundary, with

H_{r}(∂C′) = H_{r+1}(ϕ′_{0} : C^{′n−∗}⟶C′) = 0 (2r ≤ n − 3) .

In particular, this shows that every normal complex is cobordant to a connected complex. Thus L̂⟨B, β⟩^{n}(A) is also the cobordism group of connected n-dimensional (B, β)-normal complexes over A. The one-one correspondence established in §7 between connected n-dimensional normal complexes and n-dimensional (symmetric, quadratic) Poincaré pairs generalizes to a one-one correspondence between connected n-dimensional (B, β)-normal complexes over A and n-dimensional (symmetric, quadratic) (B, β)-normal Poincaré pairs over A, for any chain bundle (B, β) over A. It follows that L̂⟨B, β⟩^{n}(A) can be identified with the cobordism group of n-dimensional (symmetric, quadratic) (B, β)-normal Poincaré pairs, and that there is defined an exact sequence

… ⟶ L_{n}(A) ⟶ L⟨B, β⟩^{n}(A) ⟶ L̂⟨B, β⟩^{n}(A) ⟶^{∂} L_{n−1}(A) ⟶ …,

with ∂ defined by the quadratic boundary

∂ : L̂⟨B, β⟩^{n}(A) ⟶ L_{n−1}(A) ; (C, ϕ, γ, θ, f, χ) ⟶ ∂(C, ϕ, γ, θ) .

A map of n-dimensional normal complexes

(f, ξ, χ, η) : (C, ϕ, γ, θ) ⟶ (C′, ϕ′, γ′, θ′)

determines an (n+1)-dimensional symmetric pair ((f 1) : C ⊕ C′⟶C′, (ξ, ϕ ⊕ −ϕ′)), bundle maps

(f, χ) : (C, γ) ⟶ (C′, γ′) , (1, 0) : (C′, γ′) ⟶ (C′, γ′)

and an equivalence of hyperquadratic structures on C′

η : J(ξ) − (ξ_{0}; fϕ_{0}f^{∗}, ϕ′_{0})^{%}(S^{n}γ′) + f^{%}(θ − (ϕ_{0})^{%}(S^{n}γ)) − θ′ ⟶ 0 ,

defining a cobordism between (C, ϕ, γ, θ) and (C′, ϕ′, γ′, θ′) by analogy with the mapping cylinder construction of geometric normal bordisms. Similarly for maps of (B, β)- normal complexes. It follows that the abelian group morphisms

\begin{matrix}L̂⟨B, β⟩^{n}(A) ⟶ Q_{n}(B, β) ; (C, ϕ, γ, θ, f, χ) ⟶ (f, χ)_{%}(ϕ, θ) , \\ Q_{n}(B, β) ⟶ L̂⟨B, β⟩^{n}(A) ; (ϕ, θ) ⟶ (B, ϕ, β, θ, 1, 0) \end{matrix}

are inverse isomorphisms.

For example, if A = ℤ and (B(∞), β(∞)) is the universal chain bundle over ℤ (as constructed at the end of §6) then

L⟨B(∞), β(∞)⟩^{n}(ℤ) = L^{n}(ℤ)

= \{ \begin{matrix}ℤ & if n ≡ 0(mod 4) \\ ℤ_{2} & if n ≡ 1(mod 4) \\ 0 & if n ≡ 2, 3(mod 4) , \end{matrix}

L_{n}(ℤ) = \{ \begin{matrix}ℤ & if n ≡ 0(mod 4) \\ ℤ_{2} & if n ≡ 2(mod 4) \\ 0 & if n ≡ 1, 3(mod 4) , \end{matrix}

L̂⟨B(∞), β(∞)⟩^{n}(ℤ) = Q_{n}(B(∞), β(∞))

= \{ \begin{matrix}ℤ_{8} & if n ≡ 0(mod 4) \\ ℤ_{2} & if n ≡ 1, 3(mod 4) \\ 0 & if n ≡ 2(mod 4) . \end{matrix}

§9. Normal Wu classes

The Wu classes of the symmetric structure ϕ and the bundles β, γ in an n-dimensional (B, β)-normal complex (C, ϕ, γ, θ, f, χ) are related by a commutative diagram

\begin{matrix}H^{n−r}(C)  w ϕ_{0} u v_{r}(ϕ) \\ H_{r}(C) uv̂_{r}(γ)  w f_{∗} \\ H_{r}(B) ` ` ` ` `˜v̂_{r}(β) \end{matrix} ; \begin{matrix}H^{n−2r}(ℤ_{2}; A, (−)^{n−r})  w J \\ Ĥ^{r}(ℤ_{2}; A) . \end{matrix}

For any chain bundle (B, β) and any chain complex C we shall now define symmetric (B, β)-structure groups Q⟨B, β⟩^{n}(C) (n ≥ 0) to fit into an exact sequence

\begin{aligned}… ⟶ Q⟨B, β⟩^{n}(C) & ⟶ Q^{n}(C) ⊕ H_{n}(B^{t} ⊗_{A} C) ⟶ Q̂^{n}(C) \\ & ⟶ Q⟨B, β⟩^{n−1}(C) ⟶ …. \end{aligned}

The Wu classes v_{r}(ϕ) of a symmetric complex (C, ϕ) will then be refined to the normal Wu classes of a (B, β)-normal complex (C, ϕ, γ, θ, f, χ)

v_{r}(ϕ, γ, θ, f, χ) : H^{n−r}(C) ⟶ Q⟨B, β⟩^{n}(S^{n−r}A) ,

with

\begin{matrix}v_{r}(ϕ) : H^{n−r}(C) ⟶^{v_{r}(ϕ,γ,θ,f,χ)} Q⟨B, β⟩^{n}(S^{n−r}A) \\ ⟶ Q^{n}(S^{n−r}A) = H^{n−2r}(ℤ_{2}; A, (−)^{n−r}) . \end{matrix}

In §11 below the normal Wu classes will be used to define a ℤ_{4}-valued quadratic function on H^{n}(C) for a 2n-dimensional symmetric Poincaré complex (C, ϕ) over ℤ_{2} with normal (v_{n+1} = 0)-structure, as required to define the ℤ_{8}-valued invariant of Brown [ ].

Let (B, β) be a chain bundle over A, and let C be a finite-dimensional A-module chain complex. An n-dimensional symmetric (B, β)-structure on C (ϕ, θ, f) is defined by an n-dimensional symmetric structure ϕ ∈ (W^{%}C)_{n} together with a chain θ ∈ (Ŵ^{%}C)_{n+1} and a chain map f : B^{n−∗}⟶C such that

J(ϕ) − f̂^{%}(S^{n}β) = dθ ∈ (Ŵ^{%}C)_{n} .

An n-dimensional (B, β)-normal structure (ϕ, γ, θ, g, χ) on C determines the n-dimen-sional symmetric (B, β)-structure (ϕ, θ + (ϕ_{0})^{%}(S^{n}χ), ϕ_{0}g^{∗}) on C. Conversely, if f^{∗} : C^{n−∗}⟶B is a composite

f^{∗} : C^{n−∗} ⟶^{ϕ_{0}} C ⟶^{g} B

(as is always the case up to chain homotopy if (C, ϕ) is a Poincaré complex) the symmetric (B, β)-structure (ϕ, θ, f) determines the n-dimensional (B, β)-normal structure (ϕ, g^{∗}γ, θ, g, 0).

An n-dimensional symmetric (B, β)-structure (Poincaré) complex over A (C, ϕ, θ, f) is an n-dimensional A-module chain complex C together with an n-dimensional symmetric (B, β)-structure (ϕ, θ, f) (such that ϕ_{0} : C^{n−∗}⟶C is a chain equivalence). As for symmetric (Poincaré) pairs there is also the analogous notion of symmetric (B, β)-structure (Poincaré) pair. There is essentially no difference between symmetric (B, β)-structure Poincaré complexes and (B, β)-normal Poincaré complexes, so that the L-groups L⟨B, β⟩^{n}(A) (n ≥ 0) can also be regarded as the cobordism groups of n-dimensional symmetric (B, β)-structure Poincaré complexes over A.

An equivalence of n-dimensional symmetric (B, β)-structures on C

(ξ, η, g) : (ϕ, θ, f) ⟶ (ϕ′, θ′, f′)

is defined by an equivalence of symmetric structures ξ : ϕ⟶ϕ′ together with a chain η ∈ (Ŵ^{%}C)_{n+2} and a chain homotopy g : f ≃ f′ : B^{n−∗}⟶C such that

J(ξ) − (g; f, f′)^{%}(S^{n}β) − θ′ + θ = dη ∈ (Ŵ^{%}C)_{n+1} .

The n-dimensional symmetric (B, β)-structure group of C Q⟨B, β⟩^{n}(C) is the abelian group of equivalence classes of n-dimensional symmetric (B, β)-symmetric structures on C, with addition by

(ϕ, θ, f) + (ϕ′, θ′, f′) = (ϕ + ϕ′, θ + θ′ + [f, f′](S^{n}β), f + f′) ∈ Q⟨B, β⟩^{n}(C) .

There is also a more economical description of Q⟨B, β⟩^{n}(C) as the abelian group of equivalence classes of pairs (ψ, f) defined by an n-dimensional quadratic structure ψ ∈ (W_{%}C)_{n} and a chain map f : B^{n−∗}⟶C such that

f_{%}H(S^{n}β) = dψ ∈ (W_{%}C)_{n−1} ,

(so that

fβ_{−n−s−1}f^{∗} = dψ_{s} + ψ_{s}d^{∗} + ψ_{s+1} + ψ_{s+1}^{∗} ∈ Hom_{A}(C^{∗}, C)_{n−s} (s ≥ 0)) subject to the equivalence relation

(ψ, f) ∼ (ψ′, f′) if there exist a chain homotopy g : f ≃ f′ : B^{n−∗}⟶C

and an equivalence of quadratic structures χ : ψ′ − ψ ⟶ (g; f, f′)_{%}H(S^{n}β),

with addition by

(ψ, f) + (ψ′, f′) = (ψ + ψ′ + H([f, f′](S^{n}β)), f + f′) .

The pair (ψ, f) determines the triple (ϕ, θ, f) with

ϕ_{s} = \{ \begin{matrix}fβ_{s−n}f^{∗} & if s ≥ 1 \\ fβ_{−n}f^{∗} + (1 + T)ψ_{0} & if s = 0, \end{matrix}

θ_{s} = \{ \begin{matrix}0 & if s ≥ 0 \\ ψ_{−s−1} & if s ≤ −1. \end{matrix}

Conversely, a triple (ϕ, θ, f) determines the pair (ψ, f) with ψ_{s} = θ_{−s−1} (s ≥ 0) .

Given an n-dimensional symmetric (B, β)-structure (ϕ, θ, f) on C, a chain bundle map (g, χ) : (B, β)⟶(B′, β′) and a chain map h : C⟶C′ define the pushforward n-dimensional symmetric (B′, β′)-structure on C’

⟨g, χ⟩(h)^{%}(ϕ, θ, f) = (h^{%}(ϕ), h^{%}(θ + S^{n}(f̂^{%}χ), hfg^{∗}) .

Thus the groups Q⟨B, β⟩^{∗}(C) are covariant in both (B, β) and C, with pushforward abelian group morphisms

\begin{aligned}⟨g, χ⟩(h)^{%} : & Q⟨B, β⟩^{n}(C) ⟶ Q⟨B′, β′⟩^{n}(C′) ; \\ & (ϕ, θ, f) ⟶ (h^{%}(ϕ), ĥ^{%}(θ + f̂^{%}(S^{n}χ)), hfg^{∗}) \end{aligned}

depending only on the homotopy classes of (g, χ) and h.

An n-dimensional symmetric (B, β)-structure (ϕ, θ, f) on C determines an n-dimen-sional symmetric structure ϕ ∈ (W^{%}C)_{n} on C, so that there is defined a forgetful map

s : Q⟨B, β⟩^{n}(C) ⟶ Q^{n}(C) ; (ϕ, θ, f) ⟶ ϕ .

An n-dimensional quadratic structure ψ ∈ (W_{%}C)_{n} on C determines an n-dimensional symmetric (B, β)-structure ((1 + T)ψ, θ, 0) on C for any (B, β), with

θ_{s} = \{ \begin{matrix}ψ_{−s−1} & if s ≤ −1 \\ 0 & if s ≥ 0 . \end{matrix}

Thus there are also defined forgetful maps

s : Q_{n}(C) ⟶ Q⟨B, β⟩^{n}(C) ; ψ ⟶ ((1 + T)ψ, θ, 0) ,

and sr = 1 + T : Q_{n}(C)⟶Q^{n}(C).

Let P ⟨B, β⟩^{n}(C) be the abelian group of equivalence classes of n-dimensional symmetric (B, β)-structures (ϕ, θ, f) with ϕ = 0, to be denoted (θ, f), subject to the equivalence relation

(θ, f) ∼ (θ′, f′) if there exists an equivalence of (B, β)-structures (0, η, g) : (0, θ, f)⟶(0, θ′, f′).

The symmetric (B, β)-structure groups Q⟨B, β⟩^{∗}(C) and the groups P ⟨B, β⟩^{∗}(C) are related by a commutative braid of exact sequences of abelian groups

N N N N N N ȷ ȷ ȷ ȷ ȷȷ fi  N N N N N N ȷ ȷ ȷ ȷ ȷȷ fi

Q_{n}(C)  Q^{n}(C)  P ⟨B, β⟩^{n−1}(B, β)

Q⟨B, β⟩^{n}(C)^{fi ȷ ȷ ȷ ȷ ȷ}  ^{N N N NN P}  Q̂^{n}(C)^{fi ȷ ȷ ȷ ȷ ȷ}  ^{N N N NN P}

P ⟨B, β⟩^{n}(C)^{N N N NN P}  H_{n}(B^{t} ⊗_{A} C)^{fi ȷ ȷ ȷ ȷ ȷ}  ^{N N N NN P}  Q_{n−1}(C)^{fi ȷ ȷ ȷ ȷ ȷ}

’’ ’ ’ ’ ’ ^{[ [ [ [ [[]}  ^{’’ ’ ’ ’ ’} ^{[ [ [ [ [[]}

If (B, β) is the \{ \begin{matrix}universal \\ zero \end{matrix} chain bundle \{ \begin{matrix}(B(∞), β(∞)) \\ (0, 0) \end{matrix} the forgetful map

\{ \begin{matrix}Q⟨B(∞), β(∞)⟩^{n}(C) ⟶ Q^{n}(C) ; (ϕ, θ, f) ⟶ ϕ \\ Q_{n}(C) ⟶ Q⟨0, 0⟩^{n}(C) ; ψ ⟶ ((1 + T)ψ, θ) , \end{matrix}

is an isomorphism and

\{ \begin{matrix}P ⟨B(∞), β(∞)⟩^{n}(C) = 0 \\ P ⟨0, 0⟩^{n}(C) = Q̂^{n+1}(C) . \end{matrix}

The Wu classes of an n-dimensional symmetric (B, β)-structure (ϕ, θ, f) on C are the A-morphisms

\begin{aligned}v_{r}(ϕ, θ, f) : & H^{n−r}(C) ⟶ Q⟨B, β⟩^{n}(S^{n−r}A) ; \\ & (x : C⟶S^{n−r}A) ⟶ ⟨1, 0⟩(x)^{%}(ϕ, θ, f) . \end{aligned}

Now

\begin{matrix}Q⟨B, β⟩^{n}(S^{n−r}A) \\ = \{ \begin{matrix}H_{r}(B) & if 2r < n \\ \{(a, b) ∈ A ⊕ B_{r}|db = 0 ∈ B_{r−1}\}/∼ & if 2r = n \\ \{(a, b) ∈ A ⊕ B_{r}|a + (−)^{r}\overline{a} + β_{−2r}(b)(b) = 0 ∈ A, db = 0 ∈ B_{r−1}\}/ ∼ & if 2r > n \end{matrix} \end{matrix}

with the equivalence relation ∼ defined by

(a, b) ∼ (a′, b′) if there exists (x, y) ∈ A ⊕ B_{r+1} such that
a′ − a = x + (−)^{r+1}\overline{x} + β_{−2r−2}(y)(y)
+β_{−2r−1}(y)(b) + β_{−2r−1}(b′)(y) ∈ A,
b′ − b = dy ∈ B_{r}

and addition by

(a, b) + (a′, b′) = (a + a′ + β_{−2r}(b)(b′), b + b′) .

The map to the symmetric Q-group is given by

\begin{matrix}Q⟨B, β⟩^{n}(S^{n−r}A) ⟶ Q^{n}(S^{n−r}A) = H^{n−2r}(ℤ_{2}; A, (−)^{n−r}) ; \\ \{ & \begin{matrix}b ⟶ β_{−2r}(b)(b) & if 2r < n , \\ (a, b) ⟶ a + (−)^{r}\overline{a} + β_{−2r}(b)(b) & if 2r = n \\ 0 & if 2r > n . \end{matrix} \end{matrix}

The Wu classes are given by

\begin{aligned}v_{r}(ϕ, θ, f) : & H^{n−r}(C) ⟶ Q⟨B, β⟩^{n}(S^{n−r}A) ; \\ & z ⟶ \{ \begin{matrix}f^{∗}(z) & if 2r < n \\ (θ_{n−2r−1}(z)(z), f^{∗}(z)) & if 2r ≥ n \end{matrix} (z ∈ C^{n−r}, d^{∗}z = 0) . \end{aligned}

§10. Forms

In Ranicki [ ] the even-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} L-groups \{ \begin{matrix}L^{2n}(A) \\ L_{2n}(A) \end{matrix} (n ≥ 0) were

related to the Witt groups \{ \begin{matrix}W^{(−)^{n}}(A) \\ W_{(−)^{n}}(A) \end{matrix} (n ≥ 0) of nonsingular \{ \begin{matrix}(−)^{n}-symmetric \\ (−)^{n}-quadratic \end{matrix}

forms over A. In particular, it was shown that

\{ \begin{matrix}L^{0}(A) = W^{+1}(A) \\ L_{2n}(A) = W_{(−)^{n}}(A) \end{matrix} (n ≥ 0) .

This relationship between L-groups and Witt groups will now be generalized to the even-dimensional symmetric (B, β)-structure L-groups L⟨B, β⟩^{2n}(A) and the Witt groups W_{Q(n)}(A) of nonsingular Q(n)-quadratic forms over A, with (B, β) any chain bundle over A and

Q(n) = Q⟨B, β⟩^{2n}(S^{n}A) .

Let ϵ = ±1. An ϵ-symmetric form over A (M, λ) is a f.g. projective A-module M together with an element λ ∈ Hom_{A}(M, M^{∗}) such that

ϵλ^{∗} = λ : M ⟶ M^{∗} .

Equivalently, the form is defined by a pairing

λ : M × M ⟶ A ; (x, y) ⟶ λ(x, y) = λ(x)(y)

such that

λ(ax, by) = bλ(x, y)\overline{a} ,
λ(x + x′, y) = λ(x, y) + λ(x′, y) ,
ϵ\overline{λ(y, x)} = λ(x, y) (x, y ∈ M, a, b ∈ A) .

Let Q(ϵ) be an A-group together with A-morphisms

\begin{matrix}r : Q(ϵ) ⟶ H^{0}(ℤ_{2}; A, ϵ) = \{a ∈ A | ϵ\overline{a} = a\} , \\ s : H_{0}(ℤ_{2}; A, ϵ) = A/\{b − ϵ\overline{b} | b ∈ A\} ⟶ Q(ϵ) \end{matrix}

such that

rs = 1 + T_{ϵ} : H_{0}(ℤ_{2}; A, ϵ) ⟶ H^{0}(ℤ_{2}; A, ϵ) .

A Q(ϵ)-quadratic form over A (M, λ, μ) is an ϵ-symmetric form (M, λ) together with an A-morphism μ : M⟶Q(ϵ) such that

\begin{matrix}r(μ(x)) = λ(x, x) ∈ H^{0}(ℤ_{2}; A, ϵ) , \\ μ(x + y) − μ(x) − μ(y) = s(λ(x, y)) ∈ Q(ϵ) (x, y ∈ M) . \end{matrix}

There is an evident notion of isomorphism of Q(ϵ)-quadratic forms.

A Q(ϵ)-quadratic form (M, λ, μ) is nonsingular if λ ∈ Hom_{A}(M, M^{∗}) is an isomorphism of A-modules.

A nonsingular Q(ϵ)-quadratic form (M, λ, μ) is hyperbolic if there exists a direct summand L ⊂ M such that

(i) the inclusion j ∈ Hom_{A}(L, M) fits into an exact sequence

0 ⟶ L ⟶^{j} M ⟶^{j^{∗}λ} L^{∗} ⟶ 0 ,

(ii) μj = 0 : L ⟶ Q(ϵ) .

The Q(ϵ)-quadratic Witt group of A W_{Q(ϵ)}(A) is the abelian group of equivalence classes of nonsingular Q(ϵ)-quadratic forms (M, λ, μ), subject to the equivalence relation

(M, λ, μ) ∼ (M′, λ′, μ′) if there exists an isomorphism

(M, λ, μ) ⊕ (N, ν, ρ) \underrightarrow{≃} (M′, λ′, μ′) ⊕ (N′, ν′, ρ′) for some hyperbolic Q(ϵ)-quadratic forms (N, ν, ρ), (N′, ν′, ρ′).

For Q(ϵ) = H^{0}(ℤ_{2}; A, ϵ), r = 1, s = 1 + T_{ϵ} a Q(ϵ)-quadratic form (M, λ, μ) may be identified with the ϵ-symmetric form (M, λ), since λ determines μ by

μ(x) = λ(x, x) ∈ H^{0}(ℤ_{2}; A, ϵ) (x ∈ M) .

The Witt group of ϵ-symmetric forms W_{Q(ϵ)}(A) is denoted by W^{ϵ}(A).

For Q(ϵ) = H_{0}(ℤ_{2}; A, ϵ), r = 1 + T_{ϵ}, s = 1 a Q(ϵ)-quadratic form (M, λ, μ) is just a ϵ-quadratic form in the sense of Wall [ ]. The Witt group of ϵ-quadratic forms W_{Q(ϵ)}(A) is denoted by W_{ϵ}(A).

For Q(ϵ) = im(1 + T_{ϵ} : H_{0}(ℤ_{2}; A, ϵ)⟶H^{0}(ℤ_{2}; A, ϵ)), r = projection, s = injection a Q(ϵ)-quadratic form (M, λ, μ) is just an ϵ-symmetric form (M, λ) for which there exists an ϵ-quadratic form (M, λ, μ : M⟶H_{0}(ℤ_{2}; A, ϵ)). Such an ϵ-symmetric form is even. The Witt group of even ϵ-symmetric forms W_{Q(ϵ)}(A) is denoted by W ⟨v_{0}⟩^{ϵ}(A).

For ϵ = +1 \{ \begin{matrix}ϵ-symmetric \\ ϵ-quadratic \end{matrix} is abbreviated to \{ \begin{matrix}symmetric \\ quadratic. \end{matrix}

A 2n-dimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} (Poincaré) complex over A \{ \begin{matrix}(C, ϕ) \\ (C, ψ) \end{matrix} with H^{n}(C) a

f.g. projective A-module determines a (nonsingular) \{ \begin{matrix}(−)^{n}-symmetric \\ (−)^{n}-quadratic \end{matrix} form over A

\{ \begin{matrix}(H^{n}(C), ϕ_{0}, v_{n}(ϕ)) \\ (H^{n}(C), (1 + T)ψ_{0}, v^{n}(ψ)) \end{matrix}, with

\{ \begin{matrix}v_{n}(ϕ) : H^{n}(C) ⟶ H^{0}(ℤ_{2}; A, (−)^{n}) ; x ⟶ ϕ_{0}(x)(x) \\ v^{n}(ψ) : H^{n}(C) ⟶ H_{0}(ℤ_{2}; A, (−)^{n}) ; x ⟶ ψ_{0}(x)(x) . \end{matrix}

Conversely, a (nonsingular) \{ \begin{matrix}(−)^{n}-symmetric \\ (−)^{n}-quadratic \end{matrix} form \{ \begin{matrix}(M, λ) \\ (M, λ, μ) \end{matrix} determines a 2ndimensional \{ \begin{matrix}symmetric \\ quadratic \end{matrix} (Poincaré) complex \{ \begin{matrix}(C, ϕ) \\ (C, ψ) \end{matrix} such that

\{ \begin{matrix}ϕ_{0} \\ (1 + T)ψ_{0} \end{matrix} = λ : C^{n} = M ⟶ C_{n} = M^{∗} ,

C_{r} = 0 (r ≠ n) ,

v^{n}(ψ) = μ : H^{n}(C) = M ⟶ H_{0}(ℤ_{2}; A, (−)^{n}) .

The corresponding morphisms from the Witt groups to the L-groups

\{ \begin{matrix}W^{(−)^{n}}(A) ⟶ L^{2n}(A) ; (M, λ) ⟶ (C, ϕ) \\ W_{(−)^{n}}(A) ⟶ L_{2n}(A) ; (M, λ, μ) ⟶ (C, ψ) \end{matrix}

were shown in Ranicki [ ] to be isomorphisms for n = 0 if A is any ring, and for all n ≥ 0 if A is \{ \begin{matrix}a Dedekind \\ any \end{matrix} ring. For A a field the inverse isomorphisms are defined by

\{ \begin{matrix}L^{2n}(A) ⟶ W^{(−)^{n}}(A) ; (C, ϕ) ⟶ (H^{n}(C), ϕ_{0}) \\ L_{2n}(A) ⟶ W_{(−)^{n}}(A) ; (C, ψ) ⟶ (H^{n}(C), (1 + T)ψ_{0}, v^{n}(ψ)) \end{matrix} (n ≥ 0) .

Given A-groups M, N and a symmetric bilinear pairing

ϕ : N × N ⟶ M

such that

ϕ(ay, ay′) = aϕ(y, y′) ∈ M (a ∈ A, y, y′ ∈ N)

let M ×_{ϕ} N be the A-group of pairs (x ∈ M, y ∈ N), with addition by

(x, y) + (x′, y′) = (x + x′ + ϕ(y, y′), y + y′) ∈ M ×_{ϕ} N

and A acting by

A × (M ×_{ϕ} N) ⟶ M ×_{ϕ} N ; (a, (x, y)) ⟶ (ax, ay) .

There is then defined a short exact sequence of A-groups and A-morphisms 0 ⟶ M ⟶ M ×_{ϕ} N ⟶ N ⟶ 0 with

\begin{matrix}M ⟶ M ×_{ϕ} N ; x ⟶ (x, 0) \\ M ×_{ϕ} N ⟶ N ; (x, y) ⟶ y . \end{matrix}

Given a chain bundle (B, β) over A define the A-group

\begin{aligned}Q(n) & = Q⟨B, β⟩^{2n}(S^{n}A) \\ & = \{(a, b) ∈ A ⊕ B_{n} | db = 0 ∈ B_{n−1}\}/∼ , \end{aligned}

with (a, b) ∼ (a′, b′) if there exist (x, y) ∈ A ⊕ B_{n+1} such that

a′ − a = x + (−)^{n+1}\overline{x} + β_{−2n−2}(y)(y)

+ β_{−2n−1}(y)(b) + β_{−2n−1}(b′)(y) ,
b′ − b = dy ,

with addition by

(a, b) + (a′, b′) = (a + a′ + β_{−2n}(b)(b′), b + b′)

and A-action by

A × Q(n) ⟶ Q(n) ; (x, (a, b)) ⟶ x(a, b) = (xa\overline{x}, xb) .

The A-morphisms

r : Q_{2n}(S^{n}A) = H_{0}(ℤ_{2}; A, (−)^{n}) ⟶ Q(n) ; a ⟶ (a, 0) ,

s : Q(n) ⟶ Q^{2n}(S^{n}A) = H^{0}(ℤ_{2}; A, (−)^{n}) ;

(a, b) ⟶ a + (−)^{n}\overline{a} + β_{−2n}(b)(b)

are such that there is defined a commutative braid of exact sequences

[ [ [ [ [ ’ ’ ’ ’ ’)  [ [ [ [ [ ’ ’ ’ ’ ’) 1 + T [ [ [ [ [ ’ ’ ’ ’ ’)

H_{n+1}(B)  Q_{2n}(S^{n}A)  Q^{2n}(S^{n}A)  0

Q̂^{2n+1}(S^{n}A)^{)’ ’ ’ ’}  ^{h h hh j} H  Q(n)^{64 4 4 4} r  ^{h h hh j} s  Q̂^{2n}(S^{n}A)^{64 4 4 4} J  ^{````}ˆ

0 ^{h h hh j}  P ⟨B, β⟩^{2n}(S^{n}A)^{64 4 4 4}  ^{h h hh j}  H_{n}(B)^{64 4 4 4}  ^{h h hh j}  0^{64 4 4 4}

N N N N N^{‚ ‚ ‚ ‚‚ ›}  ^{N N N N N}^{‚ ‚ ‚ ‚‚ ›}  ^{N N N N N}^{‚ ‚ ‚ ‚‚ ›}

If (B, β) is such that for all y ∈ B_{n+1} there exists x ∈ A such that

(β_{−2n−2} + β_{−2n−1}d)(y)(y) = x + (−)^{n}\overline{x} ∈ A

(e.g. if d = 0 : B_{n+1}⟶B_{n} and v_{n+1}(β) = 0 : H_{n+1}(B)⟶Ĥ^{n+1}(ℤ_{2}; A)) then there is a natural identification of A-groups

Q(n) = Q_{2n}(S^{n}A) ×_{β_{−2n}} H_{n}(B)

with

β_{−2n} : H_{n}(B) × H_{n}(B) ⟶ Q_{2n}(S^{n}A) ; (b, b′)⟶β_{−2n}(b)(b′) .

For any chain bundle (B, β) and any Q(n)-quadratic form (M, λ, μ) there exist Amodule morphisms

g : M ⟶ B_{n} , ψ : M ⟶ M^{∗}

such that

dg = 0 : M ⟶ B_{n−1} ,
λ − g^{∗}β_{−2n}g = ψ + (−)^{n}ψ^{∗} : M ⟶ M^{∗} ,
μ : M ⟶ Q(n) ; x ⟶ (ψ(x)(x), g(x)) .

If (M, λ, μ) is a nonsingular form there is thus defined a 2n-dimensional symmetric (B, β)-structure Poincaré complex (C, ϕ, θ, f) with

ϕ_{0} = λ : C^{n} = M⟶C_{n} = M^{∗} ,

C_{r} = 0 (r ≠ n) ,

θ_{−1} = ψ : C^{n} = M ⟶ C_{n} = M^{∗} ,

f = gλ^{−1} : C_{n} = M^{∗} ⟶^{λ^{−1}} M ⟶^{g} B_{n} ,

v_{n}(ϕ, θ, f) = μ : H^{n}(C) = M ⟶ Q(n) .

The construction defines a morphism of abelian groups

W_{Q(n)}(A) ⟶ L⟨B, β⟩^{2n}(A) ; (M, λ, μ) ⟶ (C, ϕ, θ, f) .

Conversely, if (C, ϕ, θ, f) is a 2n-dimensional symmetric (B, β)-structure Poincaré complex such that H^{n}(C) is a f.g. projective A-module there is defined a nonsingular Q(n)-quadratic form (H^{n}(C), ϕ_{0}, v_{n}(ϕ, θ, f)), with

v_{n}(ϕ, θ, f) : H^{n}(C) ⟶ Q(n) ; x ⟶ (θ_{−1}(x)(x), f(x)) .

It follows that for any ring A there is a natural identification of the 0-dimensional L-group with the Witt group

L⟨B, β⟩^{0}(A) = W_{Q(0)}(A) .

For a field A the morphisms

W_{Q(n)}(A) ⟶ L⟨B, β⟩^{2n}(A) (n ≥ 0)

are injections, which are split by

L⟨B, β⟩^{2n}(A) ⟶ W_{Q(n)}(A) ; (C, ϕ, θ, f) ⟶ (H^{n}(C), ϕ_{0}, v_{n}(ϕ, θ, f)) (n ≥ 0) .

For n ≥ 0 let (B⟨n + 1⟩, β⟨n + 1⟩) be the v_{n+1} = 0 universal chain bundle over A, characterized up to equivalence by the properties

(i) v̂_{r}(β⟨n + 1⟩) : H_{r}(B⟨n + 1⟩)⟶Ĥ^{r}(ℤ_{2}; A) is an isomorphism for r ≠ n + 1, (ii) H_{n+1}(B⟨n + 1⟩) = 0. The (v_{n+1} = 0)-symmetric L-groups of A are defined by

L⟨v_{n+1}⟩^{m}(A) = L⟨B⟨n + 1⟩, β⟨n + 1⟩⟩^{m}(A) (m ≥ 0) .

Define the A-group

Q⟨v_{n+1}⟩ = Q⟨B⟨n + 1⟩, β⟨n + 1⟩⟩^{2n}(S^{n}A) ,

to fit into the commutative braid of exact sequences

[ [ [ [ [ ’ ’ ’ ’ ’)  [ [ [ [ [ ’ ’ ’ ’ ’) 1 + T [ [ [ [ [ ’ ’ ’ ’ ’)

0  Q_{2n}(S^{n}A)  Q^{2n}(S^{n}A)  0

Q̂^{2n+1}(S^{n}A)^{C A A A A} ^{h h hh j} H  Q⟨v_{n+1}⟩^{64 4 4 4} r  ^{h h hh j} s  Q̂^{2n}(S^{n}A)^{64 4 4 4} J  ^{````}ˆ

0^{h h hh j}  Q^{2n+1}(S^{n}A)^{64 4 4 4} 1  ^{h h hh j}  Q̂^{2n}(S^{n}A)^{64 4 4 4}  ^{h h hh j} 1  0^{64 4 4 4}

N N N N N^{‚ ‚ ‚ ‚‚ ›}  ^{N N N N N}^{‚ ‚ ‚ ‚‚ ›} 0  ^{N N N N N}^{‚ ‚ ‚ ‚‚ ›}

In §11 below we shall make use of the surjections

L⟨v_{n+1}⟩^{2n}(A) ⟶ W_{Q⟨v_{n+1}⟩}(A) ; (C, ϕ, θ, f) ⟶ (H^{n}(C), ϕ_{0}, v_{n}(ϕ, θ, f)) (n ≥ 0)

defined for a field A.

§11. An example

As an illustration of the exact sequence

… ⟶ L_{n}(A) ⟶ L⟨B, β⟩^{n}(A) ⟶ Q_{n}(B, β) ⟶^{∂} L_{n−1}(A) ⟶ …

we compute the Witt groups L^{0}(A), L_{0}(A), L⟨v_{1}⟩^{0}(A) for A a perfect field of characteristic 2, without appealing to the theorem of Arf [ ] on the classification of quadratic forms over such A.

For any field A \{ \begin{matrix}L^{2n}(A) \\ L_{2n}(A) \end{matrix} (n ≥ 0) is the Witt group of nonsingular \{ \begin{matrix}(−)^{n}-symmetric \\ (−)^{n}-quadratic \end{matrix}

forms over A, and \{ \begin{matrix}L^{2n+1}(A) = 0 \\ L_{2n+1}(A) = 0 \end{matrix} (n ≥ 0) - see Ranicki [ ] for details.

Let then A be a perfect field of characteristic 2, so that there is defined an automorphism

A ⟶ A ; a ⟶ a^{2} .

Let A have the identity involution

: A ⟶ A ; a ⟶ \overline{a} = a .

As an additive group

Ĥ^{r}(ℤ_{2}; A) = A (r ∈ ℤ)

with A acting by

A × Ĥ^{r}(ℤ_{2}; A) ⟶ Ĥ^{r}(ℤ_{2}; A) ; (a, x) ⟶ a^{2}x ,

and there is defined an isomorphism of A-modules

A ⟶ Ĥ^{r}(ℤ_{2}; A) ; a ⟶ a^{2} .

The chain bundle over A (B(∞), β(∞)) defined by

\begin{matrix}d_{B(∞)} = 0 : B(∞)_{r} = A ⟶ B(∞)_{r−1} = A , \\ β(∞)_{s} = \{ \begin{matrix}1 \\ 0 \end{matrix} : B(∞)_{r} = A ⟶ B(∞)^{−r−s} = A if \{ \begin{matrix}s = −2r \\ s ≠ −2r \end{matrix} \end{matrix}

is universal. The twisted quadratic groups of (B(∞), β(∞)) are given up to isomorphism by

\begin{matrix}Q_{2n}(B(∞), β(∞)) = Q^{•}(A) , \\ Q_{2n+1}(B(∞), β(∞)) = Q_{•}(A) , \end{matrix}

with the abelian groups Q^{•}(A), Q_{•}(A) defined by

\begin{matrix}Q^{•}(A) = \{a ∈ A | a + a^{2} = 0\} = ℤ_{2} , \\ Q_{•}(A) = A/\{b + b^{2} | b ∈ A\} , \end{matrix}

and isomorphisms defined by

Q_{2n}(B(∞), β(∞)) \underrightarrow{≃} Q^{•}(A) ; (ϕ, θ) ⟶ ϕ_{0}(1)(1) ,
ϕ_{0} : B(∞)^{n} = A ⟶ B(∞)_{n} = A ,
Q_{2n+1}(B(∞), β(∞)) \underrightarrow{≃} Q_{•}(A) ; (ϕ, θ) ⟶ θ_{−1}(1)(1) ,
θ_{−1} : B(∞)^{n+1} = A ⟶ B(∞)_{n+1} = A .

A symmetric form over A (M, λ) is even if and only if

λ(x, x) = 0 ∈ A (x ∈ M) .

A nonsingular even symmetric form over A (M, λ) is hyperbolic, since for any x ∈ M there exists y ∈ M such that λ(x)(y) = 1 ∈ A, so that a hyperbolic summand may be split off (M, λ)

(M, λ) = (Ax ⊕ Ay, ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )) ⊕ (M′, λ′) ,

with

rank_{A}M′ = (rank_{A}M) − 2 .

Thus

L⟨v_{0}⟩^{0}(A) = W ⟨v_{0}⟩(A) = 0

and the symmetrization maps

1 + T : L_{2n}(A) = L_{0}(A) ⟶ L⟨v_{0}⟩^{0}(A) ⟶ L^{2n}(A)

are zero. It is now immediate from the exact sequence

… ⟶ L_{n}(A) ⟶^{1+T} L^{n}(A) ⟶ Q_{n}(B(∞), β(∞)) ⟶^{∂} L_{n−1}(A) ⟶ …

that

L^{2n}(A) = Q^{•}(A) , L_{2n}(A) = Q_{•}(A) .

In the symmetric case there is defined an isomorphism

L^{2n}(A) \underrightarrow{≃} Q^{•}(A) = ℤ_{2} ; (C, ϕ) ⟶ ϕ_{0}(v)(v) = rank_{A}H^{n}(C) ,

sending a 2n-dimensional symmetric Poincaré complex (C, ϕ) over A to the element ϕ_{0}(v)(v) ∈ Q^{•}(A), with v ∈ H^{n}(C) the unique cohomology class such that

ϕ_{0}(x)(v) = ϕ_{0}(x)(x) ∈ Ĥ^{n}(ℤ_{2}; A) (x ∈ H^{n}(C)) .

The inverse isomorphism

Q^{•}(A) = ℤ_{2} \underrightarrow{≃} L^{2n}(A)

sends 1 ∈ Q^{•}(A) to the 2n-dimensional symmetric Poincaré complex (C, ϕ) defined by

ϕ_{0} = 1 : C^{n} = A ⟶ C_{n} = A , C_{r} = 0 (r ≠ 0) .

In the quadratic case there is defined an isomorphism

∂ : Q_{•}(A) \underrightarrow{≃} L_{2n}(A) ; a ⟶ (C, ψ)

with (C, ψ) the 2n-dimensional quadratic Poincaré complex over A given by

ψ_{0} = ( \begin{matrix}a & 1 \\ 0 & 1 \end{matrix} ) : C^{n} = A ⊕ A ⟶ C_{n} = A ⊕ A , C_{r} = 0 (r ≠ n) .

The inverse isomorphism L_{2n}(A)\underrightarrow{≃}Q_{•}(A) sends a 2n-dimensional quadratic Poincaré complex (C, ψ) over A to the Arf invariant c ∈ Q_{•}(A) of the nonsingular quadratic form (H^{n}(C), (1 + T)ψ_{0}, v^{n}(ψ)) over A, as defined by

c = ∑_{i=1}^{m} v^{n}(ψ)(x_{2i}).v^{n}(ψ)(x_{2i+1}) ∈ Q_{•}(A)

with {x_{i} | 1 ≤ i ≤ m} any base for H^{n}(C) such that

(1 + T)ψ_{0}(x_{i}, x_{j}) = \{ \begin{matrix}1 & if (i, j) = (2r, 2r + 1) or (2r + 1, 2r) \\ 0 & otherwise . \end{matrix}

The chain bundle over A (B⟨v_{n+1}⟩, β⟨v_{n+1}⟩) (n ≥ 0) defined by

B⟨v_{n+1}⟩_{r} = \{ \begin{matrix}A & if r ≠ n + 1 \\ 0 & if r = n + 1 \end{matrix} ,

d = 0 : B⟨v_{n+1}⟩_{r} ⟶ B⟨v_{n+1}⟩_{r−1} ,

β⟨v_{n+1}⟩_{s} = \{ \begin{matrix}1 \\ 0 \end{matrix} : B⟨v_{n+1}⟩_{r} ⟶ B⟨v_{n+1}⟩^{−r−s} if \{ \begin{matrix}s = −2r, r ≠ n + 1 \\ otherwise \end{matrix}

is (v_{n+1} = 0)-universal. Define symmetric bilinear pairings

\begin{matrix}ρ : A × A ⟶ A ; (a, b) ⟶ ab , \\ σ : Q^{•}(A) × Q^{•}(A) ⟶ A ; (1, 1) ⟶ 1 \end{matrix}

such that

Q⟨B⟨v_{n+1}⟩, β⟨v_{n+1}⟩⟩^{2n}(S^{n}A) = A ×_{ρ} A ,
Q_{2n}(B⟨v_{n+1}⟩, β⟨v_{n+1}⟩) = A ×_{σ} Q^{•}(A),
Q_{2n+1}(B⟨v_{n+1}⟩, β⟨v_{n+1}⟩) = 0 .

Let

\begin{aligned}Q⟨v_{1}⟩ & = Q⟨B⟨v_{1}⟩, β⟨v_{1}⟩⟩^{0}(A) \\ & = A ×_{ρ} A . \end{aligned}

Given a nonsingular Q⟨v_{1}⟩-quadratic form (M, λ, μ) over A there exist v ∈ M, ψ ∈ Hom_{A}(M, M^{∗}) such that

\begin{matrix}λ(x, y) = λ(x, v)λ(y, v) + ψ(x)(y) + ψ(y)(x) ∈ A (x, y ∈ M) \\ μ : M ⟶ Q⟨v_{1}⟩ = A ×_{ρ} A ; x ⟶ (ψ(x)(x), λ(x, v)) . \end{matrix}

The morphism

\begin{matrix}L⟨v_{1}⟩^{0}(A) = W_{Q⟨v_{1}⟩}(A) ⟶ Q_{0}(B⟨v_{1}⟩, β⟨v_{1}⟩) ; \\ (M, λ : M × M⟶A, μ : M⟶Q⟨v_{1}⟩) ⟶ μ(v) = (ψ(v)(v), λ(v, v)) \end{matrix}

fits into a short exact sequence

0 ⟶ L_{0}(A) ⟶ L⟨v_{1}⟩^{0}(A) ⟶ Q_{0}(B⟨v_{1}⟩, β⟨v_{1}⟩) ⟶ 0 .

The injection

L_{2n}(A) ⟶ L⟨v_{n+1}⟩^{2n}(A) \underrightarrow{≃} W_{Q⟨v_{1}⟩}(A) = L⟨v_{1}⟩^{0}(A) sends the cobordism class of a 2n-dimensional quadratic Poincaré complex over A (C, ψ) to the Witt class of the nonsingular Q⟨v_{1}⟩-quadratic form (H^{n}(C), (1 + T)ψ_{0}, iv^{n}(ψ)), with i the canonical injection

i : H_{0}(ℤ_{2}; A, (−)^{n}) = A ⟶ Q⟨v_{1}⟩ = A ×_{ρ} A ; a ⟶ (a, 0) .

In the special case A = ℤ_{2}

\begin{matrix}L_{2n}(ℤ_{2}) = Q_{•}(ℤ_{2}) = ℤ_{2} , L^{2n}(ℤ_{2}) = Q^{•}(ℤ_{2}) = ℤ_{2} , \\ Q⟨v_{n+1}⟩ = ℤ_{4} , Q_{0}(B⟨v_{1}⟩, β⟨v_{1}⟩) = ℤ_{4} , \end{matrix}

with

L⟨v_{n+1}⟩^{2n}(ℤ_{2}) = W_{ℤ_{4}}(ℤ_{2}) = ℤ_{8}

the Witt group of nonsingular ℤ_{4}-valued quadratic forms over ℤ_{2} appearing in the work of Brown [ ].