45 SLIDES ON CHAIN DUALITY ANDREW RANICKI

Abstract The texts of 45 slides^{1} on the applications of chain duality to the homological analysis of the singularities of Poincaré complexes, the double points of maps of manifolds, and to surgery theory.

1. Introduction

• Poincaré duality

H^{n−∗}(M) ≅ H_{∗}(M)

is the basic algebraic property of an n-dimensional manifold M.

• A chain complex C with n-dimensional Poincaré duality

H^{n−∗}(C) ≅ H_{∗}(C)

is an algebraic model for an n-dimensional manifold, generalizing the \underline{intersection form}.

• Spaces with Poincaré duality (such as manifolds) determine Poincaré duality chain complexes in \underline{additive categories with chain duality}, giving rise to interesting invariants, old and new.

2. What is chain duality?

• 𝔸 = additive category.

• 𝔹(𝔸) = additive category of finite chain complexes in 𝔸 .

• A contravariant additive functor T : 𝔸 → 𝔹 (𝔸) extends to

T : 𝔹 (𝔸) → 𝔹 (𝔸) ; C → T (C)

by the total double complex

T (C)_{n} = ∑_{p+q=n} T (C_{−p})_{q} .

1The lecture at the conference on Surgery and Geometric Topology, Josai University, Japan on 17 September, 1996 used slides 1.–36.

• Definition: A \underline{chain duality} (T, e) on 𝔸 is a contravariant additive functor T : 𝔸 → 𝔹 (𝔸), together with a natural transformation e : T^{2} → 1 : 𝔸 → 𝔹 (𝔸) such that for each object A in 𝔸 :

– e(T (A)) . T (e(A)) = 1 : T (A) → T (A) ,

– e(A): T^{2}(A) → A is a chain equivalence.

3. Properties of chain duality

• The dual of an \underline{object} A is a \underline{chain complex} T (A).

• The dual of a \underline{chain complex} C is a \underline{chain complex} T (C).

• Motivated by \underline{Verdier duality} in sheaf theory.

• A.Ranicki, \underline{Algebraic L-theory and topological manifolds}, Tracts in Mathematics 102, Cambridge (1992)

4. Involutions

• An \underline{involution} (T, e) on an additive category 𝔸 is a chain duality such that T (A) is a 0-dimensional chain complex (= object) for each object A in 𝔸, with e(A) : T^{2}(A) → A an isomorphism.

• Example: An involution R → R; r → \overline{r} on a ring R determines the involution (T, e) on the additive category 𝔸(R) of f.g. free left Rmodules:

– T (A) = Hom_{R}(A, R)

– R × T (A) → T (A) ; (r, f) → (x → f(x)\overline{r})

– e(A)^{−1} : A → T^{2}(A) ; x → (f → \overline{f(x)}).

5. Manifolds and homeomorphisms up to homotopy

• Traditional questions of surgery theory:

– Is a space with Poincaré duality homotopy equivalent to a manifold?

– Is a homotopy equivalence of manifolds homotopic to a homeomorphism?

• Answered for dimensions ≥ 5 by \underline{surgery exact sequence} in terms of the \underline{assembly} map

A : H_{∗}(X; 𝕃_{•}(ℤ)) → L_{∗}(ℤ[π_{1}(X)]) .

• L-theory of additive categories with involution suffices for surgery groups L_{∗}(ℤ[π_{1}(X)]).

• Need chain duality for the generalized homology groups H_{∗}(X; 𝕃_{•}(ℤ)) and A.

6. Manifolds and homeomorphisms

• Will use chain duality to answer questions of the type:

– Is a space with Poincaré duality a manifold?

– Is a homotopy equivalence of manifolds a homeomorphism?

7. Controlled topology

• Controlled topology (Chapman-Ferry-Quinn) considers:

– the approximation of manifolds by Poincaré complexes,

– the approximation of homeomorphisms of manifolds by homotopy equivalences.

• \underline{Philosophy} of controlled topology, with control map 1 : X → X :

– A Poincaré complex X is a homology manifold if and only if it is an ϵ-controlled Poincaré complex for all ϵ > 0.

– A map of homology manifolds f : M → X has contractible point inverses if and only if it is an ϵ-controlled homotopy equivalence for all ϵ > 0.

8. Simplicial complexes

• In dealing with applications of chain duality to topology will only work with (connected, finite) \underline{simplicial complexes} and (oriented) polyhedral homology manifolds and Poincaré complexes.

• Can also work with \underline{Δ-sets} and \underline{topological spaces}, using the methods of:

– M.Weiss, \underline{Visible L-theory}, Forum Math. 4, 465–498 (1992)

– S.Hutt, \underline{Poincaré sheaves on topological spaces}, Trans. A.M.S. (1996)

9. Simplicial control

• Additive category 𝔸(ℤ, X) of \underline{X-controlled ℤ-modules} for a simplicial complex X.

– A.Ranicki and M.Weiss, \underline{Chain complexes and assembly}, Math. Z. 204, 157–186 (1990)

• Will use chain duality on 𝔸(ℤ, X) to obtain \underline{homological obstructions} for deciding:

– Is a simplicial Poincaré complex X a homology manifold?

(Singularities)

– Does a degree 1 map f : M → X of polyhedral homology manifolds have acyclic point inverses? (Double points)

• \underline{Acyclic point inverses} H̃_{∗}(f^{−1}(x)) = 0 is analogue of homeomorphism in the world of homology.

10. The X-controlled ℤ-module category 𝔸(ℤ, X)

• X = simplicial complex.

• A \underline{(ℤ, X)-module} is a finitely generated free ℤ-module A with direct

sum decomposition

A = ∑_{σ∈X} A(σ) .

• A (ℤ, X)-module \underline{morphism} f : A → B is a ℤ-module morphism such

that

f(A(σ)) ⊆ ∑_{τ≥σ} B(τ) .

• Proposition: A (ℤ, X)-module chain map f : C → D is a chain

equivalence if and only if the ℤ-module chain maps

f(σ, σ) : C(σ) → D(σ) (σ ∈ X)

are chain equivalences.

11. Functorial formulation

• Regard simplicial complex X as the category with:

– objects: simplexes σ ∈ X

– morphisms: face inclusions σ ≤ τ.

• A (ℤ, X)-module A = ∑_{σ∈X} A(σ) determines a \underline{contravariant} functor

[A] : X → 𝔸(ℤ) = {f.g. free abelian groups} ; σ → [A][σ] = ∑_{τ≥σ} A(τ) .

• The (ℤ, X)-module category 𝔸(ℤ, X) is a full subcategory of the cate-

gory of contravariant functors X → 𝔸(ℤ).

12. Dual cells

• The \underline{barycentric subdivision} X′ of X is the simplicial complex with one

n-simplex σ̂_{0}σ̂_{1} … σ̂_{n} for each sequence of simplexes in X

σ_{0} < σ_{1} < ⋯ < σ_{n} .

• The \underline{dual cell} of a simplex σ ∈ X is the contractible subcomplex

D(σ, X) = {σ̂_{0}σ̂_{1} … σ̂_{n} | σ ≤ σ_{0}} ⊆ X′ ,

with boundary

∂D(σ, X) = {σ̂_{0}σ̂_{1} … σ̂_{n} | σ < σ_{0}} ⊆ D(σ, X) .

• Introduced by Poincaré to prove duality.

• A simplicial map f : M → X′ has acyclic point inverses if and only if (f|)_{∗} : H_{∗}(f^{−1}D(σ, X)) ≅ H_{∗}(D(σ, X)) (σ ∈ X) .

13. Where do (ℤ, X)-module chain complexes come from?

• For any simplicial map f : M → X′ the simplicial chain complex Δ(M) is a (ℤ, X)-module chain complex:

Δ(M)(σ) = Δ(f^{−1}D(σ, X), f^{−1}∂D(σ, X))

with a degreewise direct sum decomposition

[Δ(M)][σ] = ∑_{τ≥σ} Δ(M)(τ) = Δ(f^{−1}D(σ, X)) .

• The simplicial cochain complex Δ(X)^{−∗} is a (ℤ, X)-module chain complex with:

Δ(X)^{−∗}(σ)_{r} = \begin{cases}ℤ & if r = −|σ| \\ 0 & otherwise. \end{cases}

14. The (ℤ, X)-module chain duality

• Proposition: The additive category 𝔸(ℤ, X) of (ℤ, X)-modules has a chain duality (T, e) with

T (A) = Hom_{ℤ}(Hom_{(ℤ,X)}(Δ(X)^{−∗}, A), ℤ)

• T A(σ) = [A][σ]^{|σ|−∗}

• T (A)_{r}(σ) = \begin{cases}∑_{τ≥σ} Hom_{ℤ}(A(τ), ℤ) & if r = −|σ| \\ 0 & if r ≠ −|σ| \end{cases}

• T (C) ≃_{ℤ} Hom_{(ℤ,X)}(C, Δ(X′))^{−∗} ≃_{ℤ} Hom_{ℤ}(C, ℤ)^{−∗}

• T (Δ(X′)) ≃_{(ℤ,X)} Δ(X)^{−∗}.

15. Products

• The \underline{product} of (ℤ, X)-modules A, B is the (ℤ, X)-module

A ⊗_{(ℤ,X)} B = ∑_{λ,μ∈X,λ∩μ≠∅} A(λ) ⊗_{ℤ} B(μ) ⊆ A ⊗_{ℤ} B ,

(A ⊗_{(ℤ,X)} B)(σ) = ∑_{λ,μ∈X,λ∩μ=σ} A(λ) ⊗_{ℤ} B(μ) .

• C ⊗_{(ℤ,X)} Δ(X′) ≃_{(ℤ,X)} C .

• T C ⊗_{(ℤ,X)} D ≃_{ℤ} Hom_{(ℤ,X)}(C, D) .

• For simplicial maps f : M → X′, g : N → X′ – Δ(M) ⊗_{(ℤ,X)} Δ(N) ≃_{(ℤ,X)} Δ((f × g)^{−1}Δ_{X})

– T Δ(M) ⊗_{(ℤ,X)} T Δ(N) ≃_{ℤ} Δ(M × N, M × N\(f × g)^{−1}Δ_{X})^{−∗} .

16. Cap product

• The \underline{Alexander-Whitney} diagonal chain approximation

Δ : Δ(X′) → Δ(X′) ⊗_{ℤ} Δ(X′) ;

(x̂_{0} … x̂_{n}) → ∑_{i=0}^{n}(x̂_{0} … x̂_{i}) ⊗ (x̂_{i} … x̂_{n})

is the composite of a chain equivalence

Δ(X′) ≃_{(ℤ,X)} Δ(X′) ⊗_{(ℤ,X)} Δ(X′)

and the inclusion

Δ(X′) ⊗_{(ℤ,X)} Δ(X′) ⊆ Δ(X′) ⊗_{ℤ} Δ(X′) .

• Homology classes [X] ∈ H_{n}(X) are in one-one correspondence with the

chain homotopy classes of (ℤ, X)-module chain maps [X] ∩ − : Δ(X)^{n−∗} → Δ(X′) .

17. Homology manifolds

• Definition: A simplicial complex X is an \underline{n-dimensional homology} \underline{manifold} if

H_{∗}(X, X\σ̂) = \begin{cases}ℤ & if ∗ = n \\ 0 & otherwise \end{cases} (σ ∈ X) .

• Proposition: A simplicial complex X is an n-dimensional homology manifold if and only if there exists a homology class [X] ∈ H_{n}(X) such that the cap product

[X] ∩ − : Δ(X)^{n−∗} → Δ(X′)

is a (ℤ, X)-module chain equivalence.

• Proof: For any simplicial complex X

H_{∗}(X, X\σ̂) = H_{∗−|σ|}(D(σ, X), ∂D(σ, X)) ,

H^{n−∗}(D(σ, X)) = \begin{cases}ℤ & if ∗ = n \\ 0 & otherwise \end{cases} (σ ∈ X) .

18. Poincaré complexes

• Definition: An \underline{n-dimensional Poincaré complex} X is a simplicial complex with a homology class [X] ∈ H_{n}(X) such that

[X] ∩ − : H^{n−∗}(X) ≅ H_{∗}(X) .

• Poincaré duality theorem: An n-dimensional homology manifold X is an n-dimensional Poincaré complex.

• Proof: A (ℤ, X)-module chain equivalence

[X] ∩ − : Δ(X)^{n−∗} → Δ(X′)

is a ℤ-module chain equivalence.

• There is also a ℤ[π_{1}(X)]-version.

19. McCrory’s Theorem

• X = n-dimensional Poincaré complex

– X × X is a 2n-dimensional Poincaré complex.

– Let V ∈ H^{n}(X ×X) be the Poincaré dual of Δ_{∗}[X] ∈ H_{n}(X ×X).

– Exact sequence

H^{n}(X × X, X × X\Δ_{X}) → H^{n}(X × X) → H^{n}(X × X\Δ_{X}) .

Theorem (McCrory) X is an n-dimensional homology manifold if and only if V has image 0 ∈ H^{n}(X × X\Δ_{X}).

• \underline{A characterization of homology manifolds}, J. Lond. Math. Soc. 16 (2), 149–159 (1977)

20. Chain duality proof of McCrory’s Theorem

• V has image 0 ∈ H^{n}(X × X\Δ_{X}) if and only if there exists U ∈ H^{n}(X × X, X × X\Δ_{X}) with image V.

• U is a chain homotopy class of (ℤ, X)-module chain maps Δ(X′) → Δ(X)^{n−∗}, since

H^{n}(X × X, X × X\Δ_{X}) = H_{n}(T Δ(X) ⊗_{(ℤ,X)} T Δ(X))
= H_{0}(Hom_{(ℤ,X)}(Δ(X′), Δ(X)^{n−∗})) .

• U is a chain homotopy inverse of

ϕ = [X] ∩ − : Δ(X)^{n−∗} → Δ(X′)

with

ϕU = 1 ∈ H_{0}(Hom_{(ℤ,X)}(Δ(X′), Δ(X′))) = H^{0}(X) ,
ϕ = T ϕ , (T U)ϕ = (T U)(T ϕ) = T (ϕU) = 1 .

21. The homology tangent bundle

• The tangent bundle τ_{X} of a manifold X is the normal bundle of the

diagonal embedding

Δ : X → X × X ; x → (x, x) .

• The \underline{homology tangent bundle} τ_{X} of an n-dimensional homology man-

ifold X is the fibration

(X, X\{∗}) ⟶ (X × X, X × X\Δ_{X}) ⟶ X

with X × X → X; (x, y) → x.

• \underline{Thom space} of τ_{X}

T (τ_{X}) = (X × X)/(X × X\Δ_{X}) .

• \underline{Thom class} of τ_{X}

U ∈ H̃^{n}(T (τ_{X})) = H^{n}(X × X, X × X\Δ_{X})

has image V ∈ H^{n}(X × X).

22. Euler

• The \underline{Euler characteristic} of a simplicial complex X is

χ(X) = ∑_{r=0}^{∞}(−)^{r}dim_{ℝ}H_{r}(X; ℝ) ∈ ℤ .

• For an n-dimensional Poincaré complex X χ(X) = Δ^{∗}(V) ∈ H^{n}(X) = ℤ .

• The \underline{Euler class} of n-plane bundle η over X e(η) = [U] ∈ im(H̃^{n}(T (η)) → H^{n}(X)) .

• \underline{Reformulation of McCrory’s Theorem}:

an n-dimensional Poincaré complex X is a homology manifold if and only if V ∈ H^{n}(X × X) is the image of Thom class U ∈ H̃^{n}(T (τ_{X})), in which case

χ(X) = e(τ_{X}) ∈ H^{n}(X) = ℤ .

23. Degree 1 maps

• A map f : M → X of n-dimensional Poincaré complexes has \underline{degree 1}

if

f_{∗}[M] = [X] ∈ H_{n}(X) .

• A homology equivalence has degree 1.

• The \underline{Umkehr} ℤ-module chain map of a degree 1 map f : M → X

f^{!} : Δ(X) ≃ Δ(X)^{n−∗} ⟶^{f^{∗}} Δ(M)^{n−∗} ≃ Δ(M)

is such that ff^{!} ≃ 1 : Δ(X) → Δ(X).

• A degree 1 map f is a homology equivalence if and only if

f^{!}f ≃ 1 : Δ(M) → Δ(M) ,

if and only if

(f^{!} ⊗ f^{!})Δ_{∗}[X] = Δ_{∗}[M] ∈ H_{n}(M × M) .

24. The double point set

• Does a degree 1 map of n-dimensional homology manifolds f : M → X have acyclic point inverses?

• Obstruction in homology of \underline{double point set}

(f × f)^{−1}Δ_{X} = {(x, y) ∈ M × M | f(x) = f(y) ∈ X}.

• Define maps

i : M → (f × f)^{−1}Δ_{X} ; a → (a, a) ,
j : (f × f)^{−1}Δ_{X} → X ; (x, y) → f(x) = f(y)

such that f = ji : M → X.

• The Umkehr map

j^{!} : H_{n}(X) ≅ H^{n}(X × X, X × X\Δ_{X})
→ H^{n}(M × M, M × M\(f × f)^{−1}Δ_{X})
≅ H_{n}((f × f)^{−1}Δ_{X}) (Lefschetz duality)

is such that j_{∗}j^{!} = 1.

25. Lefschetz

• \underline{Lefschetz duality}: If W is an m-dimensional homology manifold and

A ⊆ W is a subcomplex then

H^{∗}(W, W \A) ≅ H_{m−∗}(A) .

• \underline{Proof}: For any regular neighbourhood (V, ∂V) of A in W there are

defined isomorphisms

H^{∗}(W, W \A) ≅ H^{∗}(W, W \V) (homotopy invariance)

≅ H^{∗}(W, \overline{W \V}) (collaring) ≅ H^{∗}(V, ∂V) (excision) ≅ H_{m−∗}(V) (Poincaré-Lefschetz duality) ≅ H_{m−∗}(A) (homotopy invariance).

• Alexander duality is the special case W = S^{m}.

26. Acyclic Point Inverse Theorem

Theorem A degree 1 map f : M → X of n-dimensional homology manifolds has acyclic point inverses if and only if

i_{∗}[M] = j^{!}[X] ∈ H_{n}((f × f)^{−1}Δ_{X}) .

• Equivalent conditions:

– i_{∗} : H_{n}(M) ≅ H_{n}((f × f)^{−1}Δ_{X}) ,

– i_{∗} : H_{∗}(M) ≅ H_{∗}((f × f)^{−1}Δ_{X}) ,

– H_{∗}^{lf}((f × f)^{−1}Δ_{X}\Δ_{M}) = 0 .

• Conditions satisfied if f : M → X is injective, with (f × f)^{−1}Δ_{X} = Δ_{M} .

• In general, i_{∗} ≠ j^{!}f_{∗} and i_{∗}[M] ≠ j^{!}[X] .

27. Proof of Theorem - Part I

• A simplicial map f : M → X′ has acyclic point inverses if and only if f : Δ(M) → Δ(X′) is a (ℤ, X)-module chain equivalence.

• For degree 1 map f : M → X′ of n-dimensional homology manifolds

define the Umkehr (ℤ, X)-module chain map

f^{!} : Δ(X′) ≃ Δ(X′)^{n−∗} ⟶^{f^{∗}} Δ(M)^{n−∗} ≃ Δ(M) .

• f^{!} is a chain homotopy right inverse for f ff^{!} ≃ 1 : Δ(X′) → Δ(X′) .

• f^{!} is also a chain homotopy left inverse for f if and only if f^{!}f = 1 ∈ H_{0}(Hom_{(ℤ,X)}(Δ(M), Δ(M))) .

28. Proof of Theorem - Part II

• Use the (ℤ, X)-Poincaré duality

Δ(M)^{n−∗} ≃ Δ(M)

and the properties of chain duality in 𝔸(ℤ, X) to identify

1 = i_{∗}[M] , f^{!}f = j^{!}[X] ∈ H_{0}(Hom_{(ℤ,X)}(Δ(M), Δ(M)))

= H_{0}(Hom_{(ℤ,X)}(Δ(M)^{n−∗}, Δ(M))) = H_{n}(Δ(M) ⊗_{(ℤ,X)} Δ(M)) = H_{n}((f × f)^{−1}Δ_{X}) .

29. Cohomology version of Theorem

Theorem^{∗} A degree 1 map f : M → X of n-dimensional homology manifolds has acyclic point inverses if and only if the Thom classes U_{M} ∈ H^{n}(M ×M, M ×M\Δ_{M}), U_{X} ∈ H^{n}(X ×X, X ×X\Δ_{X}) have the same image in H^{n}(M × M, M × M\(f × f)^{−1}Δ_{X}).

• Same proof as homology version, after Lefschetz duality identifications

U_{M} = [M] ∈ H^{n}(M × M, M × M\Δ_{M}) = H_{n}(M) , U_{X} = [X] ∈ H^{n}(X × X, X × X\Δ_{X}) = H_{n}(X) , H^{n}(M × M, M × M\(f × f)^{−1}Δ_{X}) = H_{n}((f × f)^{−1}Δ_{X}) .

30. The double point obstruction

• The \underline{double point obstruction} of a degree 1 map f : M → X of homology manifolds

i_{∗}[M] − j^{!}[X] ∈ H_{n}((f × f)^{−1}Δ_{X})

is 0 if and only if f has acyclic point inverses.

• The obstruction has image

χ(M) − χ(X) ∈ H^{n}(M) = ℤ .

• If f is covered by a map of homology tangent bundles

b : (M × M, M × M\Δ_{M}) → (X × X, X × X\Δ_{X})

then

– U_{M} = b^{∗}U_{X} ∈ H^{n}(M × M, M × M\Δ_{M}),

– the double point obstruction is 0, and f has acyclic point inverses.

31. Normal maps

• A degree 1 map f : M → X of n-dimensional homology manifolds is \underline{normal} if it is covered by a map b : τ_{M} ⊕ ϵ^{∞} → τ_{X} ⊕ ϵ^{∞} of the stable tangent bundles.

• The stable map of Thom spaces

T (b) : Σ^{∞}T (τ_{M}) → Σ^{∞}T (τ_{X})

induces a map in cohomology

T (b)^{∗} : H̃^{n}(T (τ_{X})) = H^{n}(X × X, X × X\Δ_{X})
→ H̃^{n}(T (τ_{M})) = H^{n}(M × M, M × M\Δ_{M})

which sends the Thom class U_{X} to U_{M}.

• However, Theorem^{∗} may not apply to a normal map (f, b) : M → X, since in general

(f × f)^{∗} ≠ (inclusion)^{∗}T (b)^{∗} : H̃^{n}(T (τ_{X}))
→ H^{n}(M × M, M × M\(f × f)^{−1}Δ_{X})

(dual of i_{∗} ≠ j^{!}f_{∗}).

32. The surgery obstruction

• The Wall surgery obstruction of a degree 1 normal map (f, b) : M → X of n-dimensional homology manifolds

σ_{∗}(f, b) ∈ L_{n}(ℤ[π_{1}(X)])

is 0 if (and for n ≥ 5 only if) (f, b) is \underline{normal bordant} to a homotopy equivalence.

• A degree 1 map f : M → X with acyclic point inverses is a normal map with zero surgery obstruction.

• What is the relationship between the double point obstruction of a degree 1 normal map (f, b) : M → X and the surgery obstruction?

• Use chain level surgery obstruction theory:

A.Ranicki, \underline{The algebraic theory of surgery}, P. Lond. Math. Soc. (3) 40, 87–283 (1980)

33. Quadratic Poincaré complexes

• The simply-connected surgery obstruction σ_{∗}(f, b) ∈ L_{n}(ℤ) is the cobor-

dism class of the n-dimensional \underline{quadratic} Poincaré complex

(C, ψ) = (C(f^{!}), (e ⊗ e)ψ_{b})

where

– e : Δ(M) → C(f^{!}) is the inclusion in the algebraic mapping cone of the ℤ-module chain map f^{!} : Δ(X) → Δ(M),

– the \underline{quadratic structure} ψ is the image of ψ_{b} ∈ H_{n}(EΣ_{2} ×_{Σ_{2}} (M × M)) = H_{n}(W ⊗_{ℤ[Σ_{2}]} (Δ(M) ⊗_{ℤ} Δ(M))) ,

– EΣ_{2} = S^{∞}, a contractible space with a free Σ_{2}-action,

– W = Δ(EΣ_{2}).

• There is also a ℤ[π_{1}(X)]-version.

34. The double point and surgery obstructions - Part I

• For any degree 1 map f : M → X of n-dimensional homology manifolds

the composite of

i_{∗}f^{!} − j^{!} : H_{∗}(X) → H_{∗}((f × f)^{−1}Δ_{X})

and H_{∗}((f × f)^{−1}Δ_{X}) → H_{∗}(M × M) is

Δ_{∗}f^{!} − (f^{!} ⊗ f^{!})Δ_{∗} : H_{∗}(X) → H_{∗}(M × M) .

• For a degree 1 normal map (f, b) : M → X

H_{n}((f × f)^{−1}Δ_{X}) → H_{n}(M × M)

sends the double point obstruction i_{∗}[M] − j^{!}[X] to

(1 + T)ψ_{b} = Δ_{∗}[M] − (f^{!} ⊗ f^{!})Δ_{∗}[X] ∈ H_{n}(M × M) .

• (1 + T)ψ_{b} = 0 if and only if f is a homology equivalence.

35. The double point and surgery obstructions - Part II

• A degree 1 normal map (f, b) : M → X of n-dimensional homology manifolds determines the \underline{X-controlled quadratic structure}

ψ_{b,X} ∈ H_{n}(EΣ_{2} ×_{Σ_{2}} (f × f)^{−1}Δ_{X})
= H_{n}(W ⊗_{ℤ[Σ_{2}]} (Δ(M) ⊗_{(ℤ,X)} Δ(M))) .

• ψ_{b,X} has images

– the quadratic structure

[ψ_{b,X}] = ψ_{b} ∈ H_{n}(EΣ_{2} ×_{Σ_{2}} (M × M)) ,

– the double point obstruction

(1 + T)ψ_{b,X} = i_{∗}[M] − j^{!}[X] ∈ H_{n}((f × f)^{−1}Δ_{X}) .

36. The normal invariant

• The X-controlled quadratic Poincaré cobordism class

σ_{∗}^{X}(f, b) = (C(f^{!}), (e ⊗ e)ψ_{b,X}) ∈ L_{n}(𝔸(ℤ, X)) = H_{n}(X; 𝕃_{•}(ℤ))

is the \underline{normal invariant} of an n-dimensional degree 1 normal map (f, b) : M → X.

• σ_{∗}^{X}(f, b) = 0 if (and for n ≥ 5 only if) (f, b) is normal bordant to a map with acyclic point inverses.

• The non-simply-connected surgery obstruction of (f, b) is the assembly of the normal invariant

σ_{∗}(f, b) = Aσ_{∗}^{X}(f, b) ∈ L_{n}(ℤ[π_{1}(X)]) .

37. Hom and Derived Hom

• For (ℤ, X)-modules A, B the additive group Hom_{(ℤ,X)}(A, B) does not have a natural (ℤ, X)-module structure, but the chain duality determines a natural (ℤ, X)-module resolution.

• \underline{Derived Hom} of (ℤ, X)-module chain complexes C, D

RHom_{(ℤ,X)}(C, D) = T (C) ⊗_{(ℤ,X)} D .

• \underline{Adjoint properties}:

RHom_{(ℤ,X)}(C, D) ≃_{ℤ} Hom_{(ℤ,X)}(C, D)
RHom_{(ℤ,X)}(T C, D) ≃_{(ℤ,X)} C ⊗_{(ℤ,X)} D .

• D = Δ(X′) is the \underline{dualizing complex} for chain duality

T (C) ≃_{(ℤ,X)} RHom_{(ℤ,X)}(C, Δ(X′))

as for Verdier duality in sheaf theory.

38. When is a Poincaré complex homotopy equivalent to a manifold?

• Every n-dimensional topological manifold is homotopy equivalent to an n-dimensional Poincaré complex

• Is every n-dimensional Poincaré complex homotopy equivalent to an n-dimensional topological manifold?

•

From now on n ≥ 5

• \underline{Browder-Novikov-Sullivan-Wall} obstruction theory has been reformulated in terms of \underline{chain duality}

– the total surgery obstruction.

39. Browder-Novikov-Sullivan-Wall theory

• An n-dimensional Poincaré complex X is homotopy equivalent to an n-dimensional topological manifold if and only if

1. the Spivak normal fibration of X admits a topological reduction,

2. there exists a reduction such that the corresponding normal map M → X has surgery obstruction

σ_{∗}(f, b) = 0 ∈ L_{n}(ℤ[π_{1}(X)]) .

40. Algebraic Poincaré cobordism

• Λ = ring with involution.

• L_{n}(Λ) = Wall \underline{surgery obstruction} group

= the cobordism group of n-dimensional \underline{quadratic Poincaré}

complexes over Λ

– n-dimensional f.g. free Λ-module chain complexes C with H^{n−∗}(C) ≅ H_{∗}(C) ,

– uses \underline{ordinary duality} C^{n−∗} = Hom_{Λ}(C, Λ)_{∗−n} .

41. Assembly

• X = connected simplicial complex – X̃ = universal cover

– p : X̃ → X covering projection.

• \underline{Assembly} functor

A : 𝔸(ℤ, X) = {(ℤ, X)-modules} → 𝔸(ℤ[π_{1}(X)]) = {ℤ[π_{1}(X)]-modules};

M = ∑_{σ∈X} M(σ) → M(X̃) = ∑_{σ̃∈X̃} M(pσ̃) .

• The assembly A(T (M)) of dual (ℤ, X)-module chain complex

T (M) ≃_{ℤ} Hom_{(ℤ,X)}(M, Δ(X′))

is chain equivalent to dual ℤ[π_{1}(X)]-module M(X̃)^{∗} = Hom_{ℤ[π_{1}(X)]}(M(X̃), ℤ[π_{1}(X)]) .

42. The algebraic surgery exact sequence

• For any simplicial complex X exact sequence

⋯ → H_{n}(X; 𝕃_{•}(ℤ)) →^{A} L_{n}(ℤ[π_{1}(X)]) → 𝕊_{n}(X) → H_{n−1}(X; 𝕃_{•}(ℤ)) → …

with

• A = assembly,

• 𝕃_{•}(ℤ) = the 1-connective simply-connected surgery spectrum – π_{∗}(𝕃_{•}(ℤ)) = L_{∗}(ℤ) ,

• H_{n}(X; 𝕃_{•}(ℤ)) = \underline{generalized homology} group

– cobordism group of n-dimensional quadratic Poincaré (ℤ, X)-module complexes C ≃ T C^{n−∗}

– uses \underline{chain duality} T C^{n−∗} ≃_{(ℤ,X)} RHom_{(ℤ,X)}(C, Δ(X′))_{∗−n} .

43. The structure group

• X = simplicial complex.

• 𝕊_{n}(X) = \underline{structure} group.

• 𝕊_{n}(X) = cobordism group of – (n − 1)-dimensional quadratic Poincaré (ℤ, X)-module complexes – with contractible ℤ[π_{1}(X)]-module assembly.

44. Local and global Poincaré duality

• X = n-dimensional Poincaré complex.

• The cap product [X] ∩ − : Δ(X)^{n−∗} → Δ(X′):

– is a (ℤ, X)-module chain map, – assembles to ℤ[π_{1}(X)]-module chain equivalence

[X] ∩ − : Δ(X̃)^{n−∗} → Δ(X̃′) .

• The algebraic mapping cone

C = C([X] ∩ − : Δ(X)^{n−∗} → Δ(X′))_{∗−1}

– is an (n − 1)-dimensional quadratic Poincaré (ℤ, X)-module complex,

– with contractible ℤ[π_{1}(X)]-assembly.

• X is a homology manifold if and only if C is (ℤ, X)-contractible.

45. The total surgery obstruction

• X = n-dimensional Poincaré complex.

• The \underline{total surgery obstruction} of X is the cobordism class

s(X) = C([X] ∩ −)_{∗−1} ∈ 𝕊_{n}(X) .

• Theorem 1: X is homotopy equivalent to an n-dimensional topological manifold if and only if s(X) = 0 ∈ 𝕊_{n}(X).

• Theorem 2: A homotopy equivalence f : M → N of n-dimensional topological manifolds has a total surgery obstruction s(f) ∈ 𝕊_{n+1}(N) such that f is homotopic to a homeomorphism if and only if s(f) = 0.

– Should also consider Whitehead torsion.

Department of Mathematics and Statistics University of Edinburgh Edinburgh EH9 3JZ Scotland, UK e-mail: a.ranicki@ed.ac.uk