Chain duality for categories over complexes

We show that the additive category of chain complexes parametrized by a finite simplicial complex $K$ forms a category with chain duality. This fact, never fully proven in the original reference, is fundamental for Ranicki's algebraic formulation of the surgery exact sequence of Sullivan and Wall, and his interpretation of the surgery obstruction map as the passage from local Poincar\'e duality to global Poincar\'e duality. Our paper also gives a new, conceptual, and geometric treatment of chain duality on $K$-based chain complexes.


Introduction
Kervaire and Milnor [KM63] developed and applied the new field of surgery to classify exotic smooth structures on spheres.Browder and Novikov independently extended and relativized the theory.Sullivan in his thesis [Sul66] investigated the obstruction theory for deforming a homotopy equivalence to a homeomorphism.In seminar notes [Sul96] written shortly after his thesis, Sullivan's Theorem 3 packaged this in what is now called the surgery exact sequence.(We will be ahistorical and concentrate on topological manifolds; Kervaire-Milnor concentrated on smooth manifolds and Sullivan on PL-manifolds.The extension to topological manifolds is due to the deep work of Kirby and Siebenmann [KS77].).It was extended to the nonsimply-connected case and to the case of compact manifolds by Wall [Wal99].The surgery exact sequence for a closed n-dimensional manifold X with n ≥ 5 is The object one wants to compute is the structure set S T OP (X), first defined by Sullivan.Representatives of the structure set are given by (simple) homotopy equivalences from a closed n-manifold to X. Computing the structure set is the key ingredient in computing the manifold moduli set, the set of homeomorphism types of n-manifolds homotopy equivalent to X.The beauty of the surgery exact sequence is marred by many flaws.One is that it is an exact sequence of pointed sets.Another flaw is standard with a long exact sequence, to do computations one needs to compute the surgery obstruction map σ, including its domain, the normal invariants N T OP (X), and its codomain, the L-groups.For many fundamental groups (e.g.finite groups or finitely generated abelian groups) one can compute the L-groups algebraically.Sullivan, in his thesis, analyzed the normal invariants, using transversality to establish a bijection N T OP (X) ∼ = [X, G/T OP ].He also computed the homotopy groups π i (G/T OP ) which vanish for i odd, have order 2 for i ≡ 2 (mod 4), and are infinite cyclic for i ≡ 0 (mod 4) (this follows from the generalized Poincaré conjecture).Furthermore, Sullivan analyzed the homotopy type of G/T OP and established Sullivan periodicity, Ω 4 (Z × G/T OP ) ≃ Z × G/T OP .This then determines an Ω-spectrum L .and its 1-connective cover L . 1 .There is a formal identification [X, G/T OP ] = H 0 (X; L . 1 ).
The surgery exact sequence is now becoming more presentable, but it is still marred by functoriality issues: the L-groups are covariant in X, the normal invariants are contravariant in X, and the structure set has no obvious variance at all.Furthermore it is only defined for manifolds, and one would like an abelian group structure.These flaws make computing the surgery obstruction map difficult.Quinn's [Qui70] vision (largely carried out by Ranicki [Ran92], see also [KMM13]) is to find a bijection between the surgery exact sequence and a long exact sequences of abelian groups defined for every space X and fully covariant in X.In more detail, there is the following commutative diagram where the vertical maps labelled ∼ = are bijections when X is a closed n-manifold and the bottom two horizontal lines are exact sequences of abelian groups, defined for any space X.These two lines are called the 1-connective algebraic surgery exact sequence and the algebraic surgery exact sequence, respectively.The maps A 1 and A are called assembly maps; they are defined at the spectrum level.Hence there is a long exact sequence of homotopy groups, where the algebraic structure groups are defined be to the homotopy groups of the cofiber of the assembly maps.The map A is conjectured to be an isomorphism when X = Bπ with π torsionfree.There are myriad ways of constructing the assembly maps (the construction in [DL98] seems best for computations).Different constructions are identified via axiomatics (see [WW95], also [DL98]).Ranicki's version of assembly, needed for his approach to the above diagram, was motivated by his earlier work with Weiss [RW90] viewing the assembly maps as a passage from local to global Poincaré duality.Much earlier Ranicki [Ran80] reinterpreted Wall's algebraic L-groups as bordism groups of algebraic Poincaré complexes over the group ring Z[π 1 X].This is the global Poincaré duality.The local Poincaré duality comes from making a geometric degree one normal map transverse to the dual cones of X (see Section 3 for the definition).These degree one normal maps to the cones are then assembled to give the original degree one normal map.
More precisely, Ranicki [Ran92] defined the notion of an additive category with chain duality A, the associated algebraic bordism category Λ(A) (see his Example 3.3), and the corresponding L-groups L n (A) (see his Definition 1.8).In his notation, the assembly map is given by establishing a map of algebraic bordism categories (see his Proposition 9.11) and defining the assembly map to be the induced map on L-groups.However, one flaw in his argument is that he never provided a proof that (Z, X) -mod is an additive category with chain duality, despite his assertion in Proposition 5.1 of [Ran92].Our modest contribution to this saga is to provide a self-contained, conceptual, and geometric proof that (Z, X) -mod is an additive category with chain duality.
We are not the first to provide a proof of this result -one is given in Section 5 of [AM18].However, we found the proof and its notation rather dense.Another account of this result is given in a recent preprint of Frank Connolly [Con22].Although his aims are quite similar to ours, the approach is different, the reader may wish to compare.
We now outline our paper.In Section 2 we review Ranicki's notion of an additive category with chain duality, this is an additive category with a chain duality functor satisfying a chain homotopy equivalence condition.In Section 3 we fix a finite simplicial complex K (e.g. a triangulation of a compact manifold), and we define Ranicki's additive categories of K-based chain complexes.Here we need to warn the reader that we have deviated from Ranicki's notation in [Ran92], which we found difficult to use.A comparison between our notation and Ranicki's is given in Remark 14.The two key additive categories are Ch(Z(K) -mod) and Ch(Z(K op ) -mod).The latter category is the one whose L-theory gives the normal invariants, so is perhaps more important.The simplicial chain complex ∆K gives an object of Ch(Z(K) -mod) and the simplicial cochain complex ∆K − * gives an object of Ch(Z(K op ) -mod).More generally, given a CW-complex X with a K-dissection, the cellular chains C(X) give an object of Ch(Z(K) -mod) and given a CW-complex X with a K op -dissection, the cellular chains C(X) give an object of Ch(Z(K op ) -mod).We related this to dual cell decompositions, defined even when K is not a manifold.In Section 4, we develop homological algebra necessary for our proof that these categories admit a chain duality.Section 5 may be of independent interest.For a finite simplicial complex K, we define the dual cell decomposition DK which is a regular CW-complex refining the simplicial structure on K. Corollary 32 and Remark 35 say that this, in some sense, gives a two-sided bar resolution for the category of posets of K.
Finally, in Section 6 we define chain duality functors (T, τ ) on Ch(Z(K op ) -mod) and Ch(Z(K) -mod) and prove our main theorem.

Chain Duality
For a category K, write σ ∈ K when σ is an object of K and K(σ, τ ) for the set of morphisms from σ to τ .A preadditive category is a category where all morphism sets are abelian groups and composition is bilinear.An additive category is a preadditive category which admits finite products and coproducts.An example of an additive category is the category of finitely generated free abelian groups.
Let A be an additive category and let Ch(A) be the category of finite chain complexes over A where finite means that C n = 0 for all but a finite number of n.Homotopy notions make sense in this category: the notions of two chain maps being chain homotopic, a chain map being a chain homotopy equivalence, two chain complexes being chain homotopy equivalent, and a chain complex being contractible.The notion of homology of a chain complex over an additive category does not make sense.
Let Ch •,• (A) be the category of finite bigraded chain complexes over A. There are functors where Tot(C •,• ) n = p+q=n C p+q and Hom(C, D) p,q = A(C −p , D q ).(Throughout this paper, if the differentials are standard or can be easily determined, we omit them for readability).If C and D are finite chain complexes over an additive category A, then is a chain complex of abelian groups with differentials A 0-cycle is a chain map; the difference of chain maps is a boundary if and only if the chain maps are chain homotopic.In particular, there is a monomorphism of abelian groups Ch(A)(C, D) → Hom A (C, D) 0 .
If C and D are chain complexes of abelian groups, then there is a chain complex (C ⊗ D, d ⊗ ) with differentials  Proof.Suppose α : T U → V and β : V → W are chain maps.Then naturality of τ implies that It is also true, conversely, that an additive functor T : Ch(A) → Ch(A) op and natural transformation e : T 2 → Id satisfying e T C • T (e C ) = Id T C for all C ∈ Ch(A) determines a chain duality functor (T, τ ) where τ (f : T C → D) := e C • T (f ), but we omit the proof of this fact.Definition 5. A chain duality on an additive category A is a chain duality functor (T, τ ) so that e C : T 2 C → C is a chain homotopy equivalence for all C ∈ Ch(A).This is equivalent to the definition in Andrew Ranicki's book [Ran92, Definition 1.1].Notice that a chain duality functor does not necessarily give a chain duality, because of the extra condition that e C is a chain homotopy equivalence.We separately defined a chain duality functor because there can be uses for the weaker notion, for example, see the thesis of Christopher Palmer [Pal15].

K-based chain complexes
Let K be a finite set.
A morphism f : M → N of K-based abelian groups is simply a homomorphism of the underlying abelian groups M and N. Equivalently, it is a collection of homomorphisms In our exposition, we choose to work with M being an abelian group.However, everything we say (and everything Ranicki says in [Ran92]) generalizes to the context of R-modules where R is a ring with involution.
When the set K is a finite poset, we are interested in a subcategory of the K-based abelian groups.
Definition 7. Let K be a finite poset.
The objects of Z(K) -mod are the K-based abelian groups where M(σ) is a finitely generated free abelian group for each The slogan for morphisms is "bigger to smaller."Let K be a finite simplicial complex.There is an associated poset, also called K, whose objects are the simplices of K and whose morphisms are inclusions: σ ≤ τ means σ ⊆ τ .Our quintessential examples of a poset will be either K or K op .Our convention will be that σ ≤ τ means σ ≤ K τ and we will try and minimize the use of τ ≤ K op σ.
The simplicial chain complex ∆(K) ∈ Ch(Z(K) -mod) illustrates the bigger-to-smaller slogan.Here ∆(K Since duality is the fundamental feature of this paper, we introduce it immediately.
Definition 8. Let K be a finite poset.The duality functor * : is defined on objects by where M(σ) * = Hom Z (M(σ), Z).There is a natural isomorphism The duality functor and natural isomorphism extend to chain complexes This definition illustrates some of our notational conventions.We write C (and not C * ) to denote a chain complex.We use C − * (and not C * ) so that the dual is also a chain complex, whose differential has degree minus one.There are also sign conventions on the differential; we follow the sign conventions of Dold [Dol80] The simplicial cochain complex ∆K − * ∈ Ch(Z(K op ) -mod) of a finite simplicial complex illustrates the K op -slogan "smaller-to-bigger."Definition 9. Let K be a finite simplicial complex and let X be a finite CW-complex.
Remark 10.The K op -dissections described here are K-dissections in Ranicki's terminology.
Example 11.The geometric realization of a finite simplicial complex K has both a K-dissection given by the geometric realization of the simplices and a K op -dissection given by the dual cones of simplices.We describe the latter in order to fix notation.
Let K ′ be the barycentric subdivision of K.The vertices of K ′ are the barycenters σ i of the geometric realization of the simplices σ i ∈ K.An r-simplex in K ′ is given by a sequence σ 0 σ 1 . . .σ r where σ i < σ i+1 , and K ′ is a subdivision of K (see Chapter 3, Section 3 of Spanier [Spa81] for the definition of a subdivision); in particular there is a PL-homeomorphism |K ′ | → |K|.For σ ≤ τ ∈ K, the dual cell D τ σ is the union of the geometric realization of all simplices σ 0 σ 1 . . .σ r of the barycentric subdivision so that With K a 2-simplex, Figure 1 shows a K and K op dissection of the geometric realization of a 2-simplex.We will thus consider C n (Y, B) to be a subgroup of C n Y .More generally, given a CW-triple (Z, Y, B), we consider C n (Y, B) to be a subgroup of C n Z.
Corresponding assertions hold in the dual case.One defines Example 12.In Figure 2, we give examples of a K-dissection and a K op -dissection.
Note that if X is an n-dimensional manifold, then a map f : X → K ′ can be made transverse to the dual cells The notions of K and K op -dissections are special cases of the notion of a free C-CW-complex for a category C defined in Section 3 of [DL98].
4 The categorical point of view Z(K) -mod is an additive category, but to use homological algebra, one needs to embed it in an abelian category.In this section we develop this point of view.
Let K be a small category.
Recall that σ ∈ K means σ is an object of K and that K(σ, τ ) is the set of morphisms from σ to τ .Given a morphism α : σ → τ , let s α = σ, the source, and let t α = τ , the target.Let Mor K be the set of all morphisms in K.
Let Z[K] -mod be the category whose objects are functors M : K → Z -mod and whose morphisms are natural transformations.An example of a Z[K]-module is the trivial module Z K where Z K (σ) = Z for every object σ and Z K (α) = Id Z for every morphism α.The category Z[K] -mod is an abelian category.Its morphism sets are abelian groups.
for some σ ∈ K (the σ can vary and repeat in the direct sum).A basis for a Z[K]-module M is a collection of subsets B σ ⊂ M(σ) σ∈K so that any b ∈ M(τ ) for any τ can be expressed uniquely as b = M(α i )(b i ) for morphisms α i and basis elements b i .
We write Hom Z[K] (M, N) for the morphism set Z[K] -mod(M, N) given the structure of an abelian group as a subgroup of σ∈K Hom Z (M(σ), This is an abelian group.For an abelian group A, there is the adjoint isomorphism natural in M, N, and A. Yoneda's Lemma gives isomorphisms of abelian groups Here for a set S, Z[S] is the free abelian group with basis S (elements are where F * is exact, and F * takes projective objects to projective objects.They satisfy the adjoint property (see Lemma 1.9 of [DL98]).
In the special case c : K → 1 where 1 is the trivial category, We now generalize Definition 7.
Definition 13.Let K be a small category.Let Z(K) -mod be the category whose objects are K-based abelian groups with M a finitely generated free abelian group and whose morphisms f : M → N are a collection of homomorphisms of abelian groups {f α : M(t α ) → N(s α )} α∈Mor K , which are zero for all but a finite number of α.The composition law is given by An example of a Z(K)-module is given by Z τ where Every Z(K)-module is isomorphic to a finite direct sum of such modules.is given by the sum of all f ??→? where ?? →? satisfies ?? → σ = (?→ σ) • (?? →?).

An equivalent definition of [ ] is
The functor [ ] is a full, additive embedding.Note that there is a canonical identification Note that there is an embedding M(σ) ⊂ [M](σ), corresponding to the identity summand.[M] is a finitely generated free Z[K]-module; if {B σ } σ∈K is a collection of bases for the abelian groups M(σ), then {B σ } σ∈K is also a basis for the Remark 14.In the case where K is the category of simplices of a simplicial complex, here is the comparison of our notation with that of [Ran92]: Lemma 15.Let F : J → K be a functor.Then the functor F * : Z[J] -mod → Z[K] -mod restricts to the functor F * : Z(J) -mod → Z(K) -mod with Proof.It suffices to verify this for the module M = Z τ in which case it follows from Yoneda's Lemma.
Warning: the functor Applying the above lemma to the constant functor K → 1 gives the following corollary.

Corollary 16. The functor colim
In other words the colimit forgets the grading.
Example 17.Let K be a finite simplicial complex.Let {X(σ) | σ ∈ K} be a Kdissection (or K op -dissection) of a CW complex X.Then colim K [C(X)] = C(|X|), the usual cellular chain complex of the underlying CW-complex.And [C(X)](σ) = C(X(σ)), the cellular chain complex of X(σ) ⊂ X.A Z[K]-basis for C(X) is given by the (oriented) cells of X.
In particular, the augmentation gives a chain homotopy equivalence, where ∆K is the Z(K)-chain complex of the simplicial K-dissection of |K| and Z K is the Z[K]-chain complex given by placing Z in degree 0 at each object.Here ε(σ) = 1 for all 0-simplices of K. Thus [∆K] should be regarded as a finite free Z[K]-resolution of Z K .
Definition 18.Let F (Z[K] -mod) be the full subcategory of Z[K] -mod consisting of all modules M so that ⊕M(σ) is a finitely generated free abelian group.
Definition 19 (Round-square tensor products and Homs).There are additive functors where f : M → M ′ and g : N → N ′ are module morphisms and σ → τ is a morphism in K.
By additivity and the Tot construction, these functors extend to functors on chain complexes: C ⊗ Z(K] D and Hom Z(K] (C, D).A careful look at the definitions shows that Remark 20.We will be loose with the Z(K]-decorations in the following sense.If it is clearly labelled what M and N are, we will just write M ⊗ N or Hom(M, N).Conversely, if we write, for example, M ⊗ Z(K] N, then we will assume that M is a Z(K op )-module and N is a Z[K]-module.
Lemma 21.Hom and tensor products satisfy a myriad of identities; here are some we need: For example, we will prove part 1 for modules; then the verification for chain complexes will be clear.Let C = σ∈K C(σ) be a K-based abelian group and let D be a Z where the second and fourth equalities are due to Yoneda's Lemma.
Remark 22.The last three isomorphisms depend on the hypothesis that if M ∈ Z(K) -mod then the underlying module is finitely generated free.Without that hypothesis then one only gets a map from left to right.To remind the reader of this fact, we will often write The last two isomorphisms are called slant products.

Proposition 24. Let f : A → B be a map in Ch(Z(K) -mod). Then [f ] : [A] → [B] is a weak equivalence if and only if f is a chain homotopy equivalence.
Proof.Standard techniques from homological algebra show that, in an abelian category if the chain complexes are projective and bounded below, then a weak equivalence is a chain homotopy equivalence.Thus [f ] is a weak equivalence if and only if [f ] is a chain homotopy equivalence.But since the embedding of Z(K) -mod in Z[K] -mod is full, this occurs if and only if f is a chain homotopy equivalence.

This is proved by induction on the length of E.
The main result of this paper asserts that the additive categories of chain complexes Ch((Z(K op ) -mod) and Ch((Z(K) -mod) admit a chain duality for a finite simplicial complex K.The round-square tensor product is enough to construct the chain duality functor, but to prove that it is a chain duality we need to introduce the round tensor product and the category below.We discuss the geometric motivation in the next section.
For a simplicial complex K, let DK be the poset whose objects are 1-chains (σ ≤ τ ) and whose inequalities are (σ ≤ τ ) ≤ (σ ′ ≤ τ ′ ) if and only if σ ′ ≤ σ ≤ τ ≤ τ ′ .(A 1-chain is simply a pair of simplices σ, τ ∈ K with σ a face of τ .)There are morphisms of posets In figure 1, we see an example of the geometric realization of the posets DK (center of the figure), K, and K op (lower and upper right side respectively) when K is a 2simplex.
Definition 26 (Round tensor products and Homs).We have the following additive functors 1. over Z(K), By additivity and the Tot construction, these functors extend to functors on chain complexes: In the formula for e C : T 2 C → C, the Z(DK)-chain complex ∆K − * ⊗ Z(K) ∆K arises.We need a weak equivalence similar to the augmentation of Example 17.
The (oriented) simplices {σ} give a basis for ∆K and the dual basis { σ} gives a basis for ∆K − * .A basis of ∆K − * ⊗ Z(K) ∆K is given by { σ ⊗ τ }, where σ is a face of τ .Define ε( σ ⊗ σ) = 1 for any simplex σ and ε( σ ⊗ τ ) = 0 if σ is a proper face of τ .To verify that ε is a chain map, we need to show that if dim Thus ε is a chain map.We provide a proof that ε is a chain equivalence with the help of the geometry of the dual cell decomposition in the next section.
Note that similarly, a Z[DK op ]-basis of [∆K ⊗ Z(K op ) ∆K − * ] is given by τ ⊗ σ, where σ is a face of τ .We define ε(σ ⊗ σ) = (−1) |σ| for any simplex σ and ε(τ ⊗ σ) = 0 if σ is a proper face of τ .To verify that ε is a chain map, we need to show that if is a chain map.

The dual cell decomposition
A regular CW-complex is a CW-complex where the closure of each open n-cell is homeomorphic to an n-disk.We denote a regular CW-complex X by a pair (|X|, {σ n }) where |X| is a topological space and each σ n is a (closed) n-cell in |X|, and (|X|, {σ n }) is a subdivision of (|X|, {τ n }) if every cell of the former is a subset of a cell of the latter: σ n ⊂ τ n .Regular CW-complexes are closely related to simplicial complexes.A simplicial complex is a regular CW-complex (with simplices as cells) and every regular CWcomplex has a simplicial subdivision.
Let K = (|K|, {σ}) be a simplicial complex.We will define the dual cell subdivision DK = (|K|, {D τ σ}); its poset of cells will be the category DK mentioned earlier.The dual cell decomposition is a regular CW-complex intermediary between K and its barycentric subdivision K ′ .The utility of the dual cell decomposition is that it is useful for the proof that the category Ch(Z(K op ) -mod) admits a chain duality.
We will first remind the reader of the barycentric subdivision Here the σ i are simplices of K satisfying σ 0 < σ 1 < • • • < σ n , σ is the barycenter of σ, and σ 0 σ 1 . . .σ n is the convex hull of these barycenters.For σ ≤ τ ∈ K, the dual cell D τ σ is the union of all simplices σ 0 σ 1 . . .σ r of the barycentric subdivision so that σ ≤ σ Proof.Please refer to the Figure 3 while reading the proof.Here τ is a 2-simplex and σ is the lower left vertex.
Let σ C be the subset of τ spanned by the vertices of τ which are not vertices of σ (in the example above σ C is the blue line segment).Then σ C is a simplex of dimension dim τ − dim σ − 1.Let σ * σ C be the union of all line segments between σ and a point in σ C .Then σ * σ C is homeomorphic to the cone on σ C , and hence is homeomorphic to D dim τ −dim σ .
A radial argument using line segments starting at σ shows that Here are some details.Embed τ isometrically into R n with σ mapping to the origin.
where Cone(L) is a proper cone if L is compact, 0 ∈ L, and every ray starting at the origin intersects L in at most one point.Let S(L) = {x/ x | x ∈ L}.Two proper cones Cone(L) and Cone(M) are commensurate if every ray starting at the origin intersects L if and only if it intersects M, equivalently, S(L) = S(M).As an example, If σ is placed at the origin, D τ σ = Cone(L τ σ) and Cone(σ C ) are commensurate.
Proof.A finite regular CW-complex is a compact Hausdorff space covered by a finite collection of closed cells, whose interiors partition the space, and so that the boundary of each cell is a union of closed cells.These conditions are satisfied, since For a regular CW-complex Z, one can define a Z-dissection and a Z op -dissection, as in Definition 9. Similar to Example 11, the regular CW-complex DK has a DKdissection.Thus the chain complex C(DK) is a Z(DK)-chain complex.
An orientation on an n-cell σ of a regular CW -complex is a choice of generator of the infinite cyclic group H n (σ, ∂σ).We require that all 0-cells are positively oriented, in other words the generator is the unique singular 0-chain with image the 0-cell.An oriented regular CW -complex is a CW-complex with an orientation for each cell.We will abuse notation and, in an oriented CW-complex X, use σ to denote an n-cell, the generator of H n (σ, ∂σ), and the image element in C n (X).The cellular chains of an oriented regular CW -complex are based.In a regular CW -complex we will use the notation σ ′ ≺ σ to mean that σ ′ is a codimension one face of σ, that is, σ ′ and σ are both cells, σ ′ ⊂ ∂σ, and 1 + dim σ ′ = dim σ.If (X, {σ n }) is a oriented regular CWcomplex, and, in the cellular chain complex ∂σ = σ ′ ≺σ [σ : σ ′ ]σ ′ , then the coefficients [σ : σ ′ ] are called incidence numbers; note that [σ : σ ′ ] = ±1.
We wish to orient the dual cell decomposition DK of |K| with an orientation which only depends on the orientation of K. To quote William Massey (see page 243 of [Mas91]): "we can specify orientations for the cells of a regular CW -complex by specifying a set of incidence numbers for the complex.This is one of the most convenient ways of specifying orientations of cells."The following theorem is essentially Theorem 7.2, Chapter IX of [Mas91]: Theorem 29.Let X be a regular CW-complex.For each pair (τ n , σ n−1 ) consisting of an n-cell and an (n − 1)-cell of X, let there be given an integer α n τ σ = 0 or ±1 such that the following four conditions hold: (1) If σ n−1 is not a face of τ n , then α n τ σ = 0.
(3) If σ 0 and ρ 0 are the two vertices of the 1-cell τ 1 , then (4) Let τ n and σ n−2 be cells of X so that σ n−2 ⊂ τ n ; let ρ n−1 and ν n−1 be the unique (n − 1)-cells so that σ n−2 ⊂ ρ n−1 ⊂ τ n and σ n−2 ⊂ ν n−1 ⊂ τ n .Then Under these assumptions, there exists a unique choice of orientations for the cells so that in the cellular chain complex C(X) Thus the α's not only determine orientations on all the cells, they identify the cellular chain complex C(X).
This orients DK and shows that the map is an isomorphism of chain complexes.
Example 31.Using the assumptions in Theorem 29, figure 4 shows the orientations of the dual cells in a 2-simplex.
Figure 4: Orientation of dual cells in a 2-simplex Proof.The previous theorem shows that Lemma 27 shows that these chain complexes have the homology of a point.Since ε(σ ≤ τ ) is onto the result follows.
Lemma 33.There is an isomorphism of chain complexes Proof.First note that in this statement we are using the fact that DK ∼ = DK op with (σ ≤ τ ) ↔ (τ ≥ σ).
The isomorphism as graded groups is clear.We need to check that the map p commutes with differentials.Proof.This follows from Lemma 33 and Corollary 32.
Remark 35.Davis and Lück's [DL98, Lemma 3.17] gives, for an arbitrary category C, a free C op × C-CW-approximation of the discrete C op × C-space C(?, ??).This CWapproximation is foundational for homotopy (co)limits, and is essential for a 2-sided bar resolution.The material in this section shows that there is a much smaller model for this CW-approximation in the case where K is the poset of a finite simplicial complex, namely one can use the K op × K-space given by (σ, τ ) → D τ σ σ ≤ τ ∅ otherwise

K-based chain duality
Recall our main theorem: Ch((Z(K op ) -mod) Ch((Z(K) -mod) admits the structure (T, τ ) of a category with chain duality.Note that in what follows we will give more details for the results in the Z(K op ) -mod category.The reason for this choice is that the Z(K op ) -mod category is more relevant for applications.We now define T .
Definition 36.Let K be a finite simplicial complex.Define 1.The T functor for Ch(Z(K op ) -mod) as 2. The T functor for Ch(Z(K) -mod) as To give credence to the assertion that this is a form of duality, we pause for the following proposition, which asserts that T C is a K-based dual of the result of forgetting the K-based structure of C.
Proposition 37. The map ] is a chain homotopy equivalence.Thus the dual of the underlying chain complex of C is chain homotopy equivalent to the underlying chain complex of T C.

Proof. The augmentation
gives a weak equivalence, and hence by Proposition 25 is a weak equivalence.The chains in both the domain and codomain are free abelian groups, so this map is a chain homotopy equivalence.Thus so is its dual, By Lemma 21 parts (1) and (4), Proof.We give the proof for C, D ∈ Ch(Z(K op ) -mod) as follows: The proof with C, D ∈ Ch(Z(K) -mod) is similar.The only difference is to use the appropriate definition for T C if C is a Z(K) -mod chain complex.
By taking colimits, and restricting to 0-cycles, we obtain the following corollary.
Ranicki's book [Ran92] does prove this corollary.However, what is missing is a proof that e C is a chain homotopy equivalence, i.e. this chain duality functor produces a chain duality.By Lemma 4, there is a natural Our next task will be to show that the chain duality functor is a chain duality, i.e. that e C is a chain homotopy equivalence.But first we need a formula for e C .In the following theorem we work with Z(K op ) -mod chain complexes and use the triple tensor product.The corresponding statement for Z(K) -mod is very similar except for some details with the signs involved in the vertical isomorphism.We give the formulas in this case in Theorem 42 below.
To deal with T 2 C we have to make use of one more construction, which will allow for a convenient coordinate change.
Definition 40.Let K be a poset.Let C be a Z(K op )-chain complex, D be a Z(K)chain complex, and E be a Z[K op ]-chain complex.The triple tensor product is the Z(K op )-chain complex (see Lemma 15 and Lemma 21.1) and for c Theorem 41.The following assertions are valid.
1.There are an isomorphism of Z[K op ]-chain complexes given by where  and that e C is represented by the image of Id T C under τ (after taking colimits and 0-cycles).
However, to incorporate the triple tensor product, we factor the last isomorphism as follows:

Definition 2 .
A chain duality functor (T, τ ) on an additive category A is an additive functor T : Ch(A) → Ch(A) op together with a natural chain map τ C,D : Hom A (T C, D) → Hom A (T D, C) defined for each pair of chain complexes C, D ∈ Ch(A) so that τ 2 = Id in the sense that τ D,C • τ C,D = Id.Remark 3. By restricting to 0-cycles, the natural chain map τ induces a natural isomorphism of abelian groups τ C,D : Ch(A)(T C, D) → Ch(A)(T D, C).

Lemma 4 .
Let (T, τ ) be a chain duality functor on A. For C ∈ Ch(A), let e C : T 2 C → C be τ (Id T C ).This defines a natural transformation e : T 2 → Id : Ch(A) → Ch(A) so that for each object C ∈ Ch(A), e T C • T (e C ) = Id T C : T C → T 3 C → T C.