Chain duality for categories over complexes

Kervaire and Milnor [KM] 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 [Sullivan-thesis] investigated the obstruction theory for deforming a homotopy equivalence to a homeomorphism. In seminar notes [Sullivan-Notes] 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 [KS].). It was extended to the nonsimply-connected case and to the case of compact manifolds by Wall [Wall]. The surgery exact sequence for a closed $n$-dimensional manifold $X$ with $n\geq 5$ is

The object one wants to compute is the structure set $\mathbb{S}^{TOP}(X)$, 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 $\sigma$, including its domain, the normal invariants ${\cal N}^{TOP}(X)$, 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 ${\cal N}^{TOP}(X)\cong[X,G/TOP]$. He also computed the homotopy groups $\pi_{i}(G/TOP)$ which vanish for $i$ odd, have order 2 for $i\equiv 2\pmod{4}$, and are infinite cyclic for $i\equiv 0\pmod{4}$ (this follows from the generalized Poincaré conjecture). Furthermore, Sullivan analyzed the homotopy type of $G/TOP$ and established Sullivan periodicity, $\Omega^{4}({\mathbb{Z}}\times G/TOP)\simeq{\mathbb{Z}}\times G/TOP$. This then determines an $\Omega$-spectrum $\mathbb{L}_{.}$ and its 1-connective cover $\mathbb{L}_{.}\langle 1\rangle$. There is a formal identification $[X,G/TOP]=H^{0}(X;\mathbb{L}_{.}\langle 1\rangle)$.

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 [Quinn] vision (largely carried out by Ranicki [bluebook], see also [KMM]) 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 $\cong$ 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 $A\langle 1\rangle$ 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 $X=B\pi$ with $\pi$ 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 [WW], also [DL98]). Ranicki’s version of assembly, needed for his approach to the above diagram, was motivated by his earlier work with Weiss [RW] viewing the assembly maps as a passage from local to global Poincaré duality. Much earlier Ranicki [ats1] reinterpreted Wall’s algebraic $L$-groups as bordism groups of algebraic Poincaré complexes over the group ring ${\mathbb{Z}}[\pi_{1}X]$. 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 [bluebook] defined the notion of an additive category with chain duality ${\mathbb{A}}$, the associated algebraic bordism category $\Lambda({\mathbb{A}})$ (see his Example 3.3), and the corresponding $L$-groups $L_{n}({\mathbb{A}})$ (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 $({\mathbb{Z}},X)\operatorname{-mod}$ is an additive category with chain duality, despite his assertion in Proposition 5.1 of [bluebook]. Our modest contribution to this saga is to provide a self-contained, conceptual, and geometric proof that $({\mathbb{Z}},X)\operatorname{-mod}$ 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 [Spiros-Tibor]. However, we found the proof and its notation rather dense. Another account of this result is given in a recent preprint of Frank Connolly [Frank]. 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 [bluebook], which we found difficult to use. A comparison between our notation and Ranicki’s is given in Remark LABEL:R_notation. The two key additive categories are $\operatorname{{Ch}}({\mathbb{Z}}(K)\operatorname{-mod})$ and $\operatorname{{Ch}}({\mathbb{Z}}(K^{\operatorname{{op}}})\operatorname{-mod})$. The latter category is the one whose $L$-theory gives the normal invariants, so is perhaps more important. The simplicial chain complex $\Delta K$ gives an object of $\operatorname{{Ch}}({\mathbb{Z}}(K)\operatorname{-mod})$ and the simplicial cochain complex $\Delta K^{-*}$ gives an object of $\operatorname{{Ch}}({\mathbb{Z}}(K^{\operatorname{{op}}})\operatorname{-mod})$. More generally, given a CW-complex $X$ with a $K$-dissection, the cellular chains $C(X)$ give an object of $\operatorname{{Ch}}({\mathbb{Z}}(K)\operatorname{-mod})$ and given a CW-complex $X$ with a $K^{\operatorname{{op}}}$-dissection, the cellular chains $C(X)$ give an object of $\operatorname{{Ch}}({\mathbb{Z}}(K^{\operatorname{{op}}})\operatorname{-mod})$. We related this to dual cell decompositions, defined even when $K$ is not a manifold. In Section LABEL:sec:cat_point_of_view, we develop homological algebra necessary for our proof that these categories admit a chain duality.

Section LABEL:sec:dual_cell 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 LABEL:varepsilon_is_a_weak_eq and Remark LABEL:two-sided say that this, in some sense, gives a two-sided bar resolution for the category of posets of $K$.

Finally, in Section LABEL:K-based_chain_duality we define chain duality functors $(T,\tau)$ on $\operatorname{{Ch}}({\mathbb{Z}}(K^{\operatorname{{op}}})\operatorname{-mod})$ and $\operatorname{{Ch}}({\mathbb{Z}}(K)\operatorname{-mod})$ and prove our main theorem.

For a category $K$, write $\sigma\in K$ when $\sigma$ is an object of $K$ and $K(\sigma,\tau)$ for the set of morphisms from $\sigma$ to $\tau$. 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 ${\mathbb{A}}$ be an additive category and let $\operatorname{{Ch}}({\mathbb{A}})$ be the category of finite chain complexes over ${\mathbb{A}}$ 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 $\operatorname{{Ch}}_{\bullet,\bullet}({\mathbb{A}})$ be the category of finite bigraded chain complexes over ${\mathbb{A}}$. There are functors

where $\operatorname{{Tot}}(C_{\bullet,\bullet})_{n}=\bigoplus_{p+q=n}C_{p+q}$ and $\operatorname{{Hom}}(C,D)_{p,q}={\mathbb{A}}(C_{-p},D_{q})$. (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 ${\mathbb{A}}$, 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 $\operatorname{{Ch}}({\mathbb{A}})(C,D)\to\operatorname{{Hom}}_{{\mathbb{A}}}(C,D)_{0}$.

If $C$ and $D$ are chain complexes of abelian groups, then there is a chain complex $(C\otimes D,d_{\otimes})$ with differentials

It is also true, conversely, that an additive functor $T:\operatorname{{Ch}}({\mathbb{A}})\to\operatorname{{Ch}}({\mathbb{A}})^{\operatorname{{op}}}$ and natural transformation $e:T^{2}\to\operatorname{Id}$ satisfying $e_{TC}\circ T(e_{C})=\operatorname{Id}_{TC}$ for all $C\in\operatorname{{Ch}}({\mathbb{A}})$ determines a chain duality functor $(T,\tau)$ where $\tau(f:TC\to D):=e_{C}\circ T(f)$, but we omit the proof of this fact.

This is equivalent to the definition in Andrew Ranicki’s book [bluebook, 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 [P15].

Let $K$ be a finite set.

In our exposition, we choose to work with $M$ being an abelian group. However, everything we say (and everything Ranicki says in [bluebook]) 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.

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: $\sigma\leq\tau$ means $\sigma\subseteq\tau$. Our quintessential examples of a poset will be either $K$ or $K^{\operatorname{{op}}}$. Our convention will be that $\sigma\leq\tau$ means $\sigma\leq_{K}\tau$ and we will try and minimize the use of $\tau\leq_{K^{\operatorname{{op}}}}\sigma$. The simplicial chain complex $\Delta(K)\in\operatorname{{Ch}}({\mathbb{Z}}(K)\operatorname{-mod})$ illustrates the bigger-to-smaller slogan. Here $\Delta(K)_{n}=\oplus_{\sigma\in K^{n}}\Delta(K)_{n}(\sigma)$, with

Since duality is the fundamental feature of this paper, we introduce it immediately.

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 [Dold]: the differential $(C_{-*})_{n+1}\to(C_{-*})_{n}$ is given by $(-1)^{n+1}(\partial_{-n})^{*}$.

The simplicial cochain complex $\Delta K^{-*}\in\operatorname{{Ch}}({\mathbb{Z}}(K^{\operatorname{{op}}})\operatorname{-mod})$ of a finite simplicial complex illustrates the $K^{\operatorname{{op}}}$-slogan “smaller-to-bigger.”

Here $\sigma\cup\rho$ is the smallest simplex of $K$ which contains $\sigma$ and $\rho$, if it exists. Note that in a $K$-dissection $\sigma\leq\tau$ implies that $X(\sigma)\subset X(\tau)$, while in a $K^{\operatorname{{op}}}$-dissection, $\sigma\leq\tau$ implies that $X(\tau)\subset X(\sigma)$.