1. Introduction; description of results

This article is an addition to a theory of blocked surgery, pioneered by Ranicki, augmented by others in [Reference Adams-Florou and Macko1, Reference Connolly, Davis and Khan4, Reference Connolly, Davis and Khan5, Reference Connolly6, Reference Davis and Rovi8, Reference Farrell and Jones9, Reference Ranicki16, Reference Ranicki and Weiss17], and still in a developing state.

Let $R$ be a commutative ring; let $K$ be a finite simplicial complex. In [Reference Ranicki16] Ranicki introduced the category of $(R,K)$ chain complexes and chain maps denoted $\mathcal {B} R_K$ here. He also defined algebraically, a contravariant functor $T:\mathcal {B} R_K\to \mathcal {B} R_K$.Footnote ¹

The simplest geometric example of an $(R,K)$ chain complex arises from a

$K$-space $(X,\pi )$. This is a finite simplicial complex $X$ and a simplicial map,

$\pi :X\to K$. In that case, the simplicial cochains on $X$ (with $R$ coefficients) form an $(R,K)$ chain complex denoted $\Delta ^*X$.

At the same time, $(X,\pi )$ specifies a regular CW complex $X_K$, which is a (non-simplicial) subdivision of $X$. We show that the cellular chain complex (with $R$ coefficients) of $X_K$ forms a second $(R,K)$ chain complex $C(X_K)$.

Our main theorem, theorem 8.1, exhibits a geometrically defined chain isomorphism between $C(X_K)$ and $T\Delta ^*X$. Roughly put:

\[ T\Delta^*X= C(X_K). \]

It is also our aim to give a transparent definition of this duality functor $T$, a clear treatment of Ranicki's natural transformation $e:T^2\to id$. and a simple proof that $e_C:T^2C\to C$ is an $(R,K)$ chain equivalence for all $C$.

Our larger goal is to facilitate applications of Ranicki's theory to geometric questions such as the topological rigidity of non-positively curved groups as in [Reference Connolly, Davis and Khan4, Reference Connolly, Davis and Khan5, Reference Farrell and Jones9, Reference Farrell and Jones10] when those groups have elements of finite order.

The vehicle for such applications would be a full blown $K$-blocked surgery theory of which there are only hints in [Reference Ranicki16]. This would start with a degree-one normal map between closed manifolds, $(f, b): (M,\nu (M))\to (X,\xi )$ (as in [Reference Browder2]) together with a reference map, $\pi :X\to K$ as above. It would seek an $L$-theoretic obstruction to finding a normal cobordism of $(f,b)$ to a ‘K-blocked homotopy equivalence,’ $M'\to X$. But we will not pursue this here or even define the terms precisely.

In the classical case ($K=$ point; [Reference Browder2, Reference Kervaire and Milnor12, Reference Ranicki18, Reference Wall19, Reference Wall20]) one has the ‘surgery obstruction’ $\sigma _*(f,b)\in L_n(\mathbb {Z}[\pi _1(X)])$ to such a normal cobordism. But this functor $L_n()$, was generalized in [Reference Ranicki16] to yield obstruction groups $L_n(\mathcal {A})$ for any ‘category-with-chain-involution’ $(A, *, \epsilon )$. Here $A$ is an additive category, $\mathcal {B} A\mathop {\to }\limits ^{*} \mathcal {B} A$ is a contravariant functor satisfying certain conditions, on the category $\mathcal {B} A$, of finite chain complexes in $A$, and $\epsilon :(*)^2\to id$, is an equivalence in the homotopy category of $\mathcal {B} A$.

Ranicki, in [Reference Ranicki16], then starts with a finite complex $K$ and a category with chain involution, $\mathcal {A}= (A, *, \varepsilon )$ as above. He then constructs the additive category $A_K$ of $K$-blocked objects from $A$, and $K$-blocked $A$-maps. From $*$, and $\varepsilon$, he defines the Ranicki Duality Functor $T:\mathcal {B}(A_K)\to \mathcal {B}(A_K)$, and the natural transformation $e: T^2\to id_{\mathcal {B}(A_K)}$. This construction allows one to define the surgery obstruction groups, $L_n(\mathcal {A}_K)$ where $\mathcal {A}_K =(A_K, T, e)$.

This seems to apply directly to a $K$-blocked normal map, $M^n\mathop {\longrightarrow }\limits ^{(f,b)} X^n\mathop {\to }\limits ^{\pi } K$. Here the relevant category seems to be $A=A(R)$, the category of finitely generated free modules over a fixed commutative ring $R$. We write $A R_K$ for $(A R)_K$ and $\mathcal {B} R_K$ for $\mathcal {B}(A R_K)$. Its objects are $(R,K)$-chain complexes. So the simplicial cochain complexes of $X$ and $M$ denoted $\Delta ^*X$ and $\Delta ^*M$, and the simplicial chain complexes, $\Delta X'$ and $\Delta M'$, are $(R,K)$-chain complexes. (See § 3). Thus the $L$-groups of $(\mathcal {A} R_K, T, e)$ seem likely to be useful.

However, Ranicki's definition of $\mathcal {B} R_K \mathop {\longrightarrow }\limits ^{T}\mathcal {B} R_K$ was only a starting point. Indeed his assertion in [Reference Ranicki16] of the crucial theorem that $(\mathcal {A} R_K, T, e)$ is a category with chain involution was only proved in 2018 (by Adams-Florou and Macko, [Reference Adams-Florou and Macko1]).

This paper interprets Ranicki's notions geometrically. Section 2 fixes chain-complex conventions. Section 3 reviews Ranicki's concepts concerning $(R,K)$ complexes while attempting to simplify notation. In § 5 we introduce the $(R,K)$ chain complex $C\otimes _KD$, defined if $D$ is an $(R,K)$ complex and $C$ is an $(R,K^{op})$ complex. This complex $C\otimes _KD$ is a certain quotient of $C\otimes _RD$.

Our definition (see 6.1) of the Ranicki dual $TC$, of an $(R,K)$ complex $C$, is:

\[ TC=C^*\otimes_K \Delta^*K. \]

In § 7 we show, using work of M. Cohen [Reference Cohen3], that each $K$-space $(X,\pi )$ defines a certain regular CW-complex $X_K$, whose cellular chain complex has a natural $(R, K)$ structure. Therefore from each $K$-space $(X,\pi )$ we obtain three $(R,K)$ chain complexes:

1. (1) $\Delta ^*X$, the simplicial cochain complex of $X$ (definition 4.1).

2. (2) $C(X_K)$, the cellular chain complex of the CW complex $X_K$ (§ 8).

3. (3) $\Delta X'$, the simplicial chain complex of $X'$, the barycentric subdivision of $X$ (definition 7.2).

This paper shows that these three are closely related by $T$. Our main result, theorem 8.1, exhibits an isomorphism of $(R,K)$ chain complexes:

\[ \Phi_X: T\Delta^*X\cong C(X_K) \]

Then, using [Reference McCrory13], we prove there are $(R,K)$ chain homotopy equivalences:

\[ T\Delta X'\simeq \Delta^*X;\quad C(X_K)\simeq \Delta X'. \]

When $X$ is a pl-manifold, and $C=C(X_K)$, Poincare duality then becomes an $n$-cycle in the $(R,K^{\mathrm {op}})$ complex, $Hom_{(R,K)}(TC, C)$.

This CW complex $X_K$ is a subdivision of $X$, and $X'$ is a simplicial subdivision of $X_K$. In fact, for each simplex $S$ of $X$ and each face $\sigma$ of $\pi (S)\in K$, there is a single cell $S_\sigma$ of $X_K$. Specifically, if $D(\sigma,\pi (S))$ is the dual cell of $\sigma$ in $\pi (S)$:

\[ S_\sigma=(\pi\mid S)^{{-}1}|D(\sigma,\pi(S))| \; . \]

The author is indebted to Jim Davis for his helpful comments. He also wants to acknowledge, with thanks, the referee's suggestions for improving the text.

2. Chain complex conventions

Throughout this paper, $R$ denotes a fixed commutative ring; $AR$ is the additive category of finitely generated free $R$ modules.

For any additive category $A$ we will write $\mathcal {B} A$ for the additive category of finite chain complexes, $C=\{C_q, \partial _q\}_{q\in \mathbb {Z}}$ and chain maps $f=\{f_q:C_q\to D_q\}_{q\in \mathbb {Z}}$ from $A$. (Finite means: $C_q=0$ for all but finitely many $q$). We abbreviate $\mathcal {B} (AR)$ to $\mathcal {B} R$.

As usual two chain maps $f,g: C\to D$ are chain homotopic if there is a sequence of $A$ maps, $h=\{h_q:C_q\to D_{q+1}\}$, for which $d^D_{q+1}h_q + h_{q-1}d^C_q = g_q-f_q \ \forall q$.

We regard $A$ as the full subcategory of $\mathcal {B} A$ consisting of chain complexes concentrated in degree zero.

Let $C,D\in Ob(\mathcal {B} R)$. The complexes $C\otimes _R D$, and $Hom_R(C,D)$ in $Ob(\mathcal {B} R)$, are:

\begin{align*} \boxed{(C\otimes_R D)_q = \sum_{r\in\mathbb{Z}} C_r\otimes_R D_{q-r};\quad Hom_R(C,D)_q =\sum_{r\in \mathbb{Z}}Hom_R(C_r, D_{q+r}) }\ \text{ and:}\\ \boxed{ d^{C\otimes D}(x\otimes y)= d^Cx\otimes y +({-}1)^{|x|} x\otimes d^Dy;\quad d^{Hom}\phi =d^D\circ \phi - ({-}1)^{|\phi| } \phi\circ d^C } \end{align*}

The evaluation map, $eval_{C,D}:Hom_R(C,D)\otimes _R C\longrightarrow D$ is the R-chain map:

\[ eval_{C,D}(f\otimes x)= f(x). \]

Note that $eval_{R,D}:Hom_R(R,D)\otimes _R R\cong D$.

Write $ev_C:C^*\otimes C\to R$ for $eval_{C,R}$.

The contravariant functor $\mathcal {B} R\mathop {\to }\limits ^{*} \mathcal {B} R$ is : $C^*= Hom(C,R); \; f^* = Hom(f, 1_R)$.

Therefore we have:

\[ (C^*)_{{-}q} = Hom_R(C_q,R); \quad d^{C^*}_{{-}q} = ({-}1)^{q+1}(d^C_{q+1})^*:(C^*)_{{-}q}\to (C^*)_{{-}q-1}. \]

The functor $*$ comes with a natural equivalence, $\varepsilon : (*)^2\to 1_{\mathcal {B} R}$. Specifically, the chain isomorphism $\epsilon _C: C^{**}\to C$ is characterized by the identity:

\[ a(\varepsilon_C(\alpha)) = ({-}1)^q \alpha(a) \quad \forall \alpha\in (C^{**})_{q},\; a\in (C^*)_{{-}q} . \]

3. Basic definitions for $(R,K)$ chain complexes

4. $K$ spaces and their chain complexes

For the rest of this paper, $K$ denotes a finite simplicial complex.

A simplicial complex $K$ is a poset so the above definitions apply. In this case $\sigma \leq \tau$ means that the simplex $\sigma$ is a face (not necessarily proper) of the simplex $\tau$.

$\Delta _*(K;R)=\{ \Delta _q(K;R),\partial _q\}_{q\in \mathbb {Z}}$ denotes the simplicial chain complex of $K$.

$\Delta ^*(K;R)=Hom_R(\Delta _*(X;R), R)$ denotes the simplicial cochain complex of $K$.

One can choose a basis, $bK$ for $\Delta _*(K;R)$ consisting of one oriented $q$-simplex, $\sigma =\langle v_0\dots, v_q\rangle \in \Delta _q(K;R)$ for each $q$-simplex with vertices $v_0,\dots v_q$, of $K$. Recall: $\langle v_0,\dots,v_q\rangle = sgn(\pi )\langle v_{\pi (0)},\dots,v_{\pi (q)}\rangle$ for each $\pi \in S_{q+1}$. The oriented $q$-simplex $\sigma \in \Delta _q(K;R)$ defines a dual cochain $\sigma ^*\in \Delta ^*(K;R)_{-q}$ such that $\sigma ^*(\tau ) =0$ for all $\tau \neq \pm \sigma$, and $\sigma ^*(\sigma )=1$.

One then defines $\sigma ^{**}\in \Delta _q(K;R)^{**}$ by: $\varepsilon (\sigma ^{**}) = \sigma$.

Each simplex $\sigma \in K$ defines subcomplexes, $\overline {\sigma }$ and $\partial \sigma$, and a subset $st(\sigma )$:

\[ \overline{\sigma}=\{\tau\in K\mid \tau\leq \sigma\}; \quad \partial\sigma=\{\tau\in K\mid \tau< \sigma\}; \quad st(\sigma)=\{\tau\in K\mid \tau\geq \sigma\} \]

The incidence number $[\tau, \sigma ]\in \{1,-1, 0\}$ is defined for any oriented simplices $\sigma, \tau$ of $K$. It satisfies: $\partial _q(\sigma )= \sum _{\tau \in bK} [\sigma, \tau ] \tau$ for any basis, $bK$ of oriented simplices of $K$. $[\sigma,\tau ]\neq 0$ iff $\tau$ is a codimension-one face of $\sigma$.

The next lemma will be used in § 6.

5. $C\otimes _K D$ and the isomorphism $Hom_{(R,K)}(D, C^*) \cong (C\otimes _K D)^*$

Throughout this section, $C$ denotes an $(R,K^{op})$ complex and $D$ denotes an $(R,K)$ complex.

We will first define two $(R,K)$ complexes: $C\otimes _R D$ and a quotient of this, ${C\otimes _K D}$.

In $K$, the star of any simplex , $st(\sigma )$, as well as $K-st(\sigma )$ are full in $K$. Moreover the chain complex $C(K-st(\sigma ))$ is a subcomplex of $C(K)$ and $C(st(\sigma ))$ is a quotient complex. These fit into a short exact sequence of chain maps in $\mathcal {B} R$:

\[ 0\to C(K-st(\sigma))\mathop{\longrightarrow}\limits^{i_\sigma} C(K)\mathop{\longrightarrow}\limits^{p_{st(\sigma)}}C(st(\sigma)) \to 0 \]

Here $C(K)\mathop {\longrightarrow }\limits ^{p_{st(\sigma )}}C(st(\sigma ))$ is defined by: $p_{st(\sigma )}|_{C_q(st(\sigma ))}= 1_ {C_q(st(\sigma ))}$; and $p_{st(\sigma )}|_{C(K-st(\sigma ))}=0$.

(For the $(R,K)$ complex $D$, we get $0\to D(st(\sigma ))\to D(K)\to D(K- st(\sigma )){\to 0}$).

Definition 5.1 ($C\otimes _KD$, $C\otimes _RD$, and $C\otimes _RD \mathop {\longrightarrow }\limits ^{\pi _{C,D}}C\otimes _K D$).

Let $C$ be an $(R,K^{op})$ complex and $D$ be an $(R,K)$ complex.

1. (1) Let $C\otimes _RD$ be the $(R,K)$ complex for which:

   \begin{align*} & (C\otimes_RD)(K)= C(K)\otimes_R D(K); \\ & (C\otimes_R D)_q(\rho)= (C(K)\otimes_R D(\rho))_q \quad \forall \rho\in K, q\in \mathbb{Z} \end{align*}

2. (2) Let $C\otimes _KD$ be the $(R,K)$ complex for which:

   1. (a) $(C\otimes _K D)_q(K)= \sum _{\rho \in K} (C(st(\rho ))\otimes _R D(\rho ))_q \ \forall q\in \mathbb {Z}$

   2. (b) $(C\otimes _K D)(\rho )= C(st(\rho ))\otimes _R D(\rho ) \ \forall \;\rho \in K$

   3. (c) The map $C\otimes _RD\mathop {\longrightarrow }\limits ^{\pi _{C,D}} C\otimes _KD$ is an $(R,K)$ chain epimorphism, if we define $\pi _{C,D}$ by requiring that $\pi _{C,D}(\sigma,\rho )=0$ for $\sigma \neq \rho$ and:

      \[ \pi_{C,D}(\rho, \rho)= p_{st(\rho)}\otimes_R 1_{D(\rho) }:C(K)\otimes_R D(\rho) \longrightarrow C(st(\tau))\otimes _R D(\rho). \]

Expicitly, for any $\rho \leq \tau$ and $x\otimes _Ry\in C_r(\tau )\otimes _RD_{q-r}(\rho )\subset (C\otimes _KD)_q(\rho )$, we have

(5.1)\begin{equation} \boxed{ d^{C\otimes_K D}(x\otimes y) = \sum_{\{\sigma| \rho\leq \sigma\leq\tau\} } d^C(\sigma,\tau)x \otimes y + ({-}1)^r x\otimes d^D(\sigma, \rho) y. } \end{equation}

We now show that $(C\otimes _KD)^*$ is a convenient expression for $Hom_{(R,K)}(D, C^*)$:

Lemma 5.2 There is a natural isomorphism $\Psi$ of functors, denoted,

(5.2)\begin{equation} \Psi_{C, D}: Hom_{(R,K)} (D, C^*)\cong (C\otimes_K D)^* \end{equation}

for any $(C,D)\in Ob(\mathcal {B} R_{K^{op}}\times \mathcal {B} R_{K})$.

Proof. Suppose $f$ is in $Hom_{(R,K)}(D, C^*)_q(\sigma )$ for some $\sigma \in K$ and $q\in \mathbb {Z}$. Define an $R$-map, $\Psi (f): C(st(\sigma ))\otimes D(\sigma )_{-q}\to R$, by the formula:

\[ \Psi(f)(x\otimes y)= ({-}1)^{|x| |y|} f(y)(x) \quad \text{for }x\otimes y\in (C\otimes_KD)_{{-}q}(\sigma). \]

The same formula yields $0$, if $x\otimes y$ is in $(C\otimes _KD)(\tau )_{-q}$ for $\tau \neq \sigma$. One easily sees that this rule (i.e. $f\mapsto \Psi (f)$ gives an isomorphism,

\[ \Psi_{C,D}: Hom_{(R, K)}(D,C^*)\mathop{\longrightarrow}\limits^{{\cong}} (C\otimes_K D)^* \]

of $(R,K^{op})$ complexes for all $(C,D) \in Ob(\mathcal {B} R_{K^{op}}\times \mathcal {B} R_{K})$. Naturality is obvious.

6. Ranicki Duality and the $(R,K)$ chain equivalence $e:T^2\to 1_{\mathcal {B} R_K}$

Definition 6.1 Ranicki Duality is the contravariant functor $\mathcal {B} R_K\mathop {\longrightarrow }\limits ^{T}\mathcal {B} R_K$ defined for a chain complex $C\in Ob(\mathcal {B} R_K)$ and a $(R,K)$ chain map, $f:C\to D$ by:

\[ TC= C^*\otimes_K\Delta^*K\quad Tf= f^*\otimes_K 1_{\Delta^*K} \]

$\Delta ^*K$ comes from the $K$-space, $(K,1_K)$. After examining [Reference Ranicki16], p. 75 and p. 26, lines -6 to -4 one can see that this is in agreement with the definition indicated there, up to isomorphism and differences in sign conventions. In particular compare our formula for $d^{C\otimes _KD}$ with that on p.26, line -5 of [Reference Ranicki16].

We now want to show that $T^2C$ and $C$ are $(R,K)$-chain equivalent. See 6.5.

Definition 6.3 (of $E_C:Hom_{(R,K)}(\Delta ^*K, C)\otimes _K \Delta ^*K\to C$).

Let $C$ be an $(R,K)$ complex.

Consider the evaluation chain map, $eval_{A,B}:Hom_R(A,B)\otimes _R A\to B$, when $A=\Delta ^*K(K)$ and $B=C(K)$. Its restriction to $(Hom_{(R,K)}(\Delta ^*K, C)\otimes _R\Delta ^*K)(K),$ denoted $E'_C,$ is an $(R,K)$ chain map,

\[ E'_C:Hom_{(R,K)}(\Delta^*K, C)\otimes_R\Delta^*K\to C \]

(by definition of an $(R,K)$ map). Moreover, for each $\sigma \in K$, $E'_C$ annihilates $Hom_{(R,K)}(\Delta ^*K, C)(K-st(\sigma ))\otimes _R\Delta ^*K(\sigma )$. Therefore $E'_C$ descends uniquely to an $(R,K)$ chain map,

\[ E_C:Hom_{(R,K)}(\Delta^*K, C)\otimes_K\Delta^*K\to C,\quad E_C(f\otimes \sigma^*)=f(\sigma^*). \]

satisfying: $E'_C= E_C\circ \pi _{H, \,\Delta ^*K}$ . Here $H= Hom_{(R,K)}(\Delta ^*K, C)$ (see 5.1).

$E$ is obviously natural in $C$.

For each $(R,K)$ complex $C$, define

\[ \Psi_{C^*}= \Psi_{C^*, \Delta^*K}:Hom_{(R, K)}(\Delta^*K,C^{**})\mathop{\longrightarrow}\limits^{{\cong}} (C^*\otimes_K \Delta^*K)^* \]

In view of lemma 5.2. we have an $(R, K)$ chain isomorphism:

\[ \Psi_{C^*}\otimes 1_{\Delta^*K}: Hom_{(R,K)}(\Delta^*K, C^{**})\otimes_K\Delta^*K\mathop{\longrightarrow}\limits^{{\cong}}(C^*\otimes_K\Delta^*K)^*\otimes_K \Delta^*K= T^2C . \]

Definition 6.4 For each $(R,K)$ complex $C$ define $e_C:T^2C\to C$ by

\begin{align*} e_C& = \varepsilon_C\circ E_{C^{**}}\circ (\Psi_{C^{*}}\otimes 1_{\Delta^*K})^{{-}1}:\\ & (C^{*} \otimes_K \Delta^*K)^*\otimes_K \Delta^*K\to Hom_{(R,K)}(\Delta^*K, C^{**})\otimes_K\Delta^*K\to C^{**}\to C. \end{align*}

Note $e_C$ is an $(R,K)$ chain epimorphism and $e$ is a natural transformation.

Proof. By [Reference Ranicki16] (proposition 4.7), we need only prove that $e_C(\sigma,\sigma ): T^2C(\sigma ) \to C(\sigma )$ is an $R$-chain equivalence, for all $\sigma \in K$. (No proof of this proposition appears in [Reference Ranicki16]. A brief proof appears in Appendix 2).

Case I: Assume there is a simplex $S\in K$ for which: $C(\sigma )=0 \ \forall \;\sigma \neq S$.

We need only show $e_C(S,S)$ is a chain isomorphism, and $T^2C(\sigma )$ is contractible for $\sigma \neq S$. We compute, for all $\sigma \in K$, in view of the restriction on $C$:

\begin{align*} TC(st(\sigma)) & = (C^*\otimes_K \Delta^*K)(st(\sigma)) = (C^*\otimes_R \Delta^*\overline{S})(st(\sigma)) \\ & =C^*(S)\otimes_R\Delta^*\overline{S} (st(\sigma))\\ \text{So: } T^2C(\sigma) \cong & C^{**}(S)\otimes_R \Delta^{**}\overline{S} (st(\sigma)) \otimes_R R\sigma^* \end{align*}

So for $\sigma \neq S, \; T^2C(\sigma )$ is contractible because $\Delta ^{**}\overline {S} (st(\sigma ))$ is contractible by 4.2.

Next we prove that the map

\[ e_C(S,S)= \varepsilon_C(S,S)\circ E_{C^{**}}(S,S)\circ(\Psi_C^*\otimes 1_{\Delta^*K})^{{-}1} (S,S) \]

is an isomorphism, or equivalently that $E_C(S,S)$ is an isomorphism.

Assume $S$ has been oriented. Because $C(\sigma )=0$ for $\sigma \neq S$,

\[ E_C(S,S): [Hom_{(R,K)}(\Delta^*K, C)\otimes_K\Delta^*K](S)\to C(S) \]

is simply: $eval_{RS^*, C(S)}:Hom_R(RS^*, C(S))\otimes _R RS^*\to C(S)$.

This is a chain isomorphism as observed in § 2. So $e_C(\sigma,\sigma )$ is a chain isomorphism for $\sigma = S$ and a chain equivalence for $\sigma \neq S$. This completes the proof in Case I.

Case II (the general case): For any $C\neq 0$ in $\mathcal {B} R_K$ one can choose some $S\in K$ for which $C(S)\neq 0$, and an exact sequence $0\to C'\mathop {\longrightarrow }\limits ^{i} C\mathop {\longrightarrow }\limits ^{j} C''\to 0$ for which $i(S,S):C'(S)\to C(S)$ is an isomorphism, and $C'(\sigma )= 0$ for $\sigma \neq S$. For example, choose $S$ to be of maximum dimension among $\{\sigma \in K \mid \; C(\sigma )\neq 0\}$).

The argument is by induction on the number $n$, of $\sigma \in K$, for which $C(\sigma )\neq 0$.

If $n=1$, Case I applies. If $n>1$, by induction, $e_{C''}(\sigma,\sigma )$ and $e_{C'}(\sigma,\sigma )$ are $R$ chain equivalences. Also the commuting diagram below has exact rows.

Therefore $e_C(\sigma,\sigma )$ is an $R$-chain equivalence for all $\sigma$. This completes the proof.

Note: The first proof of the above theorem appeared in [Reference Adams-Florou and Macko1].

7. Construction of the ball complex $X_K$

The purpose of this section is to construct the complex $X_K$ advertised in the introduction and establish its properties.

Because we want to use the McCrory cap product, we follow the orderings of [Reference McCrory13] regarding simplices of $X'$.

Definition 7.2 (of $\Delta X'$): Let $(X,\pi )$ be a $K$-space. The derived complexes of $(X,\pi )$ are the simplicial subdivisions $X'$ of $X$, and $K'$ of $K$ whose vertex sets $\{b\sigma \mid \sigma \in K\}$ and $\{bS\mid S\in X\}$ are chosen as follows:

\[ \text{If } \sigma\in K, \quad b\sigma:= \hat{\sigma}\in\sigma^\circ; \]

\[ \text{If } S\in X\text{ and } \sigma=\pi(S), \quad bS:=\text{ centroid of }(S\cap \pi^{{-}1} (\hat{\sigma}))\in S^\circ. \]

By construction, $\pi (V_{X'})\subset V_{K'}$. So $\pi$ is also a simplicial map from $X'$ to $K'$, because $\pi$ is linear on each simplex of $X'$.

$X'$ provides a second geometric example, $\Delta X'$, of an $(R,K)$ complex:

We define $\Delta X'$ by,

1. (1) $\Delta X'(K)= \Delta _*(X';R)$.

2. (2) For each $\sigma \in K, p\in \mathbb {Z},\;\; (\Delta X')_p(\sigma )$ is the submodule of $\Delta _p(X';R)$ spanned by all $\langle Q^0,\dots Q^p\rangle$ in $X'$ for which $\sigma = \pi (Q^p)$.

It is straightforward to see that $\Delta X'$ is an $(R,K)$ complex.

The dual cone of a simplex $\sigma \in K$, denoted $D(\sigma, K)$, is a subcomplex of $K'$ first defined in [Reference Poincare15], § 7. It is a pl ball if $K$ is a pl-manifold). It gives rise to several ‘dual’ subcomplexes in $K'$ and $X'$ which we define now.

Of course, $D(\sigma, \tau )=\emptyset$ unless $\sigma \leq \tau$, and $D_\sigma T=\emptyset$ unless $\sigma \leq \pi (T)$.

$D_\sigma T$ is a subcomplex of $X'$. $D(\sigma, K)$ and $D(\sigma, \tau )$ are subcomplexes of $K'$.

Proof. of (1): For each vertex $v$ of $\tau$ note that,

\[ |D(v,\tau)|=\{x\in \tau \;| \;a_v(x)\geq a_w(x),\text{ for all vertices}\ w\ {\rm of\ }\tau\}. \]

where $a_v: |K|\to [0,1]$ denotes the barycentric coordinate function defined by the vertex $v$. This is a convex subset of $\tau$. So

\[ |D(\sigma, \tau)|=\cap_{v\in V(K)} |D(v, \tau)| \]

is also convex. Therefore $T_\sigma =(\pi _{|T})^{-1} (|D(\sigma, \tau )|)$ is also convex since $\pi _{|T}: T\to \tau$ is simplicial. So $T_\sigma$ is a compact convex polyhedron and therefore a pl ball.

Since $|D(\sigma, \tau )|\cap \tau ^\circ \neq \emptyset$, this operator $(\pi _{|T})^{-1}$ preserves codimension:

\[ dim(\tau) - dim(D(\sigma, \tau)) = dim (T) - dim (D_\sigma T). \]

Since $dim(D(\sigma, \tau )) = dim(\tau )- dim(\sigma )$, we get: $dim(D_\sigma T)= dim(T)-dim(\sigma )$.

Definition 7.6 Let $(X,\pi )$ be a $K$-space. We define

\[ X_K = \{T_\sigma \;\mid \sigma\in K, \; T\in X, \; \sigma\leq \pi(T)\} \]

Proof. (The induced map $f'$ means the simplicial map $f':X'\to Y'$ for which $f'(bS)= b(f(S))$ for each $S\in X.$) By lemma 7.4 the boundary of each $T_\sigma$ is a union of balls of $X_K$ with smaller dimension and

\[ T_\sigma ^\circ{=} \coprod \{A^\circ\;\mid A=\langle S_0, \ldots, S_p\rangle\in D_\sigma T, \; A\notin \partial^iD_\sigma T, \; A\notin \partial^oD_\sigma T \} . \]

This can be rewritten as:

(7.1)\begin{equation} T_\sigma^\circ{=} \coprod\{A^\circ\;\mid A=\langle S_0, \ldots, S_p\rangle\in X', \; \sigma=\pi(S_p), \; T= S_0\}, \end{equation}

By equation (7.1), for each $A\in X'$ there is a unique $T_\sigma \in X_K$ for which $A^\circ \subset T_\sigma ^\circ$. Therefore : $|X'|= \coprod \{T_\sigma ^\circ \; \mid \; \; T_\sigma \in X_K \}=|X_K|$.

This proves that $X_K$ is a ball complex and that $X'$ is a subdivision of $X_K$. Because $T_\sigma \subset T$, we see $X_K$ is a subdivision of $X$.

Now let $f:(X,\pi _X)\to (Y, \pi _Y)$ be a map of $K$-spaces. For each simplex $S\in X$ we see $f(S)\in Y$ because $f$ is simplicial. For each face $\sigma$ of $\pi _X(S)$ in $K$, we see from the definitions that $f'(D_\sigma S)= D_\sigma f(S)$. So $f'$ is a map of ball complexes, $f_K: X_K \to Y_K$.

8. The isomorphism $\Phi _X:T\Delta ^*X\cong C(X_K)$

Our main theorem is:

Theorem 8.1 For each $K$-space $(X, \pi )$ the cellular chain complex of $X_K$ with $R$ coefficients, denoted $C(X_K)$, comes with a natural $(R,K)$complex structure. There is defined (below) an isomorphism of $(R,K)$ chain complexes:

\[ \Phi_X :T\Delta^*X\cong C(X_K)\;. \]

For each map $f:(X, \pi _X)\to (Y,\pi _Y)$ of $K$-spaces, the square below commutes.

Proof. Choose a basis $bK$ of oriented cells for $\Delta _*(K;R)$. Choose next, a basis $b_*X$ of oriented cells for $\Delta _*(X;R)$. But choose the orientations in $b_*X$ so that if $T\in b_*X$ and $\sigma \in bK$ are both $q$-cells, and if $\pi _*(T)=\pm \sigma \in \Delta _q(K;R)$, then:

\[ \pi_*(T) =({-}1)^{dim(\sigma)}\sigma \in \Delta_q(K;R). \]

We call such a pair, $(bK, b_*X)$ an orientation for $(X,\pi )$.

Our first task is to construct the cellular chain complex $C_*(X_K;R)$ as the underlying $R$-complex of an $(R,K)$ complex $C(X_K)$. Define

\[ C(X_K)= \Delta X\otimes_K\Delta^*K; \quad C_*(X_K;R)=(\Delta X\otimes_K\Delta^*K)(K) \]

For each oriented simplex $\rho \in bK$ and oriented simplex $T\in b_*X$, define

\[ [T_\rho] = T\otimes_K \rho^*\in C_{|T|-|\sigma|}(X_K;R)\quad(\text{where}\ |\sigma|=dim(\sigma)). \]

(The geometric intuition for this definition is the fact that, the map $C_X$, of corollary 9.3, takes $T\otimes _K \rho ^*$ to a fundamental cycle, in $\Delta X'$, for the cell $D_\rho T$, whose underlying space is $T_\rho$).

Define $bX_K= \{ [T_\rho ]\mid T\in b_*X,\; \rho \in bK, \;T_\rho \leq \pi (T)\}$. Then $bX_K$ is an $R$-basis for $C_*(X_K;R)$ in bicorrespondence with the cells of $X_K$. Write $\partial _q$ for the boundary map in $C_*(X_K;R)$, namely: $\partial _q=(d^{\Delta X\otimes _K\Delta ^*K})_q$.

But to justify these definitions, we must check that $C_*(X_K;R)$ does compute the cellular homology of $X_K$. It suffices to check, for any $[T_\rho ]\in bX_K,$ that $\partial _q([T_\rho ])$ is a sum with $\pm 1$ coefficients of those $[S_\sigma ]\in bX_K$ which are $(q-1)$-faces of $T_\rho$. (See [Reference Cooke and Finney7], for example.)

All proper faces of $T_\rho$ have the form $T_\sigma,$ for $\rho <\sigma$, or $S_\rho$, for $S< T$.

Suppose $[T_\rho ]\in bX_K$. So $T\in b_*X$, $\rho \in bK$. Set $\tau = \pi (T)\in K$. By (5.1):

\begin{align*} \partial_q[T_\rho]& = d^{\Delta X\otimes_K\Delta^*K} (T\otimes_K\rho^*) \\ & =\sum_{\{\sigma\mid \rho\leq \sigma\leq \tau\}} \{(d^{\Delta X} (\sigma,\tau)T)\otimes\rho^*+({-}1)^{|T|} T\otimes d^{\Delta^*K}(\sigma, \rho)\rho^*\}\\ & = \sum_{S< T}[T,S][S_\rho] +({-}1)^{1+|T_\rho|} \sum_{\rho<\sigma}[\sigma,\rho][T_\sigma] \end{align*}

which is as required.

This completes the construction of the cellular chain complex of $X_K$, as an $(R,K)$ complex, $C(X_K)$.

The $(R,K)$ isomorphism, $\Phi _X: T\Delta ^* X\cong C(X_K)$ is simply:

\[ \Phi_X:= (\varepsilon_{\Delta X}\otimes_K 1_{\Delta^*K}):T\Delta^*X=\Delta^{**}X \otimes_K\Delta^*K\longrightarrow \Delta X\otimes_K\Delta^*K= C(X_K). \]

Naturality of $\Phi$ is obvious from the naturality of $\varepsilon$.

9. The McCrory cap product, $\Delta ^*X$ and $\Delta X'$

We now use the work of McCrory [Reference McCrory13] to construct, for any $K$-space, $(X, \pi )$, an $(R,K)$ chain monomorphism $C(X_K)\mathop {\longrightarrow }\limits ^{C_X} \Delta X'.$ serving two purposes.

First, it defines an $(R,K)$ chain homotopy equivalence, $T\Delta ^*X\simeq \Delta X'$.

Second, $C_X$ identifies $C(X_K)$ with that $(R, K)$ subcomplex of $\Delta X'$ which admits a basis consisting of one fundamental $q$-cycle, in $\Delta _q (D_\sigma T,\partial D_\sigma T)\subset \Delta _q(X')$, for each $q$-cell $T_\sigma$ of $X_K$. (This will complete our geometric interpretation of $T$).

Let $K$ be a finite simplicial complex. McCrory (see [Reference McCrory13], and also [Reference Flexner11]) defines a map, $c': \Delta _*(K;R)\otimes _R\Delta ^*(K;R)\to \Delta _*(K';R)$ which he shows is chain homotopic to the composite,

\[ \Delta_*(K;R)\otimes_R\Delta^*(K;R)\mathop{\longrightarrow}\limits^{{\cap}} \Delta_*(K;R)\mathop{\longrightarrow}\limits^{Sd} \Delta_*(K';R) \]

where $\cap$ denotes the Whitney-Cech cap product. We will write $c_K$ for $c'$. We repeat his definition here with appropriate sign changes because McCrory's sign conventions differ slightly from ours.

For any $q$-simplex, $Q=\langle Q^0, Q^1, \ldots Q^q\rangle \text { of } K'$ in which each $Q_i$ is oriented, McCrory then defines

\[ \varepsilon(Q) = [Q^0,Q^1][Q^1,Q^2]\ldots [Q^{q-1},Q^q]. \]

This is independent of the orientations on $Q_1, Q_2,\ldots Q_{q-1}$. If $q=0$, set $\varepsilon (Q)=0$.

For any $n$-simplex $\tau$ and $(n-q)$-simplex $\sigma$ of $K$, each simplex $Q=\langle Q^0, Q^1, \ldots Q^q\rangle$ of $D(\sigma, \tau )_q$ satisfies: $Q^0= \tau ; \;Q^q=\sigma$. Therefore, $\varepsilon (Q)$ makes sense if $\tau$ and $\sigma$ are oriented simplices chosen from some basis $bK$ of oriented simplices for $\Delta _*(K;R)$ (but not if $\sigma =-\tau ).$

The McCrory Cap Product, $\Delta _*(K;R)\otimes _R\Delta ^*(K;R)\mathop {\longrightarrow }\limits ^{c_K} \Delta _*(K';R)$ is the map defined by:

\[ \boxed{ c_K(\tau\otimes \sigma^*)= \sum_{Q\in D(\sigma,\tau)_q}({-}1)^{dim (\sigma)} \varepsilon(Q)Q } \]

for any oriented simplices $\sigma, \tau$ in some basis $bK$. Here $q= dim(\tau )-dim(\sigma )$. Note this is zero unless $\sigma \leq \tau$. Note that $c_K$ does not change if we change the basis.

$c_K$ is a chain map. We reprove this in Appendix I, § A, because of the sign changes and because McCrory's proof, [Reference McCrory13] p.155 lines 7-8, is only a sketch.

Now suppose $(X,\pi )$ is a $K$-space.

Note that if $T$ and $\sigma$ are oriented simplices of $X$ and $K$ and $q=dim(T)-dim(\sigma )\neq 0$:

\[ c_X(T\otimes_R \pi^*\sigma^*)=\sum_{Q\in (D_\sigma, T)_q}({-}1)^{dim (\sigma)} \varepsilon(Q)Q\ \in \Delta_qX'(\sigma) \]

(because $D_\sigma T=\cup \{D(S,T)\; |\; S\in X, dim(S)=dim(\sigma ), \pi (S)= \sigma \}$). This formula still makes sense and is true if $q=0$ and $\pi _*(T)\neq -\sigma$).

In this way, $c_X\circ (1\otimes \pi ^*)$ defines an $(R,K)$ chain map,

\[ c_X\circ(1\otimes \pi^*): \Delta X\otimes_R\Delta^*K\longrightarrow \Delta X' \]

Proposition 9.1 There is a unique $(R,K)$ chain map

\[ C_X: C(X_K)=\Delta X\otimes_K \Delta^*K \longrightarrow \Delta X' \]

satisfying:

\[ c_X\circ(1\otimes \pi^*)=C_X\circ \pi_{\Delta X, \Delta^*K} \]

$C_X$ is an $(R,K)$ monomorphism. For all $q$-cells $T_\sigma$ of $X_K$, with $q\neq 0$,

\[ C_X(T\otimes_K \sigma^*)= \sum_{Q\in (D_\sigma T)_q}({-}1)^{dim (\sigma)} \varepsilon(Q)Q. \]

For a $0$-cell $T_\sigma$, of $X_K$, with $T\in \Delta _n(X;R), \sigma \in \Delta _n(K;R)$ oriented so that $\pi _*(T)=\sigma$, then

\[ C_X(T\otimes_K\sigma^*)= ({-}1)^{dim(T)}\langle T \rangle, \quad (\langle T\rangle \hbox{ is the barycenter } bT \hbox{ of } T). \]

Proof. Note that $c_X(T\otimes _R \pi ^*\sigma ^*)=0$ unless $\pi (T)\geq \sigma$. Also $c_X(T\otimes _R \pi ^*\sigma ^*)\in \Delta X'(\sigma )$ for all $\sigma \in K$ and $T\in X$ because each $q$-cell $Q\in D_\sigma T$ is in $\Delta _qX'(\sigma )$ if $q= dim(T_\sigma )$.

So $c_X\,\circ (1\otimes \pi ^*):\Delta X\otimes _R \Delta ^*K\to \Delta X'$ is an $(R,K)$ chain map annihilating each $\Delta X(K-st(\sigma ))\otimes _R \Delta ^*K(\sigma )$. Hence there is a unique $(R,K)$ chain map monomorphism, $\Delta X\otimes _K \Delta ^*K\mathop {\longrightarrow }\limits ^{C_X} \Delta X'$ such that $c_X\circ (1\otimes \pi ^*) = C_X\circ \pi _{\Delta X, \Delta ^*K}$. The calculation follows if $q\neq 0$. If $q=0$, then $(\pi _{\mid T})^* \sigma ^*=T^*$, so

\[ C_X(T\otimes_K \sigma^*)= c_X(T\otimes T^*) =({-}1)^{dim (T)}\sum_{Q\in D(T,T)_0}Q= ({-}1)^{dim(T)}\langle T\rangle \]

Clearly $C_X$ is natural in $(X,\pi )$.

Proof. By 9.3, for all $T_\sigma$, $C_X$ restricts to a homotopy equivalence,

\[ C_*(T_\sigma,\partial T_\sigma; R)\to \Delta_*(D_\sigma(T),\partial D_\sigma(T);R) \]

and it takes chains on any subcomplex of $X_K$ to chains on its subdivision. By an induction-excision argument on the number of cells in the subcomplex one sees $C_X$ yields a homology equivalence and then a chain homotopy equivalence on each such subcomplex. So $C_X(\sigma,\sigma )$ is an $R$-chain equivalence for each $\sigma$. Therefore $C_X$ is an $(R,K)$ chain equivalence.

Together, 9.4 and 8.1 clearly prove:

Corollary 9.5 $T\Delta ^*X\mathop {\longrightarrow }\limits ^{ C_X \Phi _X} \Delta X'$ is an $(R,K)$ chain homotopy equivalence. Consequently $e_{\Delta ^*X}\circ T(C_X\Phi _X)$ is an explicit $(R,K)$ chain homotopy equivalence,

\[ T\Delta X'\simeq \Delta^*X. \]