Leray numbers of projections and a topological Helly type theorem

TL;DR

This paper establishes a bound on the Leray number of projections of simplicial complexes under certain conditions, leading to a topological Helly type theorem extension.

Contribution

It introduces a new inequality relating Leray numbers of complexes and their projections, extending Helly type theorems in topological combinatorics.

Findings

L((X))  r L(X)+r-1 bound established

Topological Helly type theorem extended using Leray numbers

Provides a new tool for analyzing projections of simplicial complexes

Abstract

Let X be a simplicial complex on the vertex set V. The rational Leray number L(X) of X is the minimal d such that the rational reduced homology of any induced subcomplex of X vanishes in dimensions d and above. Let \pi be a simplicial map from X to a simplex Y, such that the cardinality of the preimage of any point in |Y| is at most r. It is shown that L(\pi(X)) \leq r L(X)+r-1. One consequence is a topological extension of a Helly type result of Amenta.

Equations

$$L(\pi (X))\le rL(X)+r-1.$$

$$\mathrm{h}(\mathcal{G})\le r(d+1).$$

$$L(Y)\le rL(X)+r-1.$$

$$X_k=\Delta (A_{1k})\ast \cdots \ast \Delta (A_{k-1,k})\ast \partial \Delta (A_{kk})\ast \Delta (A_{k+1,k})\ast \cdots \ast \Delta (A_{rk}).$$

$$\mathrm{h}(\mathcal{F})\le 1+L(N(\mathcal{F})).$$

$$\mathrm{h}(\mathcal{G})\le r(d+1).$$

$$M_k=\{(x_1,\dots ,x_k)\in |X{|}^k:\pi (x_1)=\cdots =\pi (x_k)\}.$$

$$\mathrm{Alt}W=\{\mathrm{Alt}w:w\in W\}=$$

$$\{w\in W:\sigma w=\mathrm{sign}(\sigma )wforall\sigma \in S_k\}.$$

$$\mathrm{Alt}{\mathrm{H}}_{\ast }(M_k)\cong {\mathrm{H}}_{\ast }(\mathrm{Alt}\mathrm{C}(M_k)).$$

$$E_{p,q}^{1}=\left\{\begin{array}[]{ll}{\rm Alt}\,{\rm H}_{q}(M_{p+1})&0\leq p\leq r-1,0\leq q\\ 0&{\rm otherwise}\end{array}\right.~{}~{}$$

$$D^k=\mathrm{closure}\{(x_1,\dots ,x_k)\in |X{|}^k:\pi (x_1)=\cdots =\pi (x_k),x_i\ne x_j\mathrm{for}i\ne j\}.$$

$$\mathrm{Alt}{\mathrm{H}}_q(D^{p+1})\cong \mathrm{Alt}{\mathrm{H}}_q(M_{p+1})$$

$$M(X_1,\dots ,X_k)=\{(x_1,\dots ,x_k)\in |X_1|\times \cdots \times |X_k|:\pi (x_1)=\cdots =\pi (x_k)\}.$$

$$M(X_1,{\sigma }_2,\dots ,{\sigma }_k)\cong X_1[{\cap }_{i=2}^k{\tilde{\sigma }}_i].$$

$${\mathcal{S}}_p^{\prime }=\{({\sigma }_2,\dots ,{\sigma }_k)\in X_2\times \cdots \times X_k:\sum _{i=2}^k\dim {\sigma }_i\ge n-p\}$$

$$A_{\underline{\sigma }}=M(X_1,{\sigma }_2,\dots ,{\sigma }_k)\times D_{X_2}({\sigma }_2)\times \cdots \times D_{X_k}({\sigma }_k),$$

$$B_{\underline{\sigma}}=M(X_{1},\sigma_{2},\ldots,\sigma_{k})\times\Bigl{(}\bigcup_{j=2}^{k}D_{X_{2}}(\sigma_{2})\times\cdots\times\overset{.}{D}_{X_{j}}(\sigma_{j})\times\cdots\times D_{X_{k}}(\sigma_{k})\Bigr{)}~{}~{}.$$

$$E_{p,q}^{1}=\bigoplus_{\underline{\sigma}\in{\cal S}_{p}}\bigoplus_{\begin{subarray}{c}i_{1},\ldots,i_{k}\geq 0\\ i_{1}+\cdots+i_{k}=p+q\end{subarray}}{\rm H}_{i_{1}}\bigl{(}X_{1}[\cap_{i=2}^{k}\tilde{\sigma}_{i}]\bigr{)}\otimes\bigotimes_{j=2}^{k}\tilde{{\rm H}}_{i_{j}-1}\bigl{(}{\rm lk}(X_{j},\sigma_{j})\bigr{)}$$

$$K_p=\bigcup _{\underline{\sigma }\in {\mathcal{S}}_p^{\prime }}{A}_{\underline{\sigma }}\subset M(X_1,\dots ,X_k)\times \mathrm{sd}(X_2)\times \cdots \times \mathrm{sd}(X_k).$$

$$\theta^{-1}\bigl{(}(x_{1},\ldots,x_{k})\bigr{)}=\{(x_{1},\ldots,x_{k})\}\times D_{X_{2}}(\sigma_{2})\times\cdots\times D_{X_{k}}(\sigma_{k})$$

$$\bigcup _{\underline{\sigma }\in {\mathcal{S}}_p}{A}_{\underline{\sigma }}\bigcap K_{p-1}=\bigcup _{\underline{\sigma }\in {\mathcal{S}}_p}{B}_{\underline{\sigma }}.$$

$${\rm H}_{*}\Bigl{(}\bigcup_{\underline{\sigma}\in{\cal S}_{p}}A_{\underline{\sigma}},\bigcup_{\underline{\sigma}\in{\cal S}_{p}}B_{\underline{\sigma}}\Bigr{)}\cong\bigoplus_{\underline{\sigma}\in{\cal S}_{p}}{\rm H}_{*}(A_{\underline{\sigma}},B_{\underline{\sigma}})~{}~{}.$$

$$E_{p,q}^{1}={\rm H}_{p+q}(K_{p},K_{p-1})\cong{\rm H}_{p+q}\Bigl{(}\bigcup_{\underline{\sigma}\in{\cal S}_{p}}A_{\underline{\sigma}},\bigcup_{\underline{\sigma}\in{\cal S}_{p}}A_{\underline{\sigma}}\bigcap K_{p-1}\Bigr{)}\cong$$

$${\rm H}_{p+q}\Bigl{(}\bigcup_{\underline{\sigma}\in{\cal S}_{p}}A_{\underline{\sigma}},\bigcup_{\underline{\sigma}\in{\cal S}_{p}}B_{\underline{\sigma}}\Bigr{)}\cong\bigoplus_{\underline{\sigma}\in{\cal S}_{p}}{\rm H}_{p+q}(A_{\underline{\sigma}},B_{\underline{\sigma}})\cong$$

$$\bigoplus_{\underline{\sigma}\in{\cal S}_{p}}\bigoplus_{\begin{subarray}{c}i_{1},\ldots,i_{k}\geq 0\\ i_{1}+\cdots+i_{k}=p+q\end{subarray}}{\rm H}_{i_{1}}\bigl{(}M(X_{1},\sigma_{2},\ldots,\sigma_{k})\bigr{)}\otimes\bigotimes_{j=2}^{k}{\rm H}_{i_{j}}\bigl{(}D_{X_{j}}(\sigma_{j}),\overset{.}{D}_{X_{j}}(\sigma_{j})\bigr{)}\cong$$

$$\bigoplus_{\underline{\sigma}\in{\cal S}_{p}}\bigoplus_{\begin{subarray}{c}i_{1},\ldots,i_{k}\geq 0\\ i_{1}+\cdots+i_{k}=p+q\end{subarray}}{\rm H}_{i_{1}}\bigl{(}X_{1}[\cap_{i=2}^{k}\tilde{\sigma}_{i}]\bigr{)}\otimes\bigotimes_{j=2}^{k}\tilde{{\rm H}}_{i_{j}-1}\bigl{(}{\rm lk}(X_{j},\sigma_{j})\bigr{)}~{}~{}.$$

$$L(\bigcap _{j=1}^k{X}_j)\le \sum _{j=1}^{k}L(X_j).$$

$$X=N(\{F_{ij}:1\le i\le m,1\le j\le r_i\})\subset V_1\ast \cdots \ast V_m.$$

$$|{\pi }^{-1}(y)|=|\{(j_i:i\in I):\bigcap _{i\in I}{F}_{i{j}_i}\ne \varnothing \}|.$$

Full text

Suppose $V=\bigcup_{i=1}^{m}V_{i}$ is a partition of $V$ such that the induced subcomplexes $X[V_{i}]$ are all [math]-dimensional. Let $\pi$ denote the projection of $X$ into the $(m-1)$-simplex on the vertex set $\{1,\ldots,m\}$ given by $\pi(v)=i$ if $v\in V_{i}$. Let $r=\max\{|\pi^{-1}(\pi(x))|:x\in|X|\}$. It is shown that

[TABLE]

One consequence is a topological extension of a Helly type result of Amenta. Let ${\cal F}$ be a family of compact sets in ${\twelvebb R}^{d}$ such that for any ${\cal F}^{\prime}\subset{\cal F}$, the intersection $\bigcap{\cal F}^{\prime}$ is either empty or contractible.

It is shown that if ${\cal G}$ is a family of sets such that for any finite ${\cal G}^{\prime}\subset{\cal G}$, the intersection $\bigcap{\cal G}^{\prime}$ is a union of at most $r$ disjoint sets in ${\cal F}$, then the Helly number of ${\cal G}$ is at most $r(d+1)$.

1 Introduction

Let ${\cal F}$ be a family of sets. The Helly number $~{}{\rm h}({\cal F})$ of ${\cal F}$ is the minimal positive integer $h$ such that if a finite subfamily ${\cal K}\subset{\cal F}$ satisfies $\bigcap{\cal K}^{\prime}\neq\emptyset$ for all ${\cal K}^{\prime}\subset{\cal K}$ of cardinality $\leq h$, then $\bigcap{\cal K}\neq\emptyset$. Helly’s classical theorem (1913, see e.g. [3]) asserts that the Helly number of the family of convex sets in ${\twelvebb R}^{d}$ is $d+1$.

Helly’s theorem and its numerous extensions are of central importance in discrete and computational geometry (see [3, 10]). It is of considerable interest to understand the role of convexity in these results, and to find suitable topological extensions. Indeed, it is often the case that topological methods provide a deeper understanding of the underlying combinatorics behind Helly type theorems. Helly himself realized in 1930 (see [3]) that in his theorem, convex sets can be replaced by topological cells if you impose the additional requirement that all non-empty intersections of these cells are again topological cells. Helly’s topological version of his theorem also follows from the later nerve theorems of Borsuk, Leray and others (see below).

The following result was conjectured by Grünbaum and Motzkin [8] , and proved by Amenta [1]. A family of sets ${\cal G}$ is an $({\cal F},r)$-family if for any finite ${\cal G}^{\prime}\subset{\cal G}$, the intersection $\bigcap{\cal G}^{\prime}$ is a union of at most $r$ disjoint sets from ${\cal F}$.

Theorem 1.1** (Amenta).**

Let ${\cal F}$ be the family of compact convex sets in ${\twelvebb R}^{d}$. Then for any $({\cal F},r)$-family ${\cal G}$

[TABLE]

The main motivation for the present paper was to find a topological extension of Amenta’s Theorem.

Let $X$ be a simplicial complex on the vertex set $V$. The induced subcomplex on a subset of vertices $S\subset V$ is $X[S]=\{\sigma\in X:\sigma\subset S\}$. The link of a subset $A\subset V$ is ${\rm lk}(X,A)=\{\tau\in X:\tau\cup A\in X,\tau\cap A=\emptyset~{}\}~{}.$ The geometric realization of $X$ is denoted by $|X|$. We identify $X$ and $|X|$ when no confusion can arise. All homology groups considered below are with rational coefficients, i.e. ${\rm H}_{i}(X)={\rm H}_{i}(X;{\twelvebb Q})$ and $\tilde{\rm H}_{i}(X)=\tilde{\rm H}_{i}(X;{\twelvebb Q})$.

The rational Leray number $L(X)$ of $X$ is the minimal $d$ such that $\tilde{{\rm H}}_{i}(Y)=0$ for all induced subcomplexes $Y\subset X$ and $i\geq d$. The Leray number can be regarded as a simple topologically based “complexity measure” of $X$. Note that $L(X)=0$ iff $X$ is a simplex, and $L(X)\leq 1$ iff $X$ is the clique complex of a chordal graph (see [9]). It is well-known (see e.g. [7]) that $~{}L(X)\leq d~{}$ iff $~{}\tilde{{\rm H}}_{i}({\rm lk}(X,\sigma))=0~{}$ for all $\sigma\in X$ and $i\geq d$. Leray numbers have also significance in commutative algebra, since $L(X)$ is equal to the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of $X$ over ${\twelvebb Q}$ (see [7]).

From now on we assume that $V_{1},\ldots,V_{m}$ are finite disjoint [math]-dimensional complexes, and denote their join by $V_{1}*\cdots*V_{m}$. Let $\Delta_{m-1}$ be the simplex on the vertex set $[m]=\{1,\ldots,m\}$, and let $\pi$ denote the simplicial projection from $V_{1}*\cdots*V_{m}$ onto $\Delta_{m-1}$ given by $\pi(v)=i$ if $v\in V_{i}$. For a subcomplex $X\subset V_{1}*\cdots*V_{m}$, let $r(X,\pi)=\max\{|\pi^{-1}(\pi(x))|:x\in|X|\}$. Our main result is the following

Theorem 1.2**.**

Let $Y=\pi(X)$ and $r=r(X,\pi)$. Then

[TABLE]

Example: For $r\geq 1,d\geq 2$ let $m=rd$, and consider a partition $[m]=\bigcup_{k=1}^{r}A_{k}$ with $|A_{k}|=d$. For $i\in[m]$ let $V_{i}=\{i\}\times[r]$. Denote by $\Delta(A)$ the simplex on vertex set $A$, with boundary $\partial\Delta(A)\simeq S^{|A|-2}$. For $k,j\in[r]$ let $A_{kj}=A_{k}\times\{j\}$, and let

[TABLE]

Let $X=\bigcup_{k=1}^{r}X_{k}$. Then $L(X)=d-1$, and the projection $\pi:X\rightarrow\Delta_{m-1}$ satisfies $r(X,\pi)=r$. Since $\pi(X)=\partial\Delta_{m-1}$, it follows that $L(\pi(X))=m-1$. Hence equality is attained in (1).

As mentioned earlier, Theorem 1.2 is motivated by an application in combinatorial geometry. The nerve $N({\cal F})$ of a family of sets ${\cal F}$, is the simplicial complex whose vertex set is ${\cal F}$ and whose simplices are all ${\cal F}^{\prime}\subset{\cal F}$ such that $\bigcap{\cal F}^{\prime}\neq\emptyset$. It is easy to see that

[TABLE]

A finite family ${\cal F}$ of compact sets in some topological space is a good cover if for any ${\cal F}^{\prime}\subset{\cal F}$, the intersection $\bigcap{\cal F}^{\prime}$ is either empty or contractible. If ${\cal F}$ is a good cover in ${\twelvebb R}^{d}$, then by the Nerve Lemma (see e.g. [2]) $L(N({\cal F}))\leq d$, hence follows the Topological Helly’s Theorem: ${\rm h}({\cal F})\leq d+1$. Theorem 1.2 implies a similar topological generalization of Amenta’s theorem.

Theorem 1.3**.**

Let ${\cal F}$ is a good cover in ${\twelvebb R}^{d}$. Then for any $({\cal F},r)$-family ${\cal G}$

[TABLE]

The proof of Theorem 1.2 combines a vanishing theorem for the multiple point sets of a projection, with an application of the image computing spectral sequence due to Goryunov and Mond [5]. In Section 2 we describe the Goryunov-Mond result. In Section 3 we prove our main result, Proposition 3.1, which is then used to deduce Theorem 1.2. The proof of Theorem 1.3 is given in Section 4.

2 The Image Computing Spectral Sequence

For $X\subset V_{1}*\cdots*V_{m}$ and $k\geq 1$ define the multiple point set $M_{k}$ by

[TABLE]

Let $W$ be a ${\twelvebb Q}$-vector space with an action of the symmetric group $S_{k}$. Denote ${\rm Alt}=\frac{1}{k!}\sum_{\sigma\in S_{k}}{\rm sign}(\sigma)\sigma\in{\twelvebb Q}[S_{k}]$. Then

[TABLE]

[TABLE]

The natural action of $S_{k}$ on $M_{k}$ induces an action on the rational chain complex ${\rm C}_{*}(M_{k})$ and on the rational homology ${\rm H}_{*}(M_{k})$. The idempotence of ${\rm Alt}$ implies that

[TABLE]

The following result is due to Goryunov and Mond [5] (see also [4] and [6]).

Theorem 2.1** (Goryunov and Mond).**

Let $Y=\pi(X)$ and $r=r(X,\pi)$. Then there exists a homology spectral sequence $\{E_{p,q}^{r}\}$ converging to ${\rm H}_{*}(Y)$ with

[TABLE]

Remark: The $E^{1}$ terms in the original formulation of Theorem 2.1 in [5], are given by $E_{p,q}^{1}={\rm Alt}\,{\rm H}_{q}(D^{p+1})$ where

[TABLE]

The isomorphism

[TABLE]

which implies (5), is proved in Theorem 3.4 in [6]. Indeed, as noted there, the inclusion $D^{p+1}\rightarrow M_{p+1}$ induces an isomorphism ${\rm Alt}\,{\rm C}_{q}(D^{p+1})\cong{\rm Alt}\,{\rm C}_{q}(M_{p+1})~{}$ already at the alternating chains level.

3 Homology of the Multiple Point Set

In this section we study the homology of a generalization of the multiple point set. For subcomplexes $X_{1},\ldots,X_{k}\subset V_{1}*\cdots*V_{m}$, let

[TABLE]

In particular, if $X_{1}=\cdots=X_{k}=X$ then $M(X_{1},\ldots,X_{k})=M_{k}$.

We identify the generalized multiple point set $M(X_{1},\ldots,X_{k})$ with the simplicial complex whose $p$-dimensional simplices are $\{w_{i_{0}},\ldots,w_{i_{p}}\}$, where $1\leq i_{0}<\cdots<i_{p}\leq m$, $w_{i_{j}}=(v_{i_{j},1},\ldots,v_{i_{j},k})\in V_{i_{j}}^{k}$ and $\{v_{i_{0},r},\ldots,v_{i_{p},r}\}\in X_{r}$ for all $1\leq r\leq k$. The main ingredient in the proof of Theorem 1.2 is the following

Proposition 3.1**.**

\tilde{{\rm H}}_{j}\bigl{(}M(X_{1},\ldots,X_{k})\bigr{)}=0* for $j\geq\sum_{i=1}^{k}L(X_{i})$.*

The proof of Proposition 3.1 depends on a spectral sequence argument given below. We first recall some definitions. Let $K$ be a simplicial complex. The subdivision ${\rm sd}(K)$ is the order complex of the set of the non-empty simplices of $K$ ordered by inclusion. For $\sigma\in K$ let $D_{K}(\sigma)$ denote the order complex of the interval $[\sigma,\cdot]=\{\tau\in K~{}:~{}\tau\supset\sigma\}$. Let $\overset{.}{D}_{K}(\sigma)$ denote the order complex of the interval $(\sigma,\cdot]=\{\tau\in K~{}:~{}\tau\supsetneqq\sigma\}$. Note that $\overset{.}{D}_{K}(\sigma)$ is isomorphic to ${\rm sd}({\rm lk}(K,\sigma))$ via the simplicial map $\tau\rightarrow\tau-\sigma$. Since $D_{K}(\sigma)$ is contractible, it follows that {\rm H}_{i}\bigl{(}D_{K}(\sigma),\overset{.}{D}_{K}(\sigma)\bigr{)}\cong\tilde{{\rm H}}_{i-1}\bigl{(}{\rm lk}(K,\sigma)\bigr{)} for all $i\geq 0$.

For $\sigma\in V_{1}*\cdots*V_{m}$, let $\tilde{\sigma}=\bigcup_{i\in\pi(\sigma)}V_{i}$. Note that if $\sigma_{2}\in X_{2},\ldots,\sigma_{k}\in X_{k}$ then there is an isomorphism

[TABLE]

For $0\leq p\leq n=\sum_{i=2}^{k}\dim X_{i}$ let

[TABLE]

and let ${\cal S}_{p}={\cal S}^{\prime}_{p}-{\cal S}^{\prime}_{p-1}$. For $\underline{\sigma}=(\sigma_{2},\ldots,\sigma_{k})\in{\cal S}^{\prime}_{p}$ let

[TABLE]

[TABLE]

Proposition 3.2**.**

There exists a homology spectral sequence $\{E_{p,q}^{r}\}$ converging to {\rm H}_{*}\bigl{(}M(X_{1},\ldots,X_{k})\bigr{)} such that

[TABLE]

for $~{}0\leq p\leq n~{}$,$~{}0\leq q~{}$, and $E_{p,q}^{1}=0$ otherwise.

Proof: For $0\leq p\leq n$ let

[TABLE]

Write $K=K_{n}$, and consider the projection on the first coordinate $\theta:K\rightarrow M(X_{1},\ldots,X_{k})$. Let $(x_{1},\ldots,x_{k})\in M(X_{1},\ldots,X_{k})$, and let $\sigma_{i}$ denote the minimal simplex in $X_{i}$ that contains $x_{i}$. Then the fiber

[TABLE]

is a cone, hence $K$ is homotopy equivalent to $M(X_{1},\ldots,X_{k})$. The filtration $\emptyset\subset K_{0}\subset\cdots\subset K_{n}=K$ gives rise to a homology spectral sequence $\{E_{p,q}^{r}\}$ converging to ${\rm H}_{*}(K)\cong{\rm H}_{*}(M(X_{1},\ldots,X_{m}))$. The $E_{p,q}^{1}$ terms are computed as follows. First note that

[TABLE]

Secondly, \bigl{(}A_{\underline{\sigma}}-B_{\underline{\sigma}}\bigr{)}\cap A_{\underline{\sigma}^{\prime}}=\emptyset~{} for $~{}\underline{\sigma}\neq\underline{\sigma}^{\prime}\in{\cal S}_{p}$. Hence

[TABLE]

Applying excision, (8),(9), and the Künneth formula we obtain:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

$\Box$

Proof of Proposition 3.1: If $L(X_{j})=0$ for all $1\leq j\leq k$, then all the $X_{j}$’s are simplicies, say $X_{j}=\sigma_{j}$. It follows that $M(X_{1},\ldots,X_{k})$ is isomorphic to the simplex $\bigcap_{j=1}^{k}\pi(\sigma_{j})$ and thus has vanishing reduced homology in all nonnegative dimensions. Suppose then that $m=\sum_{j=1}^{k}L(X_{j})>0$. Without loss of generality we may assume that $L(X_{1})>0$. Let $i_{1},\ldots,i_{k}\geq 0$ such that $\sum_{j=1}^{k}i_{j}\geq m$. Then either $i_{1}\geq L(X_{1})$ and then {\rm H}_{i_{1}}\bigl{(}X_{1}[\cap_{i=2}^{k}\tilde{\sigma}_{i}]\bigr{)}=0, or there exists a $2\leq j\leq k$ such that $i_{j}-1\geq L(X_{j})$ and then \tilde{{\rm H}}_{i_{j}-1}\bigl{(}{\rm lk}(X_{j},\sigma_{j})\bigr{)}=0. By (7) it follows that $E_{p,q}^{1}=0$ if $p+q\geq m$, hence \tilde{{\rm H}}_{j}\bigl{(}M(X_{1},\ldots,X_{k})\bigr{)}=0 for all $j\geq m$.

$\Box$

Remark: If all the $V_{j}$’s are singletons then $M(X_{1},\ldots,X_{k})$ is isomorphic to $\bigcap_{j=1}^{k}X_{j}$. Hence Proposition 3.1 implies the following result of [7].

Corollary 3.3** ([7]).**

If $X_{1},\ldots,X_{k}$ are simplicial complexes on the same vertex set, then

[TABLE]

$\Box$**

Proof of Theorem 1.2: Let $Y=\pi(X)$ and $r=r(X,\pi)$. Assuming as we may that $L(X)>0$, we have to show that ${\rm H}_{m}(Y)=0$ for $m\geq rL(X)+r-1$. By Theorem 2.1 it suffices to show that ${\rm Alt}\,{\rm H}_{q}(M_{p+1})=0$ for all pairs $(p,q)$ such that $p\leq r-1$ and $p+q\geq rL(X)+r-1$. Indeed, $p\leq r-1$ implies that $q\geq rL(X)\geq(p+1)L(X)$, thus ${\rm H}_{q}(M_{p+1})=0$ by Proposition 3.1.

$\Box$

4 A Topological Amenta Theorem

Proof of Theorem 1.3: Suppose ${\cal G}=\{G_{1},\ldots,G_{m}\}$ is an $({\cal F},r)$-family. Write $G_{i}=\bigcup_{j=1}^{r_{i}}F_{ij}$, where $r_{i}\leq r$ and $F_{ij}\bigcap F_{ij^{\prime}}=\emptyset$ for $1\leq j\neq j^{\prime}\leq r_{i}$. Let $V_{i}=\{F_{i1},\ldots,F_{ir_{i}}\}$ and consider the nerve

[TABLE]

Let $\Delta_{m-1}$ be the simplex on the vertex set $\{G_{1},\ldots,G_{m}\}$ and let $\pi$ denote the projection of $V_{1}*\cdots*V_{m}$ into $\Delta_{m-1}$ given by $\pi(F_{ij})=G_{i}$. Then $\pi(X)=N({\cal G})$. Let $y\in|N({\cal G})|$ and let $\sigma=\{G_{i}~{}:~{}i\in I\}$ be the minimal simplex in $N({\cal G})$ such that $y\in|\sigma|$. Then

[TABLE]

On the other hand

[TABLE]

and the union on the right is a disjoint union. The assumption that ${\cal G}$ is an $({\cal F},r)$ family, together with (10) and (11), imply that $|\pi^{-1}(y)|\leq r$ for all $y\in|N({\cal G})|$. Since ${\cal F}$ is a good cover in ${\twelvebb R}^{d}$, the Leray number of the nerve satisfies $L(X)=L(N({\cal F}))\leq d$. Therefore by (2) and Theorem 1.2

[TABLE]

[TABLE]

$\Box$