2.1.1 Topological equivalences

The relation of $\ell $ -equivalence as defined above is not an equivalence relation since it is not symmetric. In order to make it symmetric, one needs to ‘formally invert’ $\ell $ -equivalences.

Definition 2.2 ( $\ell $ -equivalent and homologically $\ell $ -equivalent).

We will say that X is $\ell $ -equivalent to Y (denoted $X \sim _{\ell } Y$ ), if and only if there exists spaces, $X=X_0,X_1,\ldots ,X_n=Y$ and $\ell $ -equivalences $f_1,\ldots ,f_{n}$ as shown below:

It is clear that $\sim _{\ell }$ is an equivalence relation.

By replacing the homotopy groups, $\pi _i(\cdot )$ with homology groups $\mathrm {H}_i(\cdot )$ (resp. cohomology groups $\mathrm {H}^i(\cdot )$ with arrows reversed) in Definitions 2.1 and 2.2, we get the notion of two topological spaces $X,Y$ being homologically $\ell $ -equivalent (denoted $X \overset {h}{\sim }_{\ell } Y$ ) (resp. cohomologically $\ell $ -equivalent (denoted $X \overset {ch}{\sim }_{\ell } Y$ )).

This is a strictly weaker equivalence relation, since there are spaces X for which $\mathrm {H}_1(X) = 0$ , but $\pi _1(X) \neq 0$ .

We extend the above definitions to $\ell = -1$ by using the convention that $X \sim _{-1} Y$ (resp. $X \overset {h}{\sim }_{-1} Y$ , $X \overset {ch}{\sim }_{-1} Y$ ), if and only if $X,Y$ are both nonempty or both empty.

We extend Definition 2.1 to diagrams of topological spaces. We denote by $\mathbf {Top}$ the category of topological spaces.

Definition 2.5 ( $\ell $ -equivalence between diagrams of topological spaces).

Let J be a small category, and $X,Y: J \rightarrow \mathbf {Top}$ be two functors. We say a natural transformation $f:X \rightarrow Y$ is an $\ell $ equivalence, if the induced maps,

$$\begin{align*}f(j)_*: \pi_i(X(j)) \rightarrow \pi_i(Y(j)) \end{align*}$$

are isomorphisms for all $j \in J$ and $0 \leq i \leq \ell $ .

We will say that a diagram $X:J \rightarrow \mathbf {Top}$ is $\ell $ -equivalent to the diagram $Y:J \rightarrow \mathbf {Top}$ (denoted as before by $X \sim _{\ell } Y$ ), if and only if there exist diagrams $X=X_0,X_1,\ldots ,X_n=Y:J \rightarrow \mathbf {Top}$ and $\ell $ -equivalences $f_1,\ldots ,f_{n}$ as shown below:

It is clear that $\sim _{\ell }$ is an equivalence relation.

In the above definition, by replacing the homotopy groups with homology (resp. cohomology) groups we obtain the notion of homological (resp. cohomological) $\ell $ -equivalence between diagrams, which we will denote as before by $\overset {h}{\sim }_{\ell }$ (resp. $\overset {ch}{\sim }_{\ell }$ ).

One particular diagram will be important in what follows.

Notation 2.1 (Diagram of various unions of a finite number of subspaces).

Let J be a finite set, A a topological space, and $\mathcal {A} = (A_j)_{j \in J}$ a tuple of subspaces of A indexed by J.

For any subset $J' \subset J$ , Footnote ¹ we denote

$$ \begin{align*} \mathcal{A}^{J'} &= \bigcup_{j' \in J'} A_{j'}, \\ \mathcal{A}_{J'} &= \bigcap_{j' \in J'} A_{j'}. \end{align*} $$

We consider $2^J$ as a category whose objects are elements of $2^J$ and whose only morphisms are given by:

$$ \begin{align*} 2^J(J',J") &= \emptyset \mbox{ if } J' \not\subset J", \\ 2^J(J',J") &= \{\iota_{J',J"}\} \mbox{ if } J' \subset J". \end{align*} $$

We denote by $\mathbf {Simp}^J(\mathcal {A}):2^J \rightarrow \mathbf {Top}$ the functor (or the diagram) defined by

$$\begin{align*}\mathbf{Simp}^J(\mathcal{A})(J') = \mathcal{A}^{J'}, J' \in 2^J, \end{align*}$$

and $\mathbf {Simp}^J(\mathcal {A})(\iota _{J',J"})$ is the inclusion map $\mathcal {A}^{J'} \hookrightarrow \mathcal {A}^{J"}$ .

2.1.2 Definition of complexity of algorithms

We will use the following notion of ‘complexity of an algorithm’ in this paper. We follow the same definition as used in the book [Reference Basu, Pollack and Roy4].

2.1.3 $\mathcal {P}$ -formulas and $\mathcal {P}$ -semialgebraic sets

Notation 2.2 (Realizations, $\mathcal {P}$ -, $\mathcal {P}$ -closed semialgebraic sets).

For any finite set of polynomials $\mathcal {P} \subset \mathrm {R} [ X_{1} , \ldots ,X_{k} ]$ , we call any quantifier-free first-order formula $\phi $ with atoms, $P =0, P < 0, P>0, P \in \mathcal {P}$ , to be a $\mathcal {P}$ -formula. Given any semialgebraic subset $Z \subset \mathrm {R}^k$ , we call the realization of $\phi $ in Z, namely the semialgebraic set

$$ \begin{align*} {\mathcal R}(\phi,Z) := \{ \mathbf{x} \in Z \mid \phi (\mathbf{x})\} \end{align*} $$

a $\mathcal {P}$ -semialgebraic subset of Z.

If $Z = \mathrm {R}^k$ , we often denote the realization of $\phi $ in $\mathrm {R}^k$ by ${\mathcal R}(\phi )$ .

If $\Phi = (\phi _j)_{j \in J}$ is a tuple of formulas indexed by a finite set J, $Z \subset \mathrm {R}^k$ a semialgebraic subset, we will denote by ${\mathcal R}(\Phi ,Z)$ the tuple $({\mathcal R}(\phi _j,Z))_{j \in J}$ , and call it the realization of $\Phi $ in Z. For $J \subset J'$ , we will denote by $\Phi |_{J'}$ the tuple $(\phi _j)_{j \in J'}$ .

We say that a quantifier-free formula $\phi $ is closed if it is a formula in disjunctive normal form with no negations and with atoms of the form $P \geq 0, P \leq 0$ (resp. $P> 0, P < 0$ ), where $P \in \mathrm {D}[X_1,\ldots ,X_k]$ . If the set of polynomials appearing in a closed (resp. open) formula is contained in a finite set $\mathcal {P}$ , we will call such a formula a $\mathcal {P}$ -closed formula, and we call the realization, ${\mathcal R} \left (\phi \right )$ , a $\mathcal {P}$ -closed semialgebraic set. We say that a formula $\phi $ is a closed-formula if $\phi $ is a $\mathcal {P}$ -closed formula for some finite set of polynomials  $\mathcal {P}$ .

We will also use the following notation.

Finally, we are able to state the main results proved in this paper.

2.4.1 Covers

A standard method in algebraic topology for computing homology/cohomology of a space X is by means of an appropriately chosen cover, $(V_{\alpha } \subset X)_{\alpha \in I}$ , of X by open or closed subsets. Suppose that $X \subset \mathbb {R}^k$ is a closed or open semialgebraic set. Let $\mathcal {V} = (V_{\alpha } \subset X)_{\alpha \in I}$ be a finite cover of X by open or closed semialgebraic subsets such that for each nonempty subset $J \subset I$ , the intersection $V_J = \bigcap _{\alpha \in J} V_{\alpha }$ is either empty or contractible. We will say that such covers have the Leray property and refer to them as Leray covers. One can then associate to the cover $\mathcal {V}$ , a simplicial complex, $\mathcal {N}(\mathcal {V})$ , the nerve of $\mathcal {V}$ defined as follows.

The set of p-simplices of $\mathcal {N}(\mathcal {V})$ is defined by

$$\begin{align*}\mathcal{N}(\mathcal{V})_p = \{ \{\alpha_0,\ldots,\alpha_p\} \subset 2^I \mid V_{\alpha_0} \cap \cdots \cap V_{\alpha_p} \neq \emptyset\}. \end{align*}$$

It follows from a classical result of algebraic topology that the geometric realization $|\mathcal {N}(\mathcal {V})|$ is homotopy equivalent to X, and moreover for each $\ell \geq 0$ , the geometric realization of the $(\ell +1)$ -st skeleton of $\mathcal {N}(\mathcal {V})$ ,

$$\begin{align*}\mathrm{sk}_{\ell+1}(\mathcal{N}(\mathcal{V})) = \{ \sigma \in \mathcal{N}(\mathcal{V}) \mid \mathrm{card}(\sigma) \leq \ell+2\} \end{align*}$$

is homologically $\ell $ -equivalent (resp. $\ell $ -equivalent) to X (resp. when $\mathrm {R} = \mathbb {R}$ ).

The algorithms for computing the Betti numbers in [Reference Bürgisser, Cucker and Lairez10, Reference Bürgisser, Cucker and Tonelli-Cueto12, Reference Bürgisser, Cucker and Tonelli-Cueto11] proceeds by computing the k-skeleton of the nerve of a cover having the Leray property whose size is bounded singly exponentially in k and computing the simplicial homology groups of this complex. However, the bound on the size of the cover is not uniform but depends on a real valued parameter – namely the condition number of the input – and hence the size of the cover can become infinite. In fact, computing a singly exponential sized cover by semialgebraic subsets having the Leray property of an arbitrary semialgebraic sets is an open problem. If one solves this problem, then one would also solve immediately the problem of designing an algorithm for computing all the Betti numbers of arbitrary semialgebraic sets with singly exponential complexity in full generality.

The algorithms in [Reference Basu, Pollack and Roy5, Reference Basu1] which are able to compute some of the Betti numbers in dimensions $>0$ also depends on the existence of small covers having size bounded singly exponentially, albeit satisfying a much weaker property than the Leray property. The weaker property is that only the sets $V_{\alpha }, \alpha \in I$ (i.e., the elements of the cover) are contractible. No assumption is made on the nontrivial finite intersections amongst the sets of the cover. Covers satisfying this weaker property can indeed be computed with singly exponential complexity (this is one of the main results of [Reference Basu, Pollack and Roy5] but see Remark 3.2), and using this fact one is able to compute the first $\ell $ Betti numbers of semialgebraic subsets of $\mathrm {R}^k$ for every fixed $\ell $ with singly exponential complexity. The algorithms in [Reference Basu, Pollack and Roy5] and [Reference Basu1] do not construct a simplicial complex homotopy equivalent or $\ell $ -equivalent to the given semialgebraic set S unlike the algorithm in [Reference Bürgisser, Cucker and Lairez10].

2.4.2 Main technical contribution

The main technical result that underlies the algorithmic result of the current paper is the following. Fix $0 \leq \ell \leq k$ . Suppose for every closed and bounded semialgebraic set S one has a covering of S by closed and bounded semialgebraic subsets which are $\ell $ -connected (see Definition 2.3) and which has singly exponentially bounded complexity (meaning that the number of sets in the cover, the number of polynomials used in the quantifier-free formulas defining these sets and their degrees are all bounded singly exponentially in k). Moreover, since it is clear that contractible covers with singly exponential complexity exists, this is not a vacuous assumption. Using $\ell $ -connected covers repeatedly we build a simplicial complex of size bounded singly exponentially which is $\ell $ -equivalent to the given semialgebraic set. The definition of this simplicial complex is a bit involved (much more involved than the nerve complex of a Leray cover) and appears in Section 3. Its main properties are encapsulated in Theorem 2.

Two remarks are in order.

3.3.1 Simplified view of the definition of the poset $\mathbf {P}_{m,i}(\Phi )$

Before giving a precise definition of the poset $\mathbf {P}_{m,i}(\Phi )$ , we first give a simplified version. We make the following two simplifications in order to illustrate the key idea.

1. (a) We ignore the role of the index i in what follows. The necessity of the extra parameter i is due to the fact that the hypothesis we assume (Hypothesis 3.1 in the following paragraph) is slightly stronger than we are able to assume for designing effective algorithms for computing the poset (see Remark 3.2). The actual hypothesis that we use is encapsulated in Property 3.2 below.

2. (b) Secondly, in order to keep a geometric view of the construction, we will talk about tuples $\mathcal {S} = (S_j)_{j \in J}$ of semialgebraic sets, instead of tuples of formulas $\Phi = (\phi _j)_{j \in J}$ defining them. As above, in order to give an effective algorithms and analyzing its complexity, we need to describe the poset in terms of formulas rather than sets, which we do in the precise definition that follows this simplified version.

We make the following hypothesis.

Suppose $\mathcal {S} = (S_j)_{j \in J}$ is a finite tuple of $\ell $ -connected closed semialgebraic subsets of $\mathbb {R}^k$ .

Our goal is to define a poset $\mathbf {P}_{m}(\mathcal {S})$ such that:

Property 3.1.

$$\begin{align*}|\Delta(\mathbf{P}_{m}(\mathcal{S}))| \overset{ch}{\sim}_m \mathcal{S}^J \end{align*}$$

(see Definition 3.1). We will say that the poset $\mathbf {P}_{m}(\mathcal {S})$ satisfies Property 3.1 for the pair $(m,\mathcal {S})$ .

Remark 3.3. We use cohomological m-equivalence in Property 3.1. In the final construction, we will lose a dimension while passing from cohomological equivalence to (homological or homotopical) equivalence because of the use of the universal coefficients theorem (see the proof of Claim 3.5 inside the proof of Theorem 2), and we will end up with

$$\begin{align*}|\Delta(\mathbf{P}_{m}(\mathcal{S}))| {\sim}_{m-1} \mathcal{S}^J. \end{align*}$$

The main idea is to approximate homotopically the diagram of sets

$$\begin{align*}(\mathcal{S}_I)_{I \subset J, \mathrm{card}(I) \leq m+2} \end{align*}$$

(see Notation 2.1), and the inclusion maps

$$\begin{align*}\mathcal{S}_{I'} \hookrightarrow \mathcal{S}_I, I \subset I', \end{align*}$$

by a corresponding diagram of (the geometric realizations of the order complexes of) posets

$$\begin{align*}(\mathbf{P}_{m - \mathrm{card}(I)+1,I})_{I \subset J, \mathrm{card}(I) \leq m+2} \end{align*}$$

(where the poset $\mathbf {P}_{m - \mathrm {card}(I)+1,I}$ corresponds to $\mathcal {S}_I$ ) and poset inclusions

$$\begin{align*}\mathbf{P}_{m-\mathrm{card}(I')+1,I'} \hookrightarrow \mathbf{P}_{m - \mathrm{card}(I)+1,I}, I \subset I'. \end{align*}$$

The construction is by induction on m (we call this the global induction below).

1. 1. (Base case of the global induction, $m=-1$ .) Suppose $\mathcal {S} = (S_j)_{j \in J}$ is a finite tuple of $\ell $ -connected closed and bounded semialgebraic subsets of $\mathbb {R}^k$ . We define the poset $\mathbf {P}_{-1}(\mathcal {S})$ to be just the index set J, with no nontrivial order relations. It is depicted in Figure 3(a). It is clear that $\mathbf {P}_{-1}(\mathcal {S})$ satisfies Property 3.1 for the pair $(-1, \mathcal {S})$ .

   

   Figure 3 A simple illustration of the simplified view of the poset.

2. 2. (Induction hypothesis of the global induction.) We assume that for each $m', -1 \leq m' < m$ , and each finite tuple $\mathcal {S} = (S_j)_{j \in J}$ of $\ell $ -connected closed and bounded semialgebraic subsets of $\mathbb {R}^k$ , we have defined a poset $\mathbf {P}_{m'}(\mathcal {S})$ satisfying Property 3.1 for the pair $(m',\mathcal {S})$ .

3. 3. (Inductive step of the global induction, going from $<m$ to m.) Using the inductive hypothesis, we now define a poset $\mathbf {P}_{m}(\mathcal {S})$ satisfying Property 3.1 for the pair $(m,\mathcal {S})$ , for any tuple $\mathcal {S}$ of $\ell $ -connected closed and bounded semialgebraic subsets of $\mathbb {R}^k$ .

   Fix a finite tuple $\mathcal {S} = (S_j)_{j \in J}$ of $\ell $ -connected closed and bounded semialgebraic subsets of $\mathbb {R}^k$ . We will define $\mathbf {P}_m(\mathcal {S})$ below in steps. The poset $\mathbf {P}_m(\mathcal {S})$ as a set will be a disjoint union of the index set J, and certain subposets $\mathbf {P}_{m - \mathrm {card}(I) +1,I}$ , where I where $I \subset J, 2 \leq \mathrm {card}(I) \leq m+2$ . We define the subposets $\mathbf {P}_{m - \mathrm {card}(I) +1,I}$ by downward induction (we call this the local induction below) on $\mathrm {card}(I)$ , starting from the base case, $\mathrm {card}(I) = m+2$ .

   • (a) (Base case of the local induction, $\mathrm {card}(I) = m+2$ .) We first consider the semialgebraic sets $\mathcal {S}_I, \mathrm {card}(I)= m+2$ . Associated to each such I, we define a poset, which we denoted by $\mathbf {P}_{-1,I}$ as follows: Using Hypothesis 3.1 applied to the semialgebraic set $\mathcal {S}_I$ , we obtain a cover $(\mathcal {S}_{I,\alpha })_{\alpha \in \mathcal {C}(\mathcal {S}_I)}$ of $\mathcal {S}_I$ by closed and bounded $\ell $ -connected semialgebraic sets. We define

     $$\begin{align*}\mathbf{P}_{-1,I} = \mathbf{P}_{-1}((\mathcal{S}_{I,\alpha})_{\alpha \in \mathcal{C}(\mathcal{S}_I)}) = \mathcal{C}(\mathcal{S}_I) \end{align*}$$

     with no nontrivial order relation. It is depicted in Figure 3(a). It is clear that $\mathbf {P}_{-1,I}$ satisfies Property 3.1 for the pair $(-1, (\mathcal {S}_{I,\alpha })_{\alpha \in \mathcal {C}(\mathcal {S}_I)})$ .

   • (b) (Going from $m+2$ to $m+1$ .) Next, we consider subsets I of cardinality $m+1$ . For each such subset, we construct a poset $\mathbf {P}_{0,I}$ satisfying two conditions:

     • (i) For each set $I'$ , with $\mathrm {card}(I') = \mathrm {card}(I)+1$ , and $I \subset I'$ , the poset $\mathbf {P}_{-1,I'}$ already defined is isomorphic to a subposet of $\mathbf {P}_{0,I}$ ;

     • (ii) $|\Delta (\mathbf {P}_{0,I})|$ is cohomologically $0$ -equivalent to $\mathcal {S}_I$ .

     We apply Hypothesis 3.1 to the semialgebraic set $\mathcal {S}_I$ as input and obtain a cover $(\mathcal {S}_{I,\alpha })_{\alpha \in \mathcal {C}(\mathcal {S}_I)}$ of $\mathcal {S}_I$ by closed and bounded $\ell $ -connected semialgebraic sets. We let

     $$\begin{align*}\mathbf{P}_{-1,I} = \mathbf{P}_{-1}((\mathcal{S}_{I,\alpha})_{\alpha \in \mathcal{C}(\mathcal{S}_I)}). \end{align*}$$

     Let $J_I$ be the union of the indexing set $\mathcal {C}(\mathcal {S}_I)$ , with the posets $\mathbf {P}_{-1,I'}$ for each $I'$ with $I \subset I', \mathrm {card}(I') = \mathrm {card}(I) + 1$ . Notice that for each $\alpha \in J_I$ , there is an $\ell $ -connected closed and bounded semialgebraic set associated to it. Denote this set by $D(\alpha )$ .For every pair $\alpha ,\beta \in J_I$ , we again apply Hypothesis 3.1 to obtain a cover of $D(\alpha ) \cap D(\beta )$ by $\ell $ -connected closed and bounded semialgebraic sets, $(S_{I,\alpha ,\beta ,\gamma })_{\gamma \in I_{\alpha ,\beta }}$ , where $I_{\alpha ,\beta } = \mathcal {C}(D(\alpha ) \cap D(\beta ))$ . The poset $\mathbf {P}_{0,I}$ is defined to be the set $J_I \cup \bigcup _{\alpha ,\beta \in J_I} I_{\alpha ,\beta }$ , and the nontrivial order relations are $\gamma \prec \alpha ,\beta $ for each $\gamma \in I_{\alpha ,\beta }$ . It is depicted in Figure 3(b).

   • (c) (Local induction hypothesis.) We assume that we have already defined the posets $\mathbf {P}_{m-\mathrm {card}(I')+1,I'}$ , with $\mathrm {card}(I')> \mathrm {card}(I)$ .

   • (d) (Inductive step in general for the local induction.) We construct the poset $\mathbf {P}_{m-\mathrm {card}(I)+1,I}$ as follows. We apply Hypothesis 3.1 with the semialgebraic set $\mathcal {S}_I$ as input and obtain a cover $(\mathcal {S}_{I,\alpha })_{\alpha \in \mathcal {C}(\mathcal {S}_I)}$ of $\mathcal {S}_I$ by closed and bounded $\ell $ -connected semialgebraic sets. Let $J_I$ be the union of the indexing set $\mathcal {C}(\mathcal {S}_I)$ , with the posets $\mathbf {P}_{m-\mathrm {card}(I')+1,I'}$ for each $I'$ with $I \subset I', \mathrm {card}(I') = \mathrm {card}(I) + 1$ . Notice that for each $\alpha \in J_I$ , there is an $\ell $ -connected closed and bounded semialgebraic set associated to it. Denote this set by $D(\alpha )$ .

     We define the poset $\mathbf {P}_{m - \mathrm {card}(I) + 1,I}$ using the global induction hypothesis. The global inductive hypothesis gives us that for any finite tuple of $\ell $ -connected closed and bounded semialgebraic set (in particular, the tuple of sets $(D(\alpha ))_{\alpha \in J_I}$ ) we have defined a poset $\mathbf {P}_{m - \mathrm {card}(I)+1}((D(\alpha ))_{\alpha \in J_I})$ , which satisfies Property 3.1 for the pair $(m - \mathrm {card}(I)+1, (D(\alpha ))_{\alpha \in J_I})$ (since $m - \mathrm {card}(I) +1 < m$ ).

     We define

     $$\begin{align*}\mathbf{P}_{m -\mathrm{card}(I)+1,I} = \mathbf{P}_{m - \mathrm{card}(I)+1}((D(\alpha))_{\alpha \in J_I}). \end{align*}$$
     This finishes the local induction, and we have defined $\mathbf {P}_{m - \mathrm {card}(I) +1,I}$ , for each $I \subset J, 2 \leq \mathrm {card}(I) \leq m+2$ .

   Finally, we define

   (3.1) $$ \begin{align} \mathbf{P}_{m}(\mathcal{S}) = J \cup \bigcup_{I \subset J, 2 \leq \mathrm{card}(I) \leq m+2} \mathbf{P}_{m -\mathrm{card}(I) +1, I}. \end{align} $$

   The partial order in the poset $\mathbf {P}_{m}(\mathcal {S})$ is specified as follows. By the local induction, each of the poset $\mathbf {P}_{m -\mathrm {card}(I) +1, I}$ comes with a partial order. We extend these orders as follows:

   • (a) For each $I \subset I' \subset J$ , with $2 \leq \mathrm {card}(I)\leq \mathrm {card}(I') \leq m+2$ , there is a subposet of $\mathbf {P}_{m - \mathrm {card}(I)+1,I}$ canonically isomorphic to the poset $\mathbf {P}_{m - \mathrm {card}(I')+1,I'}$ . For each element $\alpha $ of the former and the corresponding element $\alpha '$ of the latter, we set $\alpha ' \prec \alpha $ .

   • (b) For each $j \in J$ and $\alpha \in \mathbf {P}_{m -\mathrm {card}(I)+1,I}, j \in I$ , we set the element $\alpha \prec j$ .

   This ends the definition of the poset $\mathbf {P}_m(\mathcal {S})$ completing the global induction. Figure 3(c) depicts $\mathbf {P}_{m}(\mathcal {S})$ in terms of subposets $\mathbf {P}_{m -\mathrm {card}(I)+1,I}$ . In Claim 3.11, we will show that the height of the poset $\mathbf {P}_{m}(\mathcal {S})$ is bounded by $2m+2$ .

   Notice that for any chain $\alpha _k \prec \alpha _{k-1} \prec \ldots \prec \alpha _0$ of elements in $\mathbf {P}_{m}(\mathcal {S})$ , we have a sequence of inclusion maps of semialgebraic sets $D(\alpha _k) \hookrightarrow D(\alpha _{k-1}) \hookrightarrow \ldots \hookrightarrow D(\alpha _0)$ . It is depicted in Figure 4 for a hypothetical space with four elements in the initial covering.

   

   Figure 4 Poset $\mathbf {P}_m(\mathcal {S})$ such that $|\Delta (\mathbf {P}_m(\mathcal {S}))|$ is m-equivalent to $ \bigcup _{j \in J} S_j$ with $m=2$ , $J = \{1,2,3,4\}$ .

The following two examples are illustrative.

Example 3.1. Let $\ell = \infty , m \geq 2$ , $\mathcal {S} = (S_1,S_2)$ , where $S_1, S_2$ are the closed upper and lower hemispheres of the unit sphere in $\mathbb {R}^3$ (see Figure 5(a)).

Figure 5 (a) The ideal situation, (b) $D_{m,i}'(\Phi )(.)$ and (c) $D_{m,i}(\Phi )(.)$ .

Using equation (3.1), we get

(3.2) $$ \begin{align} \mathbf{P}_m(\mathcal{S}) = \{1,2\} \cup \mathbf{P}_{m -2 +1,\{1,2\}}. \end{align} $$

Let $\mathcal {C}(\mathcal {S}_{\{1,2\}})$ be the cover of $\mathcal {S}_{\{1,2\}}$ by two closed semicircles $T_3, T_4$ , and let $\mathcal {T} = (T_3,T_4)$ .

Note that $T_3 \cap T_4$ is a set containing two points $W_5,W_6$ (say), and the only possibility for $\mathcal {C}(T_3 \cap T_4)$ , is the tuple $\mathcal {W} = (W_5,W_6)$ . Then,

(3.3) $$ \begin{align} \mathbf{P}_{m - 1}(\mathcal{T}) = \{3, 4\} \cup \mathbf{P}_{m - 2, \{3,4\}} \end{align} $$

and the subposet $\mathbf {P}_{m - 2, \{3,4\}}$ is isomorphic to the poset

(3.4) $$ \begin{align} \mathbf{P}_{m-2}(\mathcal{W}) = \{5,6\}. \end{align} $$

Substituting equation (3.4) into equation (3.3) and equation (3.3) into equation (3.2), we finally obtain that the Hasse diagram of the poset $\mathbf {P}_m(\mathcal {S})$ is

The order complex of this poset is homotopy equivalent (in fact, in this case is homeomorphic) to the sphere.

Example 3.2. Now, let $\ell = m = 2$ , $\mathcal {S} = (S_1,S_2)$ , where $S_1, S_2$ are the closed upper and lower hemispheres of the unit sphere in $\mathbb {R}^k$ , $k> 5$ . That is, $S_1$ (resp. $S_2$ ) is the intersection of the unit sphere in $\mathrm {R}^k$ , with the set defined by $X_k \geq 0$ (resp. $X_k \leq 0$ ).

Using equation (3.1), we get

$$ \begin{align*} \mathbf{P}_m(\mathcal{S}) = \{1,2\} \cup \mathbf{P}_{m -2 +1,\{1,2\}}. \end{align*} $$

Let $\mathcal {C}(\mathcal {S}_{\{1,2\}})$ be the cover of $\mathcal {S}_{\{1,2\}}$ by two closed semispheres $T_3, T_4$ , (i.e., $T_3$ (resp. $T_4$ ) is the intersection of $\mathcal {S}_{\{1,2\}}$ with $X_{k-1} \geq 0$ (resp. $X_{k-1} \leq 0$ ), and let $\mathcal {T} = (T_3,T_4)$ .

Note that $W_5 = T_3 \cap T_4$ is a $(k-3)$ -dimensional sphere, and since $k> 5$ , $W_5$ is $2$ -connected and we can take $\mathcal {C}(W_5) = (W_5)$ .

$$ \begin{align*} \mathbf{P}_{1}(\mathcal{T}) = \{3, 4\} \cup \{5 \} \end{align*} $$

with Hasse diagram

Finally we obtain that the Hasse diagram of the poset $\mathbf {P}_2(\mathcal {S})$ is

The order complex of this poset is contractible and is $2$ -equivalent (but in this case not homotopy equivalent) to $\mathbf {S}^{k-1}$ for $k>5$ .

With the definition of the poset $\mathbf {P}_m(\mathcal {S})$ , it is possible to prove the following theorem. We do not include a proof of this theorem since it is subsumed by Theorem 2^′ .

Theorem. With the same notation as in the Definition of $\mathbf {P}_m(\mathcal {S})$ defined above:

$$ \begin{align*} |\Delta(\mathbf{P}_{m}(\mathcal{S}))| {\sim}_{m-1} \bigcup_{j \in J} S_j. \end{align*} $$

More generally, we have the diagrammatic homological $(m-1)$ -equivalence

$$ \begin{align*} (J' \mapsto |\Delta(\mathbf{P}_{m}(\mathcal{S}|_{J'})|)_{J' \in 2^J} \overset{h}{\sim}_{m-1} \mathbf{Simp}^J(\mathcal{S}), \end{align*} $$

where $\mathcal {S}|_{J'} = (S_j)_{j \in J'}$ .

We now return to the precise definition of the poset $\mathbf {P}_{m,i}(\Phi )$ that we are going to the use.

3.3.2 Precise definition of $\mathbf {P}_{m,i}(\Phi )$

We begin with a few useful notation that we will use in the construction.

We also use the following convenient notation.

We are now in a position to define a poset associated to a given finite tuple of formulas that will play the key technical role in the rest of the paper.

We first fix the following.

1. A. Let $\mathrm {R} = \mathrm {R}_0 \subset \mathrm {R}_1 \subset \mathrm {R}_2 \subset \cdots $ be a sequence of extensions of real closed fields.

2. B. We also fix two sequences of functions,

   $$\begin{align*}\mathcal{I}_{i,k}: \mathcal{F}_{\mathrm{R}_i,k} \rightarrow \mathbb{Z}_{\geq -1}, \end{align*}$$

   and

   $$\begin{align*}\mathcal{C}_{i,k}: \mathcal{F}_{\mathrm{R}_i,k} \rightarrow \bigcup_{p \geq 0}(\mathcal{F}_{\mathrm{R}_{i+1},k})^{[p]}. \end{align*}$$

For each $i \geq 0$ and $-1 \leq m \leq k$ , a nonempty finite set J, and $\Phi \in (\mathcal {F}_{\mathrm {R}_i,k})^J$ , we define a poset $(\mathbf {P}_{m,i}(\Phi ), \prec )$ .

We first need an auxilliary definition which will be used in the definition of the underlying set, $\mathbf {P}_{m,i}(\Phi )$ , of the poset $(\mathbf {P}_{m,i}(\Phi ), \prec )$ .

Definition 3.2. Let J be a nonempty finite set, and $\Phi \in (\mathcal {F}_{\mathrm {R}_i,k})^J$ . We first define for each subset $I \subset _{\leq m+2} J$ , a set $J_{m,i,I,\Phi }$ , and an element $\Phi _{m,i,I,J} \in (\mathcal {F}_{\mathrm {R}_{i+1},k})^{J_{m,i,I,\Phi }}$ (using downward induction on $\mathrm {card}(I)$ ).

Base case ( $\mathrm {card}(I) = m+2$ ): In this case, we define

(3.5) $$ \begin{align} J_{m,i, I,\Phi} = \{I\} \times [ \mathcal{I}_{i,k}( \bigwedge_{j \in I} \Phi(j))], \end{align} $$

and for $(I,p) \in J_{m,i,I,\Phi }$ ,

$$\begin{align*}\Phi_{m,i, I,J}((I,p)) = \mathcal{C}_{i,k}( \bigwedge_{j \in I} \Phi(j))(p). \end{align*}$$

Inductive step: Suppose we have defined $J_{m,i,I',\Phi }$ and $\Phi _{m,i, I',J}$ for all $I'$ with $\mathrm {card}(I') = \mathrm {card}(I) +1$ . We define

(3.6) $$ \begin{align} J_{m,i,I,\Phi} = \left(\{I\} \times [ \mathcal{I}_{i,k}( \bigwedge_{j \in I} \Phi(j)] \right) \cup \bigcup_{I \subset I' \subset J, \mathrm{card}(I') = \mathrm{card}(I)+1} J_{m,i,I',\Phi}, \end{align} $$

and

$$ \begin{align*} \Phi_{m,i,I,J}(\alpha) &= \mathcal{C}_{i,k}( \bigwedge_{j \in I} \Phi(j))(p), \mbox{ if } \alpha =(I,p) \in \{I\} \times [ \mathcal{I}_{i,k}( \bigwedge_{j \in I} \Phi(j))], \\ &=\Phi_{m,i,I',J}(\alpha), \mbox{ if } \alpha \in J_{m,i,I',\Phi} \text{ for some } I' \supset I, \text{with} \\ & \hspace{2.1cm} \mathrm{card}(I') = \mathrm{card}(I)+1. \end{align*} $$

The following properties of $J_{m,i,I,\Phi }$ and $\Phi _{m,i,I,J}$ are obvious from the above definition. Using the same notation as in Definition 3.2:

We now define the set $\mathbf {P}_{m,i}(\Phi )$ .

Definition 3.3 (The underlying set of the poset $(\mathbf {P}_{m,i}(\Phi ), \prec )$ ).

We define the set $\mathbf {P}_{m,i}(\Phi )$ using induction on m.

Base case ( $m=-1$ ): For each finite set J and $\Phi \in (\mathcal {F}_{\mathrm {R}_i,k})^J$ , we define

$$\begin{align*}\mathbf{P}_{-1,i}(\Phi) = \bigcup_{j \in J} \{\{j\}\} \times \{\emptyset \}. \end{align*}$$

Inductive step: Suppose we have defined the sets $(\mathbf {P}_{m',i'}(\Phi '),\prec )$ for all $m'$ with $-1 \leq m' < m$ , $i' \geq 0$ , for all nonempty finite sets $J'$ and all $\Phi ' \in (\mathcal {F}_{\mathrm {R}_{i'},k})^{J'}$ .

We complete the inductive step by defining:

(3.7) $$ \begin{align} \mathbf{P}_{m,i}(\Phi) = \bigcup_{j \in J} \{\{j\}\} \times \{\emptyset\} \cup \bigcup_{I \subset J, 1 < \mathrm{card}(I) \leq m+2} \{I\} \times \mathbf{P}_{m -\mathrm{card}(I) +1,i+1}(\Phi_{m,i,I,J}). \end{align} $$

We now specify the partial order on $\mathbf {P}_{m,i}(\Phi )$ . For this, it will be useful to have the following alternative characterization of the elements of the poset $\mathbf {P}_{m,i}(\Phi )$ as tuples of sets. This characterization follows simply by unravelling the inductive definition of the set $\mathbf {P}_{m,i}(\Phi )$ given above.

3.3.3 Characterization of the elements of the poset $\mathbf {P}_{m,i}(\Phi )$ as tuples of sets

The elements of $\mathbf {P}_{m,i}(\Phi )$ are all finite tuples of sets (of varying lengths)

$$\begin{align*}(I_0,\ldots,I_{r},\emptyset), \end{align*}$$

satisfying the following conditions.

1. 1. $I_0$ is a subset of $J_0= J$ , $\mathrm {card}(I_0) = 1$ if $r = 0$ and $2 \leq \mathrm {card}(I_0) \leq m+2$ otherwise.

2. 2. $I_1$ is a subset of $J_1 = \left (J_0\right )_{m_0,i_0,I_0,\Phi _0}$ (see equation (3.6), Definition 3.3) with

   $$ \begin{align*} m_0 &= m,\\ i_0 &= i, \\ \Phi_0 &= \Phi, \end{align*} $$

   and

   $$\begin{align*}2 \leq \mathrm{card}(I_1) \leq (m_0 - \mathrm{card}(I_0) + 1) +2. \end{align*}$$

3. 3. $I_2$ is a subset of $J_2 = \left (J_1 \right )_{m_1,i_1,I_1, \Phi _{1}}$ , where

   $$ \begin{align*} m_1 &= m_0 - \mathrm{card}(I_0) + 1,\\ i_1 &= i_0+1, \\ \Phi_{1} &= (\Phi_0)_{m_0,i_0,I_0, J_0}, \end{align*} $$

   and

   $$\begin{align*}2 \leq \mathrm{card}(I_2) \leq (m_1- \mathrm{card}(I_1) +1) + 2. \end{align*}$$

4. 4. Continuing in the above fashion,

   (3.8) $$ \begin{align} I_{r-1} \subset J_{r-1} = \left(J_{r-2} \right)_{m_{r-2},i_{r-2}, I_{r-2}, \Phi_{r-2}}, \end{align} $$

   where

   $$ \begin{align*} m_{r-2} &= m_{r-3} - \mathrm{card}(I_{r-3}) + 1, \\ i_{r-2} &= i_{r-3}+1, \\ \Phi_{r-2} &= (\Phi_{r-3})_{m_{r-3},i_{r-3},I_{r-3}, J_{r-3}}, \end{align*} $$

   and

   (3.9) $$ \begin{align} 2 \leq \mathrm{card}(I_{r-1}) \leq m_{r-2} +2 = (m +r -1 - \sum_{j=0}^{r-2}\mathrm{card}(I_j)) + 2. \end{align} $$

5. 5. Finally,

   $$\begin{align*}I_{r} \subset J_{r} =\left(J_{r-1} \right)_{m_{r-1},i_{r-1},I_{r-1}, \Phi_{r -1}}, \end{align*}$$

   where

   $$\begin{align*}\Phi_{r -1} = (\Phi_{r -2})_{m_{r-2},i_{r-2},I_{r -2},J_{r-2}}, \end{align*}$$

   and

   $$\begin{align*}\mathrm{card}(I_r) = 1. \end{align*}$$

(We show later (see Claim 3.8) that for tuples $(I_0,\ldots ,I_r,\emptyset )$ satisfying the above conditions, $r \leq m+1$ .)

Definition 3.4 (Partial order on $\mathbf {P}_{m,i}(\Phi )$ ).

The partial order $\prec $ on $\mathbf {P}_{m,i}(\Phi )$ is defined as follows.

For $\alpha = (I^{\alpha }_0,\ldots ,I^{\alpha }_{r_{\alpha }},\emptyset ),\beta = (I^{\beta }_0,\ldots ,I^{\beta }_{r_{\beta }},\emptyset ) \in \mathbf {P}_{m,i}(\Phi ),$

(3.10) $$ \begin{align} \beta \preceq \alpha \Leftrightarrow (r_{\alpha} \leq r_{\beta}) \mbox{ and } I^{\alpha}_j \subset I^{\beta}_j, 0 \leq j \leq r_{\alpha}. \end{align} $$

3.6.1 Outline of the proof of Theorem 2

In order to prove that $|\Delta (\mathbf {P}_{m,i}(\Phi ))|$ is homologically $(m-1)$ -equivalent to ${\mathcal R}(\Phi )^J$ , we give two homological $(m-1)$ -equivalences. The source of both these maps is a semialgebraic set which is defined as the homotopy colimit of a certain functor $D_{m,i}$ from the poset category $\mathbf {P}_{m,i}(\Phi )$ to $\mathbf {Top}$ taking its values in semialgebraic subsets of $\mathrm {R}^k_{i+m+1}$ . The targets are $|\Delta (\mathbf {P}_{m,i}(\Phi ))|$ and ${\mathcal R}(\Phi )^J$ . Taken together these two homological $(m-1)$ -equivalences imply that $|\Delta (\mathbf {P}_{m,i}(\Phi ))|$ and ${\mathcal R}(\Phi )^J$ are homologically $(m-1)$ -equivalent.

In what follows, we first define the functor $D_{m,i}$ as well as an associated map $D^{\prime }_{m,i}$ , also taking values in semialgebraic sets and prove the main properties of these objects that we are going to need in the proof of Theorem 2.

3.6.2 Definition of $D_{m,i},D^{\prime }_{m,i}$

We now define for each $\alpha = (I_0,\ldots ,I_{r},\emptyset ) \in \mathbf {P}_{m,i}(\Phi )$ , a closed semialgebraic subset $D_{m,i}(\alpha ) \subset \overline {B_k(0,R)} \subset \mathrm {R}_{i+m+ 1}^k$ and also a semialgebraic set $D^{\prime }_{m,i}(\alpha ) \subset \mathrm {R}_{i+r}^k$ .

We define $D_{m,i},D^{\prime }_{m,i}$ by induction on m. For $m=-1$ , we define for $j \in J$ ,

$$\begin{align*}D_{-1,i}(\Phi)((\{j\},\emptyset)) = D^{\prime}_{-1,i}(\Phi)((\{j\},\emptyset)) = {\mathcal R}(\Phi(j),\overline{B_k(0,R)}) \subset \mathrm{R}_{i}^k. \end{align*}$$

We now define $D_{m,i}(\Phi ), D^{\prime }_{m,i}(\Phi ): \mathbf {P}_{m,i}(\Phi ) \rightarrow \mathbf {Top} $ , using the fact that they are already defined for all $-1 \leq m' < m$ . We define:

$$ \begin{align*} D_{m,i}(\Phi)((\{j\},\emptyset)) &= \mathrm{ext}({\mathcal R}(\Phi(j),\overline{B_k(0,R)}), \mathrm{R}_{i+m+1}) \cup \\ &\bigcup_{(I,\alpha) \in \mathbf{P}_{m,i}(\Phi), j \in I} \mathrm{ext}(D_{m-\mathrm{card}(I)+1,i+1}(\Phi_{m,i,I,J})(\alpha),\mathrm{R}_{i+m+1}), \\ D_{m,i}(\Phi)((I,\alpha)) &= \mathrm{ext}(D_{m - \mathrm{card}(I) +1, i+1}(\Phi_{m,i,I,J})(\alpha),\mathrm{R}_{i+m+1}), \\ &I \subset_{\leq m+2} J, \mathrm{card}(I)> 1, \alpha \in \mathbf{P}_{m-\mathrm{card}(I)+1,i+1}(\Phi_{m,i,I,J}), \end{align*} $$

(3.14) $$ \begin{align} D^{\prime}_{m,i}(\Phi)((\{j\},\emptyset)) = {\mathcal R}(\Phi(j), \overline{B_k(0,R)}), \end{align} $$

and

$$ \begin{align*} D^{\prime}_{m,i}(\Phi)((I,\alpha)) = D_{m - \mathrm{card}(I) +1,i+1}'(\Phi_{m,i,I,J})(\alpha), \end{align*} $$

for $I \subset _{\leq m+2} J, \mathrm {card}(I)> 1$ , $\alpha \in \mathbf {P}_{m-\mathrm {card}(I)+1,i+1}(\Phi _{m,i,I,J})$ .

The following lemma is obvious from the definition of $D_{m,i}(\alpha )$ given above.

Lemma 3.6. For each $\alpha \in \mathbf {P}_{m,i}(\Phi )$ ,

$$\begin{align*}D_{m,i}(\Phi)(\alpha) \searrow D^{\prime}_{m,i}(\Phi)(\alpha). \end{align*}$$

Proof. Let

$$\begin{align*}\alpha = (I_0^{\alpha},\ldots, I_{r_{\alpha}}^{\alpha} = \{j_{\alpha}\}, \emptyset) \end{align*}$$

with $I_h^{\alpha } \subset J_h^{\alpha }, 0 \leq h \leq r_{\alpha }$ following the same notation as in Section 3.3.3 (with an added superscript ${}^{\alpha }$ ).

First, observe that

(3.15) $$ \begin{align} D_{m,i}(\Phi)(\alpha) = \mathrm{ext}(D^{\prime}_{m,i}(\Phi)(\alpha), \mathrm{R}_{i+m+1}) \cup \bigcup_{\beta \prec \alpha} D_{m,i}(\Phi)(\beta). \end{align} $$

We now prove that for each $\alpha \in \mathbf {P}_{m,i}(\Phi )$ :

(3.16) $$ \begin{align} D_{m,i}(\Phi)(\alpha) \searrow D^{\prime}_{m,i}(\Phi)(\alpha), \end{align} $$

and

(3.17) $$ \begin{align} \bigcup_{\beta \prec \alpha} D_{m,i}(\Phi)(\beta) \searrow \bigcup_{\beta \prec \alpha} \lim_{\bar{\varepsilon}_{i+r_{\alpha}+1}} D_{m,i}'(\Phi)(\beta) \subset D^{\prime}_{m,i}(\Phi)(\alpha). \end{align} $$

The proof is by induction on the maximum length, $\mathrm {length}(\alpha )$ , of any chain with $\alpha $ as the maximal element.

We first note that if $\mathrm {R}' = \mathrm {R}{\langle }\bar {\varepsilon }{\rangle }$ , and $X \subset \mathrm {R}^k$ is a semialgebraic subset, then

$$\begin{align*}\lim_{\bar{\varepsilon}} \mathrm{ext}(X,\mathrm{R}') = \overline{X}. \end{align*}$$

This follows easily from the definition of $\mathrm {ext}(X,\mathrm {R}')$ and standard properties of $\lim _{\bar {\varepsilon }}$ . In particular, if X is a closed semialgebraic set, then

$$\begin{align*}\lim_{\bar{\varepsilon}} \mathrm{ext}(X,\mathrm{R}') = X. \end{align*}$$

Base case of the induction, $\mathrm {length}(\alpha ) = 1$ : It follows from equation (3.15) and the fact that that $D^{\prime }_{m,i}(\Phi )(\alpha )$ is a closed semialgebraic set, that equation (3.16) holds if $\alpha $ is a minimal element of the poset $\mathbf {P}_{m,i}(\Phi )$ (and so $\mathrm {length}(\alpha ) = 1$ ). In this case, equation (3.17) is trivially true.

Induction hypothesis: We assume now that equations (3.16) and (3.17) are true for all $\alpha \in \mathbf {P}_{m,i}(\Phi )$ , with $\mathrm {length}(\alpha ) < t$ .

Inductive step: Suppose that $\alpha \in \mathbf {P}_{m,i}(\Phi )$ , with $\mathrm {length}(\alpha ) = t$ . The inductive hypothesis implies that equations (3.16) and (3.17) both hold with $\alpha $ replaced by $\alpha '$ for all $\alpha ' \prec \alpha $ .

Using the fact that $D^{\prime }_{m,i}(\Phi )(\alpha )$ is closed, it is easy to check that equation (3.17) implies equation (3.16). So we need to prove only equation (3.17). Using the induction hypothesis, we have for each $\beta \prec \alpha $

(3.18) $$ \begin{align} \bigcup_{\beta \prec \alpha} D_{m,i}(\Phi)(\beta) \searrow \bigcup_{\beta \prec \alpha} D_{m,i}'(\Phi)(\beta). \end{align} $$

Now, observe that for any $\beta \in \mathbf {P}_{m,i}(\Phi )$ , $\beta \prec \alpha $ if and only if there exist $j_{\alpha }' \in I_{r_{\alpha } -1}^{\alpha }, j_{\alpha }' \neq j_{\alpha }$ and $j_{\alpha }" \in (J_{r_{\alpha }}^{\alpha })_{m^{\alpha }_{r_{\alpha }}, i^{\alpha }_{r_{\alpha }},\{j_{\alpha },j_{\alpha }'\},\Phi _{r_{\alpha }}}$ such that

$$\begin{align*}\beta \preceq \gamma(j_{\alpha}")= (I_0^{\alpha},\ldots,I_{r_{\alpha}-1}^{\alpha},\{j_{\alpha},j_{\alpha}'\},\{j_{\alpha}"\},\emptyset), \end{align*}$$

where we assume that $I_{-1}^{\alpha } = J$ .

Using the above observation, we have that

(3.19) $$ \begin{align} \bigcup_{\beta \prec \alpha} D_{m,i}'(\Phi)(\beta) = \bigcup_{j_{\alpha}' \in I^{\alpha}_{r_{\alpha} -1}, j_{\alpha}' \neq j_{\alpha}} \bigcup_{j_{\alpha}" \in (J_{r_{\alpha}}^{\alpha})_{m^{\alpha}_{r_{\alpha}}, i^{\alpha}_{r_{\alpha}},\{j_{\alpha},j_{\alpha}'\},\Phi_{r_{\alpha}}}} \left( \bigcup_{\beta \preceq \gamma(j_{\alpha}")} D^{\prime}_{m,i}(\Phi)(\beta) \right), \end{align} $$

where

$$\begin{align*}\gamma(j_{\alpha}")= (I_0^{\alpha},\ldots,I_{r_{\alpha}-1}^{\alpha},\{j_{\alpha},j_{\alpha}'\},\{j_{\alpha}"\},\emptyset). \end{align*}$$

Applying hypothesis (3.17), we have that

(3.20) $$ \begin{align} \left( \bigcup_{\beta \prec \gamma(j_{\alpha}")} D^{\prime}_{m,i}(\Phi)(\beta) \right) \searrow \lim_{\bar{\varepsilon}_{i+r+2}} \bigcup_{\beta \prec \gamma(j_{\alpha}")} D^{\prime}_{m,i}(\Phi)(\beta) \subset D^{\prime}_{m,i}(\Phi)(\gamma(j_{\alpha}")). \end{align} $$

Also, observe that

(3.21) $$ \begin{align} \left( \bigcup_{j_{\alpha}" \in J_{m,i,\{j_{\alpha},j_{\alpha}'\},\Phi}} D^{\prime}_{m,i}(\Phi)(\gamma(j_{\alpha}")) \right) \searrow \left(D^{\prime}_{m,i}(\Phi)(\alpha) \cap D^{\prime}_{m,i}(\Phi)(\alpha') \right) \subset D^{\prime}_{m,i}(\Phi)(\alpha), \end{align} $$

where

$$\begin{align*}\alpha' = (I_0^{\alpha},\ldots, I_{r_{\alpha}-1}^{\alpha}, \{j_{\alpha}'\}, \emptyset). \end{align*}$$

Finally, equation (3.17) now follows from equations (3.18), (3.19), (3.20) and (3.21).

Lemma 3.7.

$$\begin{align*}\left( \bigcup_{\alpha \in \mathbf{P}_{m,i}(\Phi)} D_{m,i}(\Phi)(\alpha) \right) \searrow {\mathcal R}(\Phi, \overline{B_k(0,R)})^J. \end{align*}$$

In particular, $\mathrm {ext}({\mathcal R}(\Phi , \overline {B_k(0,R)})^J, \mathrm {R}_i)$ is a semialgebraic deformation retract of $\bigcup _{\alpha \in \mathbf {P}_{m,i}(\Phi )} D_{m,i}(\Phi )(\alpha ) $ .

Proof. First, note that for each $j \in J$ , $(\{j\},\emptyset )$ is a maximal element of the poset $\mathbf {P}_{m,i}(\Phi )$ . It now follows from Lemma 3.5 that

$$\begin{align*}\bigcup_{\alpha \in \mathbf{P}_{m,i}(\Phi)} D_{m,i}(\Phi)(\alpha) = \bigcup_{j \in J} D_{m,i}(\Phi)((\{j\},\emptyset)). \end{align*}$$

The lemma now follows from Lemma 3.6 and equation (3.14).

Notation 3.9. We will denote the deformation retraction

$$\begin{align*}\bigcup_{\alpha \in \mathbf{P}_{m,i}(\Phi)} D_{m,i}(\Phi)(\alpha) \rightarrow \mathrm{ext}({\mathcal R}(\Phi, \overline{B_k(0,R)})^J, \mathrm{R}_i) \end{align*}$$

in Lemma 3.7 by $r_{m,i}(\Phi )$ .

In the proof of Theorem 2, we need the notion of the homotopy colimit of a functor which we define below.

We fix a real closed field $\mathrm {R}$ in the following definition.

Definition 3.5 (The topological standard n-simplex).

We denote by

$$\begin{align*}|\Delta^n | = \{ (t_0, \ldots, t_n) \in \mathrm{R}^{n+1}_{\geq 0} | \sum_{i=0}^n t_i = 1 \} \end{align*}$$

the standard n-simplex defined over $\mathrm {R}$ . For $0 \leq i \leq n$ , we define the face operators,

$$\begin{align*}d^i_n: |\Delta^{n-1}| \rightarrow |\Delta^{n}|, \end{align*}$$

by

$$\begin{align*}d^i_n(t_0,\dots,t_{n-1}) = (t_0,\dots,t_{i-1}, 0, t_{i}, \dots,t_{n-1}). \end{align*}$$

Definition 3.6 (Homotopy colimit).

Let $(\mathbf {P}, \prec )$ be a poset category and

$$\begin{align*}D:(\mathbf{P}, \prec) \rightarrow \mathbf{Top} \end{align*}$$

a functor taking its values in closed and bounded semialgebraic subsets of $\mathrm {R}^k$ and such that the morphisms $D(\alpha \preceq \beta )$ are inclusion maps. The homotopy colimit of D is the quotient space Footnote ⁴

$$\begin{align*}\mathrm{hocolim}(D) = \left(\coprod_{\alpha_0 \prec \cdots \prec \alpha_p} |\Delta^p| \times D(\alpha_0) \right) \Big/ \hspace{-.1cm} \sim \ , \end{align*}$$

where the equivalence relation $\sim $ is defined as follows.

For a chain $\sigma = (\alpha _0 \prec \cdots \prec \alpha _p)$ , $t \in |\Delta ^p|$ and $x \in D(\alpha _0)$ , we denote by $(t,x)_{\sigma }$ the image of $(t,x)$ under the canonical inclusion of $|\Delta ^p| \times D(\alpha _0)$ (corresponding to the chain $\sigma $ ) in the disjoint union $\coprod _{\alpha _0 \prec \cdots \prec \alpha _p} |\Delta ^p| \times D(\alpha _0)$ .

Using the above notation, the equivalence relation $\sim $ is defined by:

(3.22) $$ \begin{align} (d^i_p (t), x)_{\sigma} \sim (t, x)_{\sigma'}, \end{align} $$

for $x \in D(\alpha _0)$ and $t \in |\Delta ^{p-1}|$ , where $\sigma = (\alpha _0 \prec \cdots \prec \alpha _p)$ and

$$ \begin{align*} \sigma' = \left\{ \begin{array}{ll} (\alpha_1 \prec \cdots \prec \alpha_p) & \text{ if } i =0, \\ (\alpha_0 \prec \cdots \alpha_{i-1} \prec \alpha_{i+1} \prec \cdots \prec \alpha_p) & \text{ if } 0 < i < p, \\ (\alpha_0 \prec \cdots \prec \alpha_{p-1}) & \text{ if } i = p. \end{array} \right. \end{align*} $$

We denote by $\pi ^D_1: \mathrm {hocolim}(D) \rightarrow |\Delta (\mathbf {P})|$ , $\pi ^D_2: \mathrm {hocolim}(D) \rightarrow \mathrm {colim}(D)$ the canonical maps, where $|\Delta (\mathbf {P})|$ is the geometric realization of the order complex of $\mathbf {P}$ (see Definition 3.1). More precisely, $\pi ^D_1$ is the map induced from the projection map

$$\begin{align*}\coprod_{\alpha_0 \prec \cdots \prec \alpha_p} |\Delta^p| \times D(\alpha_0) \rightarrow \coprod_{\alpha_0 \prec \cdots \prec \alpha_p} |\Delta^p| \end{align*}$$

after taking the quotient by $\sim $ , and $\pi ^D_2$ is the composition of the map induced by the projection

$$\begin{align*}\coprod_{\alpha_0 \prec \cdots \prec \alpha_p} |\Delta^p| \times D(\alpha_0) \rightarrow \coprod_{\alpha_0 \prec \cdots \prec \alpha_p} D(\alpha_0) \end{align*}$$

and the canonical map to the colimit of the functor D, which in this case is simply the union $\bigcup _{\alpha \in \mathbf {P}} D(\alpha )$ .

The following example illustrates the definition given above.

Example 3.3. Consider the poset $\mathbf {P} = \{a,b,c\}$ , with three elements with $c \prec a, c \prec b$ as the only nontrivial ordering relation (Hasse diagram shown below).

Let $D:\mathbf {P} \rightarrow \mathbf {Top}$ be the functor, with

$$ \begin{align*} D(a) &= {\mathcal R}((X_1^2 + X_2^2 - 4 =0) \wedge (X_2 \geq 0)), \\ D(b) &= {\mathcal R}((X_1^2 + X_2^2 - 4 =0) \wedge (X_2 \leq 0)), \\ D(c) &= D(a) \cap D(b) \\ &= \{(-2,0), (2,0)\}. \end{align*} $$

The homotopy colimit of the functor D is then the quotient of the disjoint union of the spaces

$$ \begin{align*} \Delta^0 &\times D(a), \Delta^0 \times D(b), \Delta^0 \times D(c), \\ &\ \ \Delta^1 \times D(c), \Delta^1 \times D(c) \end{align*} $$

corresponding to the chains $(a), (b), (c), (c \prec a), (c \prec b)$ by the equivalence relation defined in equation (3.22). The nontrivial identifications induced by the quotienting are given by (following the notation introduced in Definition 3.6)

$$ \begin{align*} ((0,1), (-2,0))_{(c \prec a)} &\sim ((1), (-2,0))_{(c)}, \\ ((0,1), (2,0))_{(c \prec a)} &\sim ((1), (2,0))_{(c)}, \\ ((1,0), (-2,0))_{(c \prec a)} &\sim ((1), (-2,0))_{(a)}, \\ ((1,0), (2,0))_{(c \prec a)} &\sim ((1), (2,0))_{(a)}, \\ ((0,1), (-2,0))_{(c \prec b)} &\sim ((1), (-2,0))_{(c)}, \\ ((0,1), (2,0))_{(c \prec b)} &\sim ((1), (2,0))_{(c)}, \\ ((1,0), (-2,0))_{(c \prec b)} &\sim ((1), (-2,0))_{(b)}, \\ ((1,0), (2,0))_{(c \prec b)} &\sim ((1), (2,0))_{(b)}. \end{align*} $$

The quotient space (as a semialgebraic set) is shown below in Figure 7.

Figure 7 Homotopy colimit of the functor D in Example 3.3.

Proof of Theorem 2.

The theorem will follow from the following two claims.

Claim 3.1. The map $\pi _1^{D_{m,i}(\Phi )}:\mathrm {hocolim}(D_{m,i}(\Phi )) \rightarrow |\Delta (\mathbf {P}_{m,i}(\Phi ))| $ is a homological $\ell $ -equivalence (and so a homological $(m-1)$ -equivalence).

Claim 3.2. The map

$$\begin{align*}F_{m,i}(\Phi) = r_{m,i}(\Phi) \circ \pi_2^{D_{m,i}(\Phi)}:\mathrm{hocolim}(D_{m,i}(\Phi)) \rightarrow \mathrm{ext}({\mathcal R}(\Phi, \overline{B_k(0,R)})^J, \mathrm{R}_i) \end{align*}$$

is a homological $(m-1)$ -equivalence.

We first deduce the proof of the theorem from these two claims. The homological $(m-1)$ -equivalence in equation (3.11) now follows from Claims 3.1, 3.2 and Lemma 3.7.

The diagrammatic homological $(m-1)$ -equivalence in equation (3.12) follows from the commutativity of the following diagrams of maps.

For each pair $J',J" \subset J$ , with $J' \subset J"$ , we have the following commutative diagram, where the vertical arrows are inclusions and the slanted arrows induce isomorphisms in the homology groups up to dimension $m-1$ .

This implies that we have the following diagram of morphisms where both arrows are homological $(m-1)$ -equivalences:

This proves that the diagrams

$$\begin{align*}(J' \mapsto |\Delta(\mathbf{P}_{m,i}(\Phi|_{J'}))|)_{J' \in 2^J} \end{align*}$$

and

$$\begin{align*}\mathbf{Simp}^J({\mathcal R}(\Phi,\overline{B_k(0,R)})) \end{align*}$$

are homologically $(m-1)$ -equivalent.

We now proceed to prove Claims 3.1 and 3.2.

Proof of Claim 3.1.

Let $t \in |\Delta (\mathbf {P}_{m,i}(\Phi ))|$ . Then there exists a unique simplex $\sigma $ of the simplicial complex $\Delta (\mathbf {P}_{m,i}(\Phi ))$ of the smallest possible dimension such that $t \in |\sigma |$ . Let $\alpha _0 \prec \cdots \prec \alpha _p$ be the chain in $\mathbf {P}_{m,i}(\Phi )$ corresponding to $\sigma $ . Then,

$$\begin{align*}(\pi^{D_{m,i}(\Phi)}_1)^{-1}(t) = \{t\} \times D_{m,i}(\Phi)(\alpha_0). \end{align*}$$

It is clear from its definition that $D^{\prime }_{m,i}(\Phi )(\alpha )$ is homologically $\ell $ -connected. From Lemma 3.6, it follows that so is $D_{m,i}(\Phi )(\alpha )$ . It now follows from the homological version of the Vietoris–Begle theorem (see Remark 2.3) that $\pi ^{D_{m,i}(\Phi )}_1$ is a homological $\ell $ -equivalence.

Proof of Claim 3.2.

The claim will follow from the following claims. Let

$$\begin{align*}x \in \mathrm{ext}({\mathcal R}(\Phi,\overline{B_k(0,R)})^{J},\mathrm{R}_i). \end{align*}$$

We will prove that the fiber $(F_{m,i}(\Phi ))^{-1}(x)$ is homologically $(m-1)$ -connected which will suffice to prove that $F_{m,i}(\Phi )$ is a homological $(m-1)$ -equivalence by the homological version of Vietoris–Begle theorem (see Remark 2.3).

In order to study the fiber $(F_{m,i}(\Phi ))^{-1}(x)$ , we define for each $I \subset _{\leq m+2} J$ the following posets of $\mathbf {P}_{m,i}(\Phi )$ .

We define

$$ \begin{align*} \mathbf{P}_I(x) &= \{(I,\alpha) \in \{I\} \times \mathbf{P}_{m-\mathrm{card}(I)+1,i+1}(\Phi_{m,i,I,J}) \mid \\ & x \in \lim_{\bar{\varepsilon}} D_{m-\mathrm{card}(I)+1,i+1}(\Phi_{m,i,I,J})(\alpha)\} \subset \mathbf{P}_{m,i}(\Phi), \end{align*} $$

and

$$ \begin{align*} \mathbf{Q}_I(x) = \bigcup_{I \subset I' \subset_{\leq m+2} J} \mathbf{P}_{I'}(x). \end{align*} $$

The motivation behind the definition of the posets $\mathbf {P}_I(x), \mathbf {Q}_I(x)$ is as follows. First, observe that

(3.23) $$ \begin{align} (F_{m,i}(\Phi))^{-1}(x) = \left|\bigcup_{j \in J} \Delta(\mathbf{Q}_{\{j\}}(x))\right|, \end{align} $$

and

(3.24) $$ \begin{align} \bigcap_{j \in I} \mathbf{Q}_{\{j\}}(x) = \mathbf{Q}_I(x). \end{align} $$

Our strategy for proving the homological $(m-1)$ -connectedness of $(F_{m,i}(\Phi ))^{-1}(x)$ is to use the closed covering provided by equation (3.23) and then use the cohomological Mayer–Vietoris spectral sequence to reduce the problem to studying the connectivity of the various $\left |\Delta (\mathbf {Q}_I(x))\right |$ using equation (3.24). Finally, we prove (see Claim 3.5) that, for each I, $|\Delta (\mathbf {P}_I(x))|$ is homologically equivalent to $|\Delta (\mathbf {Q}_I(x))|$ . This last fact allows us to use induction on the cardinality of I to prove the required connectivity statement for the corresponding $|\Delta (\mathbf {Q}_I(x))|$ .

We now return to the proof of Claim 3.2. Since, for each $I'$ , with $I \subset I' \subset _{\leq m+2} J$ ,

$$\begin{align*}\mathbf{P}_{m -\mathrm{card}(I') +1,i+1}(\Phi_{m,i,I',J}) \subset \mathbf{P}_{m-\mathrm{card}(I) +1,i+1}(\Phi_{m,i,I,J}), \end{align*}$$

there is an injective map,

$$\begin{align*}\mathbf{P}_{I'}(x) \rightarrow \mathbf{P}_I(x), (I',\alpha) \mapsto (I,\alpha). \end{align*}$$

Thus, there is a map

$$\begin{align*}\theta_I(x): \mathbf{Q}_I(x) \rightarrow \mathbf{P}_I(x), \end{align*}$$

defined by

$$\begin{align*}\theta_I(x)((I',\alpha)) = (I,\alpha), \end{align*}$$

for each $(I',\alpha ) \in \mathbf {Q}_I(x),$ where $I \subset I' \subset _{\leq m+2} J$ .

It is obvious from the above definition and the definition of the partial order in $\mathbf {P}_{m,i}(\Phi )$ , that the map $\theta _I(x)$ is a map of posets (i.e., a map respecting the partial orders of the two posets).

Claim 3.3. The map $\theta _I(x)$ induces a simplicial map $\Theta _I(x): \Delta (\mathbf {Q}_I(x))\rightarrow \Delta (\mathbf {P}_I(x))$ . Moreover, the corresponding map $|\Theta _I(x)| : |\Delta (\mathbf {Q}_I(x))| \rightarrow |\Delta (\mathbf {P}_I(x))|$ , between the geometric realizations, is a homological equivalence.

Proof. Since the map $\theta _i(x)$ is a poset map, it carries a chain of $\mathbf {Q}_I(x)$ to a chain of $\mathbf {P}_I(x)$ . This implies that $\theta _I(x)$ induces a simplicial map $\Theta _I(x): \Delta (\mathbf {Q}_I(x))\rightarrow \Delta (\mathbf {P}_I(x))$ .

We now prove the second half of the claim. We are going to use the poset fiber theorem proved in [Reference Sturmfels and Ziegler23, Lemma 3.2] (also [Reference Björner, Wachs and Welker8, Corollary 3.4]).

For $n \geq 0$ , we denote by $\mathcal {B}_n$ the complete Boolean lattice on a set with n elements. It is a well-known fact (see, for example, [Reference Wachs27]) that $|\Delta (\mathcal {B}_n)|$ is homeomorphic to $[0,1]^n$ and is thus contractible.

Let $(I,\alpha ) \in \mathbf {P}_I(x)$ , and $I' \subset _{\leq m+2} J$ be the unique maximal subset of J such that $(I', \alpha ) \in \mathbf {P}_{I'}(x)$ (see the schematic diagram in Figure 8 which depicts a subposet of the poset shown in Figure 4).

Figure 8 $\theta _I(x)^{-1}((I,\alpha ))$ with $I = \{1,2\}$ and $I' = \{1,2,3, 4\}$ .

Then,

$$\begin{align*}\theta_I(x)^{-1}((I,\alpha)) = \{ (I",\alpha) \;\mid\; I \subset I" \subset I' \}. \end{align*}$$

Hence, the poset $\theta _I(x)^{-1}((I,\alpha ))$ is isomorphic as a poset to $\mathcal {B}_{\mathrm {card}(I') - \mathrm {card}(I)}$ . Thus, $|\Delta (\theta _I(x)^{-1}((I,\alpha )))|$ is contractible.

Moreover, for each $(I",\alpha ) \in \theta _I(x)^{-1}((I,\alpha ))$ ,

$$\begin{align*}\theta_I(x)^{-1}((I,\alpha))_{\succ (I",\alpha)} = \{ (I"',\alpha) \;\mid\; I \subset I"' \subset I" \}, \end{align*}$$

and hence $\theta _I(x)^{-1}((I,\alpha ))_{\succ (I",\alpha )}$ is isomorphic to $\mathcal {B}_{\mathrm {card}(I") - \mathrm {card}(I)}$ . This proves that $|\Delta (\theta _I(x)^{-1}((I,\alpha ))_{\succ (I",\alpha )})|$ is contractible for each $(I",\alpha ) \in \theta _I(x)^{-1}((I,\alpha ))$ .

It now follows from the poset fiber theorem [Reference Sturmfels and Ziegler23, Lemma 3.2] (also [Reference Björner, Wachs and Welker8, Corollary 3.4]) that the poset map $\theta _I(x)$ induces a homological equivalence $|\Theta _I(x)| : |\Delta (\mathbf {Q}_I(x))| \rightarrow |\Delta (\mathbf {P}_I(x))|$ .

Observe that Claim 3.3 implies in particular that if $\mathrm {card}(I) = 1$ , then $|\mathbf {Q}_I(x)|$ is contractible if nonempty.

Claim 3.4. For $ x \in \mathrm {ext}({\mathcal R}(\Phi ,\overline {B_k(0,R)})^{J"},\mathrm {R}_i) = \lim _{\bar {\varepsilon }}\bigcup _{\alpha \in \mathbf {P}_{m,i}(\Phi )} D_{m,i}(\Phi )(\alpha ), $

(3.25) $$ \begin{align} \mathrm{H}^j((F_{m,i}(\Phi))^{-1}(x)) &\cong \mathbb{Z}, \mbox{ for } j =0,\\ \nonumber &= 0, \mbox{ for } 0 < j \leq m. \end{align} $$

Proof. The proof is by induction on m starting with the case $m=0$ . The case $m=-1$ is trivial.

Base case ( $m=0$ ). We need to show that, for

$$\begin{align*}x \in \mathrm{ext}({\mathcal R}(\Phi,\overline{B_k(0,R)})^{J},\mathrm{R}_i) = \lim_{\bar{\varepsilon}}\bigcup_{\alpha \in \mathbf{P}_{0,i}(\Phi)} D_{0,i}(\Phi)(\alpha), \end{align*}$$

$(F_{0,i}(\Phi ))^{-1}(x)$ is connected.

First, note that

$$\begin{align*}F_{0,i}(\Phi) = r_{0,i}(\Phi) \circ \pi_2^{D_{0,i}(\Phi)}, \end{align*}$$

and $r_{0,i}(\Phi )$ is a semialgebraic deformation retraction. Hence, $r_{0,i}(\Phi )^{-1}(x)$ is closed and semialgebraically connected (in fact contractible).

Let $J(x) = \{j \in J \mid D_{0,i}(\Phi )((\{j\},\emptyset )) \cap r_{0,i}(\Phi )^{-1}(x) \neq \emptyset \}$ . Since the sets $D_{0,i}(\Phi )((\{j\},\emptyset )), j \in J(x)$ is a covering of the closed and semialgebraically connected set $r_{0,i}(\Phi )^{-1}(x)$ by closed sets, it follows that the union

$$\begin{align*}\bigcup_{j \in J(x)} D_{0,i}(\Phi)((\{j\},\emptyset)) \end{align*}$$

is semialgebraically connected as well. It follows that, given any $j,j' \in J(x)$ , there exists a sequence $j=j_0,j_1,\ldots ,j_N = j'$ such that, for each $h, 0 \leq h \leq N-1$ ,

$$\begin{align*}D_{0,i}(\Phi)((\{j_h\},\emptyset)) \cap D_{0,i}(\Phi)((\{j_{h+1}\},\emptyset)) \cap r_{0,i}(\Phi)^{-1}(x) \neq \emptyset. \end{align*}$$

So there exists for each $h, 0 \leq h \leq N-1$ $j" = (\{j_h,j_{h+1}\},p) \in J_{0,i,\{j_h,j_{h+1}\},\Phi }$ such that

$$\begin{align*}{\mathcal R}(\Phi_{\{j_h,j_{h+1}\}}(p)) \cap r_{m,i}(\Phi)^{-1}(x) \neq \emptyset. \end{align*}$$

So there exists $\alpha = (\{j"\},\emptyset ) \in \mathbf {P}_{-1,i+1}(\Phi _{\{j_h,_{h+1}\}})$ such that

$$\begin{align*}D_{0,i}(\Phi)((\{j_h,j_{h+1}\},\alpha)) \cap r_{0,i}(\Phi)^{-1}(x) \neq \emptyset, \end{align*}$$

and so

$$\begin{align*}(\{j_h,j_{h+1}\},\alpha) \in (F_{0,i}(\Phi))^{-1}(x). \end{align*}$$

Moreover,

$$\begin{align*}(\{j_h,j_{h+1}\},\alpha) \prec (\{j_h\},\emptyset), (\{j_{h+1}\},\emptyset) \end{align*}$$

(using Lemma 3.4). This implies that $(\{j_h\},\emptyset ), (\{j_{h+1}\},\emptyset )$ , and thus every pair of the form $(\{j\},\emptyset ), (\{j'\},\emptyset )$ in $(F_{0,i}(\Phi ))^{-1}(x)$ belongs to the same connected component of $(F_{0,i}(\Phi ))^{-1}(x)$ . Since, for every element of the form $(\{j_h,j_{h+1}\},\alpha ) \in (F_{0,i}(\Phi ))^{-1}(x)$ , we have

$$\begin{align*}(\{j_h,j_{h+1}\},\alpha) \prec (\{j_h\},\emptyset), (\{j_{h+1}\},\emptyset) \in (F_{0,i}(\Phi))^{-1}(x), \end{align*}$$

$(\{j_h,j_{h+1}\},\alpha )$ belong to the same connected component of $(F_{0,i}(\Phi ))^{-1}(x)$ as

$$\begin{align*}(\{j_h\},\emptyset), (\{j_{h+1}\},\emptyset) \end{align*}$$

as well. Together, these facts imply that $(F_{0,i}(\Phi ))^{-1}(x)$ is connected. This proves the claim in the base case.

Inductive step. Suppose we have proved the theorem for all $m', 0 \leq m' < m$ , $i \geq 0$ , all finite $J'$ , and $\Phi ' \in (\mathcal {F}_{k,\mathrm {R}_i})^{J'}$ . We now prove it for $m,i,J,\Phi $ .

$$\begin{align*}x \in \mathrm{ext}({\mathcal R}(\Phi,\overline{B_k(0,R)})^{J},R_i) = \lim_{\bar{\varepsilon}}\bigcup_{\alpha \in \mathbf{P}_{m,i}(\Phi)} D_{m,i}(\Phi)(\alpha). \end{align*}$$

Recall from equation (3.23) that

$$\begin{align*}(F_{m,i}(\Phi))^{-1}(x) = \left|\bigcup_{j \in J} \Delta(\mathbf{Q}_{\{j\}}(x))\right|. \end{align*}$$

Let

$$\begin{align*}J' = \{j \in J\;\mid \; \mathbf{Q}_{\{j\}}(x) \neq \emptyset \}. \end{align*}$$

So

$$\begin{align*}(F_{m,i}(\Phi))^{-1}(x) = |\bigcup_{j \in J'} \Delta(\mathbf{Q}_{\{j\}}(x))|. \end{align*}$$

It follows from the Mayer–Vietoris exact sequence in cohomology for closed subspaces (see, for example, [Reference Iversen17, page 148]) that there exists a spectral sequence

$$\begin{align*}E_r^{p,q} \Rightarrow \mathrm{H}^{p+q}\left(\left|\bigcup_{j \in J'} \Delta(\mathbf{Q}_{\{j\}}(x))\right|\right) \end{align*}$$

whose $E_1$ term is given by

$$\begin{align*}E_1^{p,q} = \bigoplus_{I \subset J', \mathrm{card}(I) = p+1} \mathrm{H}^q\left(\left|\bigcap_{j \in I} \Delta(\mathbf{Q}_{\{j\}}(x))\right|\right). \end{align*}$$

Notice that

$$\begin{align*}\bigcap_{j \in I} \mathbf{Q}_{\{j\}}(x) = \mathbf{Q}_I(x), \end{align*}$$

and it follows from Claim 3.3 that $ |\Delta (\mathbf {Q}_I(x))| $ is homotopy equivalent to $ |\Delta (\mathbf {P}_I(x))| $ .

So we get

$$\begin{align*}E_1^{p,q} = \bigoplus_{I \subset J', \mathrm{card}(I) = p+1} \mathrm{H}^q(|\Delta(\mathbf{P}_I(x))|). \end{align*}$$

Now, for I, with $\mathrm {card}(I)> 1$ , we can apply the induction hypothesis to deduce that

$$ \begin{align*} \mathrm{H}^j(|\Delta(\mathbf{P}_I(x))|) &\cong \mathbb{Z}, \mbox{ for } j =0, \\ &= 0, \mbox{ for } 0 < j \leq m -\mathrm{card}(I)+1. \end{align*} $$

We can deduce from this that

$$ \begin{align*} E_1^{p,0} &\cong \bigoplus_{I \subset J', \mathrm{card}(I) = p+1} \mathbb{Z}, \\ E_1^{p,q} & \cong 0, \mbox{ for } 0 < q \leq m - p. \end{align*} $$

It follows that

$$ \begin{align*} E_2^{0,0} &\cong \mathbb{Z}, \\ E_2^{p,0} &\cong 0, p> 0 \\ E_2^{p,q} & \cong 0, \mbox{ for } p \geq 0, 0 < q \leq m-p.\\[-38pt] \end{align*} $$

Note that it follows from Claim 3.5 and the Mayer–Vietoris spectral sequence argument used in its proof that for any

$$\begin{align*}J' \subset \{j \in J\;\mid \; \mathbf{Q}_{\{j\}}(x) \neq \emptyset \}, \end{align*}$$

(3.26) $$ \begin{align} \mathrm{H}^j\left(\left|\bigcup_{j \in J'} \Delta(\mathbf{Q}_{\{j\}}(x))\right|\right) &\cong \mathbb{Z}, \mbox{ for } j =0,\\ \nonumber &= 0, \mbox{ for } 0 < j \leq m. \end{align} $$

Claim 3.5. For

$$\begin{align*}x \in \mathrm{ext}({\mathcal R}(\Phi,\overline{B_k(0,R)})^{J},\mathrm{R}_i) = \lim_{\bar{\varepsilon}}\bigcup_{\alpha \in \mathbf{P}_{m,i}(\Phi)} D_{m,i}(\Phi)(\alpha), \end{align*}$$

$(F_{m,i}(\Phi ))^{-1}(x)$ is homologically $(m-1)$ -connected.

Proof. Let $X=F_{m,i}(\Phi )^{-1}(x)$ . It follows from [Reference Spanier22, Theorem 12, page 248] that there exists a short exact sequence:

$$\begin{align*}0 \rightarrow \mathrm{Ext}(\mathrm{H}^{q+1}(X),\mathbb{Z}) \rightarrow \mathrm{H}_q(X) \rightarrow \mathrm{Hom} (\mathrm{ H}^{q}(X),\mathbb{Z}) \rightarrow 0. \end{align*}$$

Thus, for each $q>0$

$$\begin{align*}\mathrm{H}^{q+1}((F_{m,i}(\Phi))^{-1}(x)) = \mathrm{H}^{q}((F_{m,i}(\Phi))^{-1}(x)) = 0 \end{align*}$$

implies that $\mathrm {H}_{q}((F_{m,i}(\Phi ))^{-1}(x)) = 0$ .

The claim now follows from equation (3.25).

Claim 3.2 now follows from Claim 3.5 and the homological version of the Vietoris-Begle theorem (see Remark 2.3).

This completes the proof of Theorem 2.