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# Intersection homotopy, refinements and coarsenings

Martintxo Saralegi-Aranguren, Daniel Tanré

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Martintxo Saralegi-Aranguren, Daniel Tanré. Intersection homotopy, refinements and coarsenings. 2025. hal-05204334

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Preprint submitted on 8 Aug 2025

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INTERSECTION HOMOTOPY, REFINEMENTS AND
COARSENINGS

MARTINTXO SARALEGI-ARANGUREN AND DANIEL TANRÉ

ABSTRACT. In previous works, we studied intersection homotopy groups associated to a Goresky and MacPherson perversity $\bar{p}$ and a filtered space $X$. They are defined as the homotopy groups of simplicial sets introduced by P. Gajer. We particularized to locally conical spaces of Siebenmann (called CS sets) and established a topological invariance for them when the regular part remains unchanged.

Here, we consider coarsenings, made of two structures of CS sets on the same topological space, the strata of one being a union of strata of the other. We endow them with a general perversity and its pushforward, where the adjective “general” means that the perversities are defined on the poset of the strata and not only according to their codimension. If the perversity verifies a growing property analogous to that of the original perversities of Goresky and MacPherson, we also find an invariance theorem for the intersection homotopy groups of a coarsening, under the above restriction on the regular parts. An invariance is shown too in some cases where singular strata become regular in the coarsening, for Thom-Mather spaces.

CONTENTS

<table><tr><td>1. Filtered spaces and CS sets</td><td>4</td></tr><tr><td>2. Coarsenings</td><td>5</td></tr><tr><td>3. Perversities</td><td>7</td></tr><tr><td>4. Intersection homotopy groups</td><td>10</td></tr><tr><td>5. Connectivity and CS-coarsenings</td><td>11</td></tr><tr><td>6. Intersection fundamental group and CS-coarsenings</td><td>12</td></tr><tr><td>7. Proofs of Theorems A and B</td><td>16</td></tr><tr><td>8. Refinement and examples</td><td>20</td></tr><tr><td>References</td><td>22</td></tr></table>

<footnote>Date: August 6, 2025.</footnote>

<footnote n="2020">2020 Mathematics Subject Classification. 55Q70, 14F43, 55N33, 32S70.</footnote>

<footnote>Key words and phrases. Intersection homotopy groups; Coarsening; Refinement invariance; Topological invariance.</footnote>

<footnote>The second author was partially supported by the Proyecto PID2020-114474GB-100 and the ANR-11-LABX-0007-01 “CEMPI”.</footnote>
This work is concerned with intersection homotopy groups of filtered spaces, $(X, S)$, endowed with a perversity $\bar{p}$. They are defined as the homotopy groups of the simplicial set $\mathcal{G}_{\bar{p}}(X, S)$ introduced by Gajer ([Gaj96, Gaj98]); i.e., $\pi_{\star}^{\bar{p}}(X, S, x_0) = \pi_{\star}(\mathcal{G}_{\bar{p}}(X, S), x_0)$. For instance, $(X, S)$ is $\bar{p}$-connected if $\mathcal{G}_{\bar{p}}(X, S)$ is connected.

In previous works ([CSAT24a, CSAT24b, CSAT25]), we studied the relations between these groups and the intersection homology groups of Goresky and MacPherson ([GM80]), establishing a Hurewicz theorem and a Van Kampen theorem. We refer the interested reader to [CSAT25] for more details. Here, we focus on the topological invariance of the $\bar{p}$-intersection homotopy groups defined from a perversity $\bar{p}$ and a topological space endowed with different filtrations.

Let's start by clarifying the situation. In [GM80, GM83], topological invariance means the existence of isomorphisms between the intersection homology groups of a given pseudomanifold for any GM-perversity (see Definition 8.4) independently of the chosen filtration. In [CSAT25], we were looking for conditions that guarantee the invariance of intersection homotopy groups within the framework of GM-perversities and locally conical filtered spaces with a structure of manifold on each stratum. These spaces are the CS sets of Siebenmann ([Sie72]) recalled in Definition 1.4. From King's work ([Kin85]) on the topological invariance of intersection homology without sheaf, there is a preferred topological pattern. King's sketch of proof is as follows: given a CS set $(X, S)$, he constructs an explicit intrinsic CS set $(X, \mathcal{X}^*)$ which is the coarsest filtration on $X$ as a CS set. The proof of topological invariance reduces to the existence of an isomorphism between the $\bar{p}$-intersection homology groups of $(X, S)$ and $(X, \mathcal{X}^*)$. In [CSAT25], we adapted this idea to the invariance of $\bar{p}$-intersection homotopy groups. The example of a double suspension on a Poincaré sphere shows that such invariance cannot be true in full generality. We established it for the CS sets $(X, S)$ having the same regular part than $(X, \mathcal{X}^*)$.

In the present work, we come back to the topological invariance of intersection homotopy groups and establish it with weaker requirements. First, we consider more general perversities, $\bar{p}: S \to \bar{Z} = Z \cup \{-\infty, \infty\}$, introduced by MacPherson ([Mac91]). Defined on the family $S$ of strata of a filtered space $X$, they take values in $\bar{Z}$. (We already addressed this situation in the context of intersection homology in [CSAT19].) Since these perversities are defined on the set of strata, they depend on the chosen filtration and we could conclude that there is no hope of topological invariance with this generality. In fact, we must not forget that, for a topological invariance, only one GM-perversity appears. Here we have two filtrations, therefore a priori two perversities, and we have to impose a link between them, that will bring the invariance. Let's explain this.

We handle with triples $(X, S, \mathcal{T})$ formed of a topological space $X$, endowed with two structures of CS sets, $(X, S)$ and $(X, \mathcal{T})$, such that any stratum of $\mathcal{T}$ is a union of strata of $S$. The condition required between the strata of $S$ and $\mathcal{T}$ is equivalent to asking that the identity map defines a stratified map, $\iota: (X, S) \to (X, \mathcal{T})$. We call $(X, S, \mathcal{T})$ a CS-coarsening. Let us call exceptional a singular stratum of $S$ which is in the regular part of $\mathcal{T}$. The absence of exceptional strata means that $(X, S)$ and $(X, \mathcal{T})$ have the same regular part.

For the starting point with perversities, we have two choices: a perversity $\bar{p}$ defined on $S$ or a perversity $\bar{q}$ on $\mathcal{T}$. As $\iota$ is a stratified map, a perversity $\bar{p}$ on $S$ generates a pushforward perversity $\iota_*\bar{p}$ on $\mathcal{T}$ and a perversity $\bar{q}$ on $\mathcal{T}$ produces a pullback perversity $\iota^*\bar{q}$ on $S$, see Definition 3.2. In conclusion, we are faced with the existence of two isomorphisms, one connecting $\bar{p}$ and $\iota_*\bar{p}$ and the other associating $\iota^*\bar{q}$ and $\bar{q}$. Before
stating them, we still need to specify the required properties on perversities. GM-perversities are defined to satisfy, in particular, the following key property,

$$ \bar{p}(i) \le \bar{p}(i+1) \le \bar{p}(i) + 1. \qquad (1) $$

We adapt these inequalities to our more general context and obtain the notion of K-perversity on a CS-coarsening, see Definition 3.3. We begin with the relation between a perversity $\bar{p}$ on $S$ and its pushforward perversity $\iota_{\star}\bar{p}$ on $T$. The first result extends [CSAT25, Theorem A] to general perversities. (Recall that $\bar{t}$ is the top perversity, defined by $\bar{t}(S) = \mathrm{codim} S - 2$, if $S$ is singular.)

**Theorem A.** Let $(X, S, T)$ be a CS-coarsening without exceptional strata and $\bar{p}$ be a K-perversity on $(X, S, T)$ of pushforward perversity $\iota_{\star}\bar{p}$ on $T$, such that $\bar{p} \le \bar{t}$. Then, for any regular point $x_0$, the identity map induces an isomorphism

$$ \pi_{\star}^{\bar{p}}(X, S, x_0) \cong \pi_{\star}^{\iota_{\star}\bar{p}}(X, T, x_0). $$

We now go beyond the context of [CSAT25], by admitting some exceptional strata. With the example of a double suspension on a Poincaré sphere ([CSAT25, Example 5.3] or its recall in Example 8.3), we know that an invariance cannot exist without additional hypotheses on the exceptional strata. We succeed for Thom-Mather spaces with a “local hypothesis” on links. In fact, we only need a weaker notion of Thom-Mather space, requiring only the existence of neighborhoods of strata, without control data, see Definition 6.2.

**Theorem B.** Let $(X, S, T)$ be a CS-coarsening such that $(X, S)$ is a normal connected Thom Mather space, with a finite number of strata. Let $\bar{p}$ be a K-perversity on $(X, S, T)$, of pushforward perversity $\iota_{\star}\bar{p}$ on $T$, such that $\bar{p} \le \bar{t}$. We suppose that the link, $L$, of any exceptional stratum verifies $\pi_{\star}^{\bar{p}}(\mathcal{c}L, \mathcal{c}S) = \{1\}$. Then, for any regular point $x_0$, the identity map induces an isomorphism

$$ \pi_{\star}^{\bar{p}}(X, S, x_0) \cong \pi_{\star}^{\iota_{\star}\bar{p}}(X, T, x_0). $$

Let $(X, S, T)$ be a CS-coarsening without exceptional strata of codimension 1. If $\bar{q}$ is a perversity on $(X, T)$, we observe (Lemma 3.7) that the pullback perversity $\iota_{\star}\bar{q}$ is a K-perversity on $(X, S, T)$ and that we have $\iota_{\star}\iota_{\star}\bar{q} = \bar{q}$. Thus the two previous results, applied to $\bar{p} := \iota_{\star}\bar{q}$, infer isomorphisms induced by the identity map,

$$ \pi_{\star}^{\iota_{\star}\bar{q}}(X, S, x_0) \cong \pi_{\star}^{\bar{q}}(X, T, x_0), $$

see Theorems 8.1 and 8.2 for explicit statements.

In Corollary 8.5, we refine the topological invariance of [CSAT25, Theorem 6.1] under the slight more general form of perversities introduced by H. King in [Kin85]. In this context, the intersection homotopy groups do not depend on the chosen stratification, if the regular part remains unchanged.

Finally, let's add a word about the method used in the proofs. We follow the scheme introduced and developed in [SA21, SA24]: any CS-coarsening $(X, S, T)$ can be decomposed in a series of CS-coarsenings in which, roughly, only two strata of $S$ are merged to give a stratum of $T$. The corresponding conical charts can be of three types which are described in the *op. cit.* and recalled in Section 2. They are the keys in the proofs of the main theorems.

Let's specify some notation and convention. Here a topological space, or more simply a space, means a compactly generated topological space and our definition of compact space includes the Hausdorff property. For a space $X$, we denote by $cX$ the quotient $cX := X \times [0, 1]/(X \times \{0\})$. The *open cone* $\mathcal{c}X$ is the quotient $\mathcal{c}X := X \times [0, 1]/(X \times \{0\})$. A point of the cone is denoted by $[x, t]$ and the apex is $v = [-, 0]$. The symbol
* is reserved for join operations between filtered spaces. Indices or superscripts are designated by ⋆.

Let $m \in \mathbb{N}$. The sphere of dimension $m$ is denoted $S^m$ and $\mathbb{S}^m$ denotes a homological sphere of dimension $m$. All stratified maps defined by the identity map are designed by $\iota$ and their induced homomorphisms, in homology or homotopy, by $\iota_\star$.

# 1. FILTERED SPACES AND CS SETS

The singular spaces of this section are general filtered spaces and the CS sets introduced by Siebenmann ([Sie72]).

**Definition 1.1.** A filtered space of formal dimension $n$ is a non-empty space $X$ with a filtration,

$$X_{-1} = \emptyset \subseteq X_0 \subseteq X_1 \subseteq \cdots \subseteq X_{n-2} \subseteq X_{n-1} \subseteq X_n = X,$$

by closed subsets. The subspaces $X_i$ are called skeleta of the filtration and the non-empty components of $X^i = X_i \setminus X_{i-1}$ are the strata of $X$. The set of strata, also called the stratification of $X$, is denoted by $S_X$ (or $\mathcal{S}$ if there is no ambiguity). The subspace $\Sigma_S = X_{n-1}$ (or $\Sigma$ if there is no ambiguity) is called the singular subspace. Its complementary, $X \setminus \Sigma_S$, is the regular subspace and the strata in $X \setminus \Sigma_S$ are called regular. The formal dimension of a stratum $S \subset X^i$ is dim $S = i$; its formal codimension is codim $S = n - i$. (This notion is not necessarily related to a topological notion of dimension.) As we are working with different filtrations on the same topological space, we often use the expression “the filtered space $(X, \mathcal{S})$” which means that $X$ is a filtered space and $\mathcal{S}$ is its set of strata.

The simplest example of filtered space is a non-empty space with the stratification by its connected components; we denote it by $(X, \mathcal{I})$ or $(X, X_{cc})$.

**Examples 1.2.** Let $(X, \mathcal{S})$ be a filtered space of dimension $n$. An open subset $U \subset X$ is a filtered space for the induced filtration, $S_U$, given by $U_i = U \cap X_i$. For the sake of simplification, if there is no ambiguity, we keep the notation $\mathcal{S}$ for the stratification on $U$ and denote by $(U, \mathcal{S})$ this induced filtered space. The product $Y \times X$ with a topological space $Y$ is a filtered space for the product filtration defined by $(Y \times X)_i = Y \times X_i$. The stratification is denoted by $\mathcal{I} \times \mathcal{S}$; its elements are products of a connected component of $Y$ with a stratum of $\mathcal{S}$. If $X$ is compact, the open cone $\mathring{c}X = X \times [0, 1[X \times \{0\}]$ is endowed with the cone filtration defined by $(\mathring{c}X)_i = \mathring{c}X_{i-1}$, $0 \le i \le n+1$. (By convention, $\mathring{c}\emptyset = \{v\}$.) The stratification, denoted by $\mathring{c}\mathcal{S}$, is constituted of the apex $\{v = [-0, 0]\}$ and the products $]0, 1[ \times S$ with a stratum $S \in \mathcal{S}$.

**Definition 1.3.** A stratified map $f: X \to Y$ between two filtered spaces is a continuous map such that, for each stratum $S$ of $X$, there exists a stratum $S^f$ of $Y$ with $f(S) \subset S^f$ and codim $S^f \le$ codim $S$. A stratified homeomorphism is a homeomorphism such that $f$ and $f^{-1}$ are stratified maps.

A stratified map $f$ which is a homeomorphism but for which the reverse application $f^{-1}$ is not stratified is called a homeomorphism and a stratified map. It is fundamental not to confuse it with a stratified homeomorphism.

**Definition 1.4.** A CS set of dimension $n$ is a Hausdorff filtered space $(X, \mathcal{S})$ of dimension $n$, whose $i$-dimensional strata are $i$-dimensional topological manifolds for each $i$, and such that for each point $x \in X_i \setminus X_{i-1}$, $i \ne n$, there exist an open neighborhood $V$ of $x$ in $X$, endowed with the induced filtration, an open neighborhood $U$ of $x$ in $X_i \setminus X_{i-1}$, a compact filtered space, $(L, \mathcal{L})$, of (formal) dimension $n - i - 1$, and a stratified homeomorphism involving stratifications recalled in Examples 1.2:

$$\varphi: (U \times \mathring{c}L, \mathcal{I} \times \mathring{c}\mathcal{L}) \to (V, \mathcal{S}).$$
If $v$ is the apex of the cone $\mathring{c}L$, the homeomorphism $\varphi$ is also required to verify $\varphi(u, v) = u$, for each $u \in U$. The pair $(V, \varphi)$ is a conical chart of $x$ and the filtered space $L$ is a link of $x$. The CS set $X$ is called normal if its links are connected.

Let us note that a CS set has a topological dimension which coincides with the formal one. We collect below some well-known basic properties of a CS set $(X, \mathcal{S})$, with a proof reference.

* Any stratum of $X$ is a connected manifold and its closure is a union of strata of lower dimension ([Fri20, Lemma 2.3.7]).

* Since the links are non-empty sets, the open subset $X\setminus\Sigma$ is dense. So, for each link $L$, the regular part $L\setminus\Sigma_L$ is dense in $L$.

* The set $\mathcal{S}$ of strata is a poset ([CSAT18, Proposition A.22]) for the relation $S_0 \leq S_1$ if $S_0 \subseteq \bar{S}_1$. (If $S_0 \neq S_1$, we note $S_0 < S_1$.) The closed strata are exactly the minimal strata. The open strata are the maximal strata so they coincide with the $n$-dimensional strata. The relation $<$ is compatible with the dimension, in the sense that $S < S'$ implies $\dim S < \dim S'$. The depth of $(X, \mathcal{S})$ is the greatest integer $m$ for which it exists a chain of strata, $S_0 < S_1 < \dots < S_m$. We denote it by depth $X = m$.

* For any singular stratum $\mathcal{S}$, there is a regular stratum $\mathcal{Q}$ such that $S < \mathcal{Q}$.

* Strata are locally closed and form a locally finite family ([Fri20, Lemma 2.3.8]) which means that every point has a neighborhood that intersects only a finite number of strata.

* The links of a stratum are not uniquely defined but if one of them is connected, so all of them are too ([Fri20, Remark 2.6.2]).

* If $X$ is a manifold, then any link is a homology sphere. So, if $X$ has no strata of codimension 1, it is normal ([CSAT25, Remark 2.5]).

## 2. COARSENINGS

In this section, we specify the notion of CS-coarsenings and their main properties. We also develop the 3 types of strata encountered in this situation, with an explicit description of the associated local structures.

**Definition 2.1.** A coarsening is a triple $(X, \mathcal{S}, \mathcal{T})$ formed of two filtered spaces, $(X, \mathcal{S})$ and $(X, \mathcal{T})$, on the same topological space $X$ such that any stratum in $\mathcal{T}$ is a union of strata of $\mathcal{S}$. We also say that $(X, \mathcal{S})$ is a refinement of $(X, \mathcal{T})$. If $S \in \mathcal{S}$, we denote by $S^u \in \mathcal{T}$ the unique stratum of $\mathcal{T}$ which contains $S$.

If both filtered spaces of a coarsening are CS sets, we speak of a CS-coarsening. A pointed CS-coarsening is a quadruplet $(X, \mathcal{S}, \mathcal{T}, x_0)$ formed of a CS-coarsening and a regular point $x_0 \in X \setminus \Sigma_{\mathcal{S}} \subset X \setminus \Sigma_{\mathcal{T}}$.

In general a coarsening of a CS set is not a CS set ([SA21, Remark 4.3]). In fact, there are even coarsenings, $(X, \mathcal{S}, \mathcal{R})$ and $(X, \mathcal{R}, \mathcal{T})$, such that $(X, \mathcal{S})$ and $(X, \mathcal{T})$ are CS sets and $(X, \mathcal{R})$ is not, see Example 8.6. In the rest of this text, we only consider CS-coarsenings. Any CS set $(X, \mathcal{S})$ has an intrinsic CS-coarsening $(X, \mathcal{S}, \mathcal{X}^*)$, where $\mathcal{X}^*$ is the coarsest filtration on $X$ as a CS set, see [Kin85] or [Fri20, Proposition 2.10.5].

Let $(X, \mathcal{S}, \mathcal{T})$ be a CS-coarsening. The condition required between the strata of $\mathcal{S}$ and $\mathcal{T}$ is equivalent to asking that the identity map defines a stratified map, $\iota: (X, \mathcal{S}) \to (X, \mathcal{T})$. A regular stratum of $\mathcal{S}$ is necessarily included in a regular stratum of $\mathcal{T}$. For a singular stratum, the situation is different: a singular stratum can be included in either a regular stratum or a singular stratum of $\mathcal{T}$. Let us begin with the first case.

**Definition 2.2.** Let $(X, \mathcal{S}, \mathcal{T})$ be a CS-coarsening. A singular stratum $S \in \mathcal{S}$ is exceptional in $(X, \mathcal{S}, \mathcal{T})$ if $S^u$ is a regular stratum in $\mathcal{T}$. Moreover, if $S$ has codimension 1, we say that it is a 1-exceptional stratum.
If $S$ is exceptional, we have $\dim S < \dim S^t$. More generally, the fact that the coarsening keeps or not the dimension of a given stratum is a fundamental distinction. So we introduce the following definitions.

**Definition 2.3.** Let $(X, S, \mathcal{T})$ be a CS-coarsening. A stratum $S \in S$ is a source stratum in $(X, S, \mathcal{T})$ if $\dim S = \dim S^t$. A non-exceptional stratum $S \in S$ is a fountain stratum in $(X, S, \mathcal{T})$ if $\dim S < \dim S^t$.

By dimensional reasons, any stratum $T \in \mathcal{T}$ admits a source stratum $S \in S$; i.e., $T = S^t$. Moreover, the family of source strata of $T$ is always an open dense subset of $T$. Exceptional strata verify the inequality on dimensions but their particularity in the conservation of intersection homotopy groups of coarsenings makes us treat them separately.

Let us specify the different local situations in a simple CS-coarsening, as it appears in [SA21, Proposition 4.4].

**Definition 2.4.** A CS-coarsening $(X, S, \mathcal{T})$ is simple if the poset $\mathcal{V}$, formed of exceptional strata and fountain strata, is of depth one.

Suppose $(X, S, \mathcal{T})$ is a simple CS-coarsening.

• Let $S \in S$ be an exceptional stratum and $x \in S \subset S^t$. We already observed that $b = \dim S^t - \dim S \ge 1$. There exist an $S$-conical chart $(\phi, W)$ of $x \in S$ whose link is $(S^{b-1}, \mathcal{I})$, a $\mathcal{T}$-conical chart $(\psi, W)$ of $x \in S^t$, and a commutative diagram where $f$ is a homeomorphism,

$$ (\mathbb{R}^a \times \check{\mathbb{S}}^{b-1}, \mathcal{I} \times \check{\mathcal{I}}) \xrightarrow{\phi} (W, S) \\ f \downarrow \qquad \downarrow \iota \\ (\mathbb{R}^{a+b}, \mathcal{I}) \xrightarrow{\psi} (W, \mathcal{T}). \qquad (2) $$

• Let $S \in S$ be a source stratum and $x \in S \subset S^t$. There exist an $S$-conical chart $(\phi, W)$ of $x \in S$ whose link is $(L, \mathcal{L})$, a $\mathcal{T}$-conical chart $(\psi, W)$ of $x \in S^t$ whose link is $(L, \mathcal{L}')$ for some coarsening $\mathcal{L}'$ of $\mathcal{L}$, and a commutative diagram,

$$ (\mathbb{R}^a \times \check{\mathcal{L}}, \mathcal{I} \times \check{\mathcal{L}}) \xrightarrow{\psi} (W, S) \\ \iota \downarrow \qquad \downarrow \iota \\ (\mathbb{R}^a \times \check{\mathcal{L}}, \mathcal{I} \times \check{\mathcal{L}}') \xrightarrow{\psi} (W, \mathcal{T}). \qquad (3) $$

For the description of the conical charts of a point in a fountain stratum, we need to recall the join of two filtered spaces in a particular case.

**Example 2.5.** Let $(X, S)$ be a filtered space and $S^m$ be a homological sphere, $m \ge 0$. We consider the join defined as a quotient, $S^m * X = cS^m \times X / \sim$, with an equivalence relation generated by $([z, 1], x) \sim ([z, 1], x')$. The elements of the join $S^m * X$ are denoted by $[[z, t], x]$. We identify $S^m$ with the subspace formed of classes $[[z, 1], x]$ and $X$ with the subspace of classes $[[z, 0], x]$. The product $S^m \times \check{\mathcal{X}}$ is also naturally included in the join through the map $(z, [x, t]) \mapsto [[z, 1 - t], x]$. We endow $S^m * X$ with the join filtration defined by

$$ S^m \subset S^m * X_0 \subset \cdots \subset S^m * X_{n-1} \subsetneq S^m * X_n. $$

The corresponding set of strata, called the join stratification and denoted by $S^m * S$, has for elements the connected components of $S^m$ and the products $cS^m \times S$ with $S \in S$. The previous inclusions of $S^m$, $X$ and $S^m \times \check{\mathcal{X}}$ in $S^m * X$ induce stratified maps.
The cone on a join can be decomposed as a product via a homeomorphism $h: \mathring{c}(\mathbb{S}^m * X) \to \mathring{c}\mathbb{S}^m \times \mathring{c}X$, see [Bro06, 5.7.4]. To understand the behavior of $h$ with respect to stratifications, we introduce the refinement of a stratification by adding a point. More specifically, let $(X, \mathcal{S})$ be a filtered space and $x$ be a point of $X$ such that $\{x\} \notin \mathcal{S}$. We define a new filtration by adding $\{x\}$ to the bottom of the filtration. We get

$$X_0 \cup \{x\} \subset X_1 \cup \{x\} \subset \dots \subset X_n \cup \{x\} = X,$$

whose associated stratification, denoted by $\mathcal{S}_x$, has for elements $\{x\}$ and the connected components of $S\backslash\{x\}$ with $S \in \mathcal{S}$.

Denote by $\mathfrak{u}$ the apex of $\mathring{c}\mathbb{S}^m$ and by $\mathfrak{v}$ that of $\mathring{c}X$. The homeomorphism $h$ induces (see [SA21, Proposition 3.3]) a stratified homeomorphism

$$h: ((\mathring{c}(\mathbb{S}^m * X), \mathring{c}(\mathbb{S}^m * \mathcal{S})) \to (\mathring{c}\mathbb{S}^m \times \mathring{c}X, (\mathcal{I} \times \mathring{c}\mathcal{S})_{(\mathfrak{u}, \mathfrak{v})}). \quad (4)$$

• Let $S \in \mathcal{S}$ be a *fountain stratum* and $x \in S \subset S^i$. Denote $b = \dim S^i - \dim S \ge 1$. Then, there exist a $\mathcal{T}$-conical chart $(\psi, W)$ of $x \in S^i$ whose link is $(L, \mathcal{L})$, an $\mathcal{S}$-conical chart $(\phi, W)$ of $x \in S$ whose link is $(\mathbb{S}^{b-1} * L, \mathbb{S}^{b-1} * \mathcal{L})$, and a commutative diagram where $J$ is a homeomorphism,

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
(\mathbb{R}^a \times \mathring{c}(\mathbb{S}^{b-1} * L), \mathcal{I} \times \mathring{c}(\mathbb{S}^{b-1} * \mathcal{L})) \arrow[r, "\phi"] & (W, S) \\
(\mathbb{R}^{a+b} \times \mathring{c}L, \mathcal{I} \times \mathring{c}\mathcal{L}) \arrow[u, "J"'] \arrow[r, "\psi"] & (W, \mathcal{T})
\end{tikzcd} \quad (5)$$

The homeomorphism $J$ is equal to ([SA21, Proposition 2.7]) $J = (f \times \mathrm{id}) \circ (\mathrm{id} \times h)$ where $h: \mathring{c}(\mathbb{S}^{b-1} * L) \to \mathring{c}\mathbb{S}^{b-1} \times \mathring{c}L$ is the stratified homeomorphism defined in (4) and the homeomorphism $f: (\mathbb{R}^a \times \mathring{c}\mathbb{S}^{b-1}, (0, \mathfrak{v})) \to (\mathbb{R}^{a+b}, 0)$ already appears in (2). Diagram (5) can therefore be presented in the following alternative form, where $\mathfrak{u}$ is the apex of $\mathring{c}\mathbb{S}^{b-1}$ and $\mathfrak{v}$ the apex of $\mathring{c}L$,

$$\begin{tikzcd}[column sep=2.8em, row sep=2.8em]
(\mathbb{R}^a \times (\mathring{c}\mathbb{S}^{b-1} \times \mathring{c}L), \mathcal{I} \times (\mathcal{I} \times \mathring{c}\mathcal{L})_{(\mathfrak{u}, \mathfrak{v})}) \arrow[r, "\phi'"] & (W, S) \\
((\mathbb{R}^a \times \mathring{c}\mathbb{S}^{b-1}) \times \mathring{c}L, (\mathcal{I} \times \mathcal{I}) \times \mathring{c}\mathcal{L}) \arrow[u, "\iota"] \arrow[r, "\psi'"] & (W, \mathcal{T})
\end{tikzcd} \quad (6)$$

Both maps $\psi' = \psi \circ (f \times \mathrm{id})$ and $\phi' = \phi \circ (\mathrm{id} \times h^{-1})$ are stratified homeomorphisms. Concerning the stratifications, the space $\mathbb{R}^a \times \mathring{c}\mathbb{S}^{b-1} \times \{\mathfrak{v}\}$ is a stratum on the bottom line. On the top line, this space is the union of the strata $\mathbb{R}^a \times \{(\mathfrak{u}, \mathfrak{v})\}$ and $\mathbb{R}^a \times (\mathring{c}\mathbb{S}^{b-1}\setminus\{\mathfrak{u}\})_{cc} \times \{\mathfrak{v}\}$. In short, we have

$$\mathbb{R}^a \times \{(\mathfrak{u}, \mathfrak{v})\} \le \mathbb{R}^a \times (\mathring{c}\mathbb{S}^{b-1}\setminus\{\mathfrak{u}\})_{cc} \times \{\mathfrak{v}\}, \quad (7)$$

$$\mathbb{R}^a \times \mathring{c}\mathbb{S}^{b-1} \times \{\mathfrak{v}\} = (\mathbb{R}^a \times \{(\mathfrak{u}, \mathfrak{v})\})^i = (\mathbb{R}^a \times (\mathring{c}\mathbb{S}^{b-1}\setminus\{\mathfrak{u}\})_{cc} \times \{\mathfrak{v}\})^i. \quad (8)$$

### 3. PERVERSITIES

The notion of perversity on a filtered space is the basis of intersection homology and homotopy. Here we recall definition and properties of general perversities of MacPherson ([Mac91]) and introduce the $K$-perversities for which a CS-coarsening invariance holds.

**Definition 3.1.** A *perversity* on a filtered space $(X, \mathcal{S})$ is a map $\bar{p}: \mathcal{S} \to \bar{\mathbb{Z}} = \mathbb{Z} \sqcup \{-\infty, \infty\}$ verifying $\bar{p}(S) = 0$ for any regular stratum. The *top perversity* is defined by $\bar{t}(S) = \mathrm{codim} S - 2$ for each singular stratum $S$. The *dual perversity* $D\bar{p}$ of a perversity $\bar{p}$ is defined by $D\bar{p} = \bar{t} - \bar{p}$. A triple $(X, \mathcal{S}, \bar{p})$ of a filtered space with a perversity on it is called a *perverse space*.
To any map $f: N \to Z$ such that $f(0) = 0$, we associate a perversity defined by $\bar{p}(S) = f(\mathrm{codim} S)$. Such perversities depend only on the codimension of the stratum and are called codimensional perversities. If $f(\ell) = k$ for any $\ell \neq 0$, the associated perversity is denoted by $\bar{k}$. The original perversities of [GM80] and those of [Kin85] are codimensional. We will come back to them in Section 8.

Perversities can be transferred through a stratified map $\varphi: (X, S) \to (Y, T)$.

**Definition 3.2.** Let $\bar{q}$ be a perversity on $(Y, T)$. The pullback perversity $\varphi^*\bar{q}$ on $(X, S)$ is defined by $\varphi^*\bar{q}(S) = \bar{q}(S^\varphi)$ for each singular $S \in S$. Let $\bar{p}$ be a perversity on $(X, S)$. The pushforward perversity $\varphi_*\bar{p}$ on $(Y, T)$ is defined by $\varphi_*\bar{p}(T) = \inf\{\bar{p}(Q) | Q^\varphi = T\}$ for each singular $T \in T$, with $\inf\emptyset = \infty$.

Notice that $\varphi_*\varphi^*\bar{q} = \bar{q}$ and $\varphi^*\varphi_*\bar{p} \leq \bar{p}$. (When $\varphi$ is a stratified homeomorphism, we have $\varphi_*\bar{p}(S^\varphi) = \bar{p}(S)$ for each $S \in S$ and $\varphi^*\varphi_*\bar{p} = \bar{p}$.) In the following, the map $\varphi$ is induced by the identity. We denote $\iota_{S, T}: (X, S) \to (X, T)$ (or $\iota$ if there is no ambiguity) the induced stratified map when $(X, S, T)$ is a CS-coarsening.

The perversities of [GM80, Kin85] verify a key property for the topological invariance of intersection homology, called the growing property. A codimensional perversity verifies the growing property if the following inequalities are verified for each integer $i$,

$$ \bar{p}(i) \leq \bar{p}(i+1) \leq \bar{p}(i+1). \qquad (9) $$

The next definition is an adaptation of these inequalities to perversities defined on the set of strata. They are the perversities for which a CS-coarsening invariance holds.

**Definition 3.3.** A K-perversity $\bar{p}$ on a CS-coarsening $(X, S, T)$ is a perversity on $(X, S)$ verifying the conditions (K1) and (K2).

(K1) For any strata $S, Q \in S$ such that $S \leq Q$ and $S^t = Q^t$, one has

$$ \bar{p}(Q) \leq \bar{p}(S) \leq \bar{p}(Q) + \bar{t}(S) - \bar{t}(Q). \qquad (10) $$

(K2) For any strata $S, Q \in S$ with $\dim S = \dim Q$ and $S^t = Q^t$, one has

$$ \bar{p}(Q) = \bar{p}(S). \qquad (11) $$

**Remark 3.4.** We list some basic observations on K-perversities.

i) Properties (K1) and (K2) are respectively equivalent to

$$ D\bar{p}(Q) \leq D\bar{p}(S) \leq D\bar{p}(Q) + \bar{t}(S) - \bar{t}(Q) \quad \text{and} \quad D\bar{p}(Q) = D\bar{p}(S). \qquad (12) $$

In other words, the perversities $\bar{p}$ and $D\bar{p}$ are simultaneously K-perversities.

ii) Two regular strata, $S$ and $Q$, always verify the condition (10). If the stratum $S$ is exceptional, as a singular stratum is always included in the closure of a regular one, say $Q$, we have $S \leq Q$ and $S^t = Q^t$. So the condition (10) gives

$$ 0 \leq \bar{p}(S) \leq \bar{t}(S), \qquad (13) $$

which implies the following properties.

* A K-perversity takes positive values on exceptional strata.

* The existence of a K-perversity implies the non-existence of 1-exceptional strata, since $0 \leq \bar{t}(S) = -1$ is impossible.

iii) Given a K-perversity $\bar{p}$ on a CS-coarsening $(X, S, T)$ and a stratum $T \in T$, the restriction of $\bar{p}$ to the stratified space $(T, S_T)$ (also denoted $\bar{p}$) is a codimensional perversity that verifies the growing property (9) for each integer $i$. In other words, the restriction of a K-perversity to the filtered space $(T, S_T)$ with $T \in T$ satisfies the key property of GM-perversities, recalled in Definition 8.4.

iv) Finally, if $(X, S, \mathcal{X}^*)$ is the intrinsic CS-coarsening ([Kin85]), our Definition 3.3 is equivalent to the definition of K-perversity given in [CSAT19, Definition 6.8].
We specify the behaviors of K-perversities with pullback and pushforward processes.

**Lemma 3.5.** Let $\bar{p}$ be a K-perversity on the $\mathcal{CS}$-coarsening $(X, S, \mathcal{T})$. Then, the following properties are verified:

i) $\iota_* \bar{p}(T) = \bar{p}(S)$ if $S \in \mathcal{S}$ is a source stratum of $T \in \mathcal{T}$,

ii) $\iota_* D_{\iota_* \bar{p}} \le D \bar{p}$,

iii) $\iota_* D \bar{p} = D_{\iota_* \bar{p}}$,

iv) if $\bar{p} \le \bar{t}$, then $\iota_* \bar{p} \le \bar{t}$.

*Proof.* Properties i) and ii) are exactly [SA21, Lemma 5.3 and 5.4]. It remains to prove iii) and iv). Let $T \in \mathcal{T}$ be a singular stratum. We know that there is a source stratum $S \in S$ of $T$. Since $D \bar{p}$ is a K-perversity, we have from i), $\iota_* D \bar{p}(T) = D \bar{p}(S)$. By replacing dual perversities with their definition and using i) a second time, we get iii) since

$$ \iota_* D \bar{p}(T) = \bar{t}(S) - \bar{p}(S) = \bar{t}(T) - \iota_* \bar{p}(T) = D_{\iota_* \bar{p}}(T). $$

Similarly, the last property is a consequence of

$$ \iota_* \bar{p}(T) = \bar{p}(S) \le \bar{t}(S) = \mathrm{codim} S - 2 = \mathrm{codim} T - 2 = \bar{t}(T). $$

□

**Lemma 3.6.** ([SA21, Lemma 5.6]). Let $(X, S, \mathcal{R})$ and $(X, \mathcal{R}, \mathcal{T})$ be two $\mathcal{CS}$-coarsenings. If $\bar{p}$ is a K-perversity on the $\mathcal{CS}$-coarsening $(X, S, \mathcal{T})$, then the following properties are verified.

i) The perversity $\bar{p}$ is a K-perversity on the $\mathcal{CS}$-coarsening $(X, S, \mathcal{R})$.

ii) The perversity $\iota_* \bar{p}$ is a K-perversity on the $\mathcal{CS}$-coarsening $(X, \mathcal{R}, \mathcal{T})$.

**Lemma 3.7.** Let $(X, S, \mathcal{T})$ be a $\mathcal{CS}$-coarsening without 1-exceptional stratum. If $\bar{q}$ is a perversity on $(X, \mathcal{T})$, then the pullback $\iota_* \bar{q}$ is a K-perversity on the $\mathcal{CS}$-coarsening $(X, S, \mathcal{T})$. Moreover, the inequality $\bar{q} \le \bar{t}$ implies $\iota_* \bar{q} \le \bar{t}$.

*Proof.* Let $S, Q \in S$ such that $S^t = Q^t$. We have to prove (K1) and (K2). Suppose $S \le Q$. With the definition of $\iota_* \bar{q}$, the inequalities (10) of (K1) become

$$ \bar{q}(Q^t) \le \bar{q}(S^t) \le \bar{q}(Q^t) + \bar{t}(S) - \bar{t}(Q), $$

which is equivalent to $\bar{t}(S) - \bar{t}(Q) \ge 0$. This last inequality is true if $S$ and $Q$ are both regular or both singular. There remains the exceptional case for $S$, where the previous inequality is reduced to $\bar{t}(S) = \mathrm{codim} S - 2 \ge 0$. This is true since there is no 1-exceptional stratum.

For (K2), we suppose $\mathrm{dim} S = \mathrm{dim} Q$. By definition of $\iota_* \bar{q}$, we have, as expected,

$$ \iota_* \bar{q}(Q) = \bar{q}(Q^t) = \bar{q}(S^t) = \iota_* \bar{q}(S). $$

Suppose $\bar{q} \le \bar{t}$ and let $S \in S$ be a singular stratum. If $S^t$ is a regular stratum, then $\iota_* \bar{q}(S) = \bar{q}(S^t) = 0 \le \bar{t}(S)$, since there is no 1-exceptional strata. If $S^t$ is singular, we have:

$$ \iota_* \bar{q}(S) = \bar{q}(S^t) \le \bar{t}(S^t) = \mathrm{codim}(S^t) - 2 \le \mathrm{codim}(S) - 2 = \bar{t}(S). $$

□

**Remark 3.8.** Let us specify how some particular induced perversities are defined.

1) The first point concerns the induced perversity on a link $L$, from a given perversity $\bar{p}$ on a $\mathcal{CS}$ set $X$. If $\varphi: \mathbb{R}^k \times \hat{c}L \to V$ is a conical chart of $x \in S \in S$, the open subset $V \setminus S$ of $X$ is given with the induced perversity still denoted by $\bar{p}$. If $T$ is a stratum of $L$, then $\mathbb{R}^k \times ]0, 1[ \times T$ is a stratum of $V \setminus S$ and we set $\bar{p}(T) = \bar{p}(\mathbb{R}^k \times [0, 1[ \times T)$. We also set $\bar{p}(\{v\}) = \bar{p}(S)$ for the apex $v$ of $\hat{c}L$.

2) The second point concerns the left vertical arrow of (6),

$$ \iota: A := (\mathbb{R}^a \times (\hat{c}\mathbb{S}^{b-1} \times \hat{c}L), \mathcal{I} \times (\mathcal{I} \times \hat{c}\mathcal{L})_{(u,v)}) \to B := ((\mathbb{R}^a \times \hat{c}\mathbb{S}^{b-1}) \times \hat{c}L, (\mathcal{I} \times \mathcal{I}) \times \hat{c}\mathcal{L}). $$
We start with a K-perversity $\bar{p}$ on $A$ and denote $\bar{q} = \iota_* \bar{p}$ its pushforward perversity on $B$. Let us notice that $D\bar{q}$ reduces to a perversity on $(\mathring{c}L, \mathring{c}\mathcal{L})$ as in the point 1). From Lemma 3.5, we know that $D\bar{q} = \iota_* D\bar{p}$. On $A\setminus\{(u, v)\} = B\setminus\{(u, v)\}$, the two perversities $D\bar{p}$ and $D\bar{q}$ coincide. Moreover, the value $D\bar{p}(\{(u, v)\})$ can be specified by taking in account (7), (8) and the property (K1) of Definition 3.3, as follows:

$$D\bar{q}(\{v\}) \le D\bar{p}(\{(u, v)\}) \le D\bar{q}(\{v\}) + b. \qquad (14)$$

### 4. INTERSECTION HOMOTOPY GROUPS

Intersection homotopy groups of a filtered space $(X, S)$, endowed with a perversity, are defined as the homotopy groups of a simplicial set introduced by P. Gajer ([Gaj96]). In this section, we recall their definitions and properties used in the following.

Let $(X, S, \bar{p})$ be a perverse space. A simplex $\sigma: \Delta \to X$ is $\bar{p}$-allowable if we have, for each singular stratum $S \in S$,

$$\dim \sigma^{-1}(S) \le \dim \Delta - D\bar{p}(S) - 2. \qquad (15)$$

Here the dimension chosen for $\dim \sigma^{-1}(S)$ is the polyhedral dimension ([Gaj96]). We do not need the definition; it is enough to know that it gives the same properties as the classical dimension by the skeleton, see [CSAT24b]. The simplex $\sigma$ is $\bar{p}$-full if each of its faces is $\bar{p}$-allowable. The set of $\bar{p}$-full simplexes is a simplicial set verifying the Kan condition, see [Gaj96, Page 946], [CSAT25, Proposition 2.3]. We denote it by $\mathcal{G}_{\bar{p}}(X, S)$ (or $\mathcal{G}_{\bar{p}}X$ if there is no ambiguity) and call it the Gajer $\bar{p}$-space associated to $(X, S)$.

**Proposition 4.1** ([CSAT25, Proposition 3.4]). Let $(X, \bar{p})$ be a perverse space and $Y$ be a topological space. The product $Y \times X$ is equipped with the product filtration (Examples 1.2) and with the perversity, still denoted by $\bar{p}$, defined by $\bar{p}(Y \times S) = \bar{p}(S)$. Then, the canonical projections, $p_Y: Y \times X \to Y$ and $p_X: Y \times X \to X$, induce an isomorphism

$$\mathcal{G}_{\bar{p}}(Y \times X) \cong \mathrm{Sing} Y \times \mathcal{G}_{\bar{p}}X,$$

where $\mathrm{Sing} Y$ is the simplicial set made up of the singular simplices of $Y$.

Let $f: (X, S, \bar{p}) \to (Y, \mathcal{T}, \bar{q})$ be a stratified map between perverse filtered spaces such that $f^*D\bar{q} \le D\bar{p}$. Then $f$ induces a simplicial map $\mathcal{G}(f): \mathcal{G}_{\bar{p}}(X, S) \to \mathcal{G}_{\bar{q}}(Y, \mathcal{T})$, see [CSAT25, Proposition 3.5]. In particular, if $(X, S, \mathcal{T})$ is a CS-coarsening and $\bar{p}$ a perversity on $(X, S)$, we have $\iota_* D\iota_* \bar{p} \le D\bar{p}$, by Lemma 3.5.ii). So the identity map induces a simplicial map

$$\mathcal{G}(\iota): \mathcal{G}_{\bar{p}}(X, S) \to \mathcal{G}_{\iota_* \bar{p}}(X, \mathcal{T}). \qquad (16)$$

A pointed perverse space is a quadruplet $(X, S, \bar{p}, x_0)$ where $(X, S, \bar{p})$ is a perverse space and $x_0 \in X \setminus \Sigma$ is a regular point, called *basepoint*. We also denote by $x_0$ the simplicial subset of $\mathrm{Sing} X$ generated by $x_0: \Delta^0 \to X$.

**Definition 4.2** ([Gaj96], [CSAT25, Definition 4.1]). Let $(X, S, \bar{p}, x_0)$ be a pointed perverse space. The $\bar{p}$-intersection homotopy groups are the homotopy groups of the simplicial set $\mathcal{G}_{\bar{p}}(X, S)$,

$$\pi_{\ell}^{\bar{p}}(X, S, x_0) := \pi_{\ell}(\mathcal{G}_{\bar{p}}(X, S), x_0).$$

If there is no ambiguity, we do not mention the basepoint and write $\pi_{\ell}^{\bar{p}}(X, S)$. Intersection homotopy groups of cones satisfy a property similar to that of intersection homology ([CSAT25, Proposition 4.5]): suppose $X$ compact and let $\mathring{c}X$ be the open cone of apex $v$, with the cone filtration and pointed by $y_0 = [x_0, 1/2]$. Let $\bar{p}$ be a perversity on $\mathring{c}X$, we denote also by $\bar{p}$ the perversity induced on $X$. Then, the $\bar{p}$-intersection
homotopy groups of $\mathring{c}X$ are given by

$$ \pi_{\ell}^{\bar{p}}(\mathring{c}X, \mathring{c}S, y_0) = \begin{cases} \pi_{\ell}^{\bar{p}}(X, S, x_0) & \text{if } \ell \le D\bar{p}(\mathbf{v}), \\ 0 & \text{if } \ell > D\bar{p}(\mathbf{v}). \end{cases} \quad (17) $$

The homology of $\mathscr{G}_{\bar{p}}(\mathring{c}X, \mathring{c}S)$ has a different behavior ([CSAT25, Proposition 3.10]). Recall that a local coefficient system (of abelian groups) on a simplicial set $K$ is a contravariant functor $\mathfrak{C}: \Pi K \to \text{Ab}$, from its fundamental groupoid, with values in the category of abelian groups. The inclusion $L \hookrightarrow \mathring{c}L$ given by $(x \mapsto [x, 1/2])$, induces

$$ \left\{ \begin{array}{ll} \text{isomorphisms} & H_{\ell}(\mathscr{G}_{\bar{p}}L; \mathfrak{C}) \xrightarrow{\cong} H_{\ell}(\mathscr{G}_{\bar{p}}\mathring{c}L; \mathfrak{C}), & \text{if } \ell \le D\bar{p}(\mathbf{v}) \text{ and} \\ \text{an epimorphism} & H_{\ell}(\mathscr{G}_{\bar{p}}L; \mathfrak{C}) \to H_{\ell}(\mathscr{G}_{\bar{p}}\mathring{c}L; \mathfrak{C}), & \text{if } \ell = D\bar{p}(\mathbf{v}) + 1. \end{array} \right. \quad (18) $$

In other words, we have

$$ H_j(\mathscr{G}_{\bar{p}}\mathring{c}L, \mathscr{G}_{\bar{p}}L; \mathfrak{C}) = 0 \quad \text{for any } j \le D\bar{p}(\mathbf{v}) + 1. \quad (19) $$

## 5. CONNECTIVITY AND CS-COARSENINGS

In this section, we study the set $\pi_0^{\bar{p}}(X)$ of connected components of the Gajer space $\mathscr{G}_{\bar{p}}X$ associated to a perverse space $(X, S, \bar{p})$. The goal is to demonstrate in Proposition 5.4 the existence of an isomorphism between the sets of connected components of the two stratifications in a CS-coarsening. This is the first condition to fill in for an application of Lemma 7.2.

**Lemma 5.1.** Let $(X, S, \bar{p}, x_0)$ be a pointed perverse space with $\bar{p} \le \bar{t}$. Then, the canonical inclusion induces an isomorphism of sets and a surjective homomorphism,

$$ \pi_0(X \setminus \Sigma) \xrightarrow{\cong} \pi_0^{\bar{p}}(X, S) \quad \text{and} \quad \pi_1(X \setminus \Sigma, x_0) \to \pi_1^{\bar{p}}(X, S, x_0). \quad (20) $$

*Proof.* This comes directly from the definition of $\bar{p}$-full simplexes: in dimension 0 or 1 they do not meet the singular part of $X$. $\square$

A perverse space is $\bar{p}$-connected if $\pi_0^{\bar{p}}(X, S) = 0$. When $(X, S, \bar{p})$ is $\bar{p}$-connected, we do not need to specify the basepoint of intersection homotopy groups and we write $\pi_{\ell}^{\bar{p}}(X, S)$ instead of $\pi_{\ell}^{\bar{p}}(X, S, x_0)$.

**Lemma 5.2.** Let $(X, S)$ be a CS set. For any point $x \in X$, there exists a path $\beta: [0, 1] \to X$ with $\beta(0) = x$ and $\beta([0, 1]) \subset X \setminus \Sigma$. This path can be chosen as small as wanted. Moreover, if $x$ is $\bar{p}$-allowable for some perversity $\bar{p}$, then $\beta$ is $\bar{p}$-allowable too.

*Proof.* If the point $x$ is regular, we choose the constant path $x$. If not, let $\varphi: \mathbb{R}^k \times \mathring{c}L \to U \subset X$ be a conical chart with $\varphi(0, \mathbf{v}) = x$. (Here, $\mathbf{v}$ is the apex of the cone $\mathring{c}L$.) We choose a point $z \in L \setminus \Sigma_L$, which exists since the regular part of $L$ is not empty. For any $\varepsilon > 0$, the map $\beta(t) = \varphi(0, [z, \varepsilon t])$ checks the required conditions. Let $S$ be the stratum of $S$ containing $x$, the last assertion is a consequence of

$$ \dim \beta^{-1}(S) = \dim\{x\} = 0 \le \dim\{x\} - D\bar{p}(S) - 2 = \dim \beta^{-1}(S) - D\bar{p}(S) - 2. \quad \square $$

The next result is demonstrated in [Fri20, Lemma 2.6.3] for pseudomanifolds. For these, a link is also a pseudomanifold and the proof can be done by induction on the dimension. Since CS sets do not have such a recursive property, we provide an adaptation of it.

**Lemma 5.3.** Let $(X, S)$ be a normal CS set. Then the space $X$ is connected if and only if the regular part $X \setminus \Sigma$ is.
**Proof.** Since the regular part is dense in $X$, we have only one implication to prove. So let $(X, S)$ be a normal, connected CS set. We proceed by induction on the depth. When depth $(X, S) = 0$, we have $\Sigma = \emptyset$, and the result is clear. Suppose the result is true for any CS set of depth $< n$ and that $(X, S)$ is a normal, connected CS set, of depth $n$, with more than one regular stratum. We denote by $S$ one regular stratum and by $T$ the union of the other regular strata. We know $X = \bar{S} \cup \bar{T}$ and the existence of $x \in \bar{S} \cap \bar{T}$ since $X$ is connected. Let $\varphi: \mathbb{R}^i \times \hat{c}L \to U$ be a conical chart of $x$. The intersection of $S$ and $T$ with $U$ are of the form $\mathbb{R}^i \times A \times ]0, 1[$ and $\mathbb{R}^i \times B \times ]0, 1[$, where $A$ and $B$ are union of regular strata of $L$. Since $L$ is a filtered space, $A$ and $B$ are open in $L$ and the CS set $\mathbb{R}^i \times L \times ]0, 1[$ is a CS set of depth $< n$ with more than one regular stratum. We get the result by contradiction. $\square$

**Proposition 5.4.** Let $\bar{p}$ be a $K$-perversity on a CS-coarsening $(X, S, \mathcal{T})$, such that $\bar{p} \le \bar{t}$. Then, the identity map induces a bijection

$$ \iota_0: \pi_0^{\bar{p}}(X, S) \xrightarrow{\cong} \pi_0^{\ell_* \bar{p}}(X, \mathcal{T}). $$

**Proof.** From (16), we know the existence of a simplicial map

$$ \mathcal{G}(\iota): \mathcal{G}_{\bar{p}}(X, S) \to \mathcal{G}_{\iota_* \bar{p}}(X, \mathcal{T}). $$

We denote the induced maps by $\iota_\ell: \pi_\ell^{\bar{p}}(X, S) \to \pi_\ell^{\ell_* \bar{p}}(X, \mathcal{T})$. By hypothesis, we have $D\bar{p} \ge \bar{0}$. From $\iota_* D\bar{p} = D\iota_* \bar{p}$ (Lemma 3.5), we deduce $D\iota_* \bar{p} \ge \bar{0}$. Thus the $\bar{p}$-full $\ell$-simplexes of $(X, S)$ and $(X, \mathcal{T})$ do not meet the respective singular parts for $\ell = 0, 1$. So, with Lemma 5.1, the desired property becomes

$$ \iota_0: \pi_0(X \setminus \Sigma_S) \to \pi_0(X \setminus \Sigma_T) \text{ is a bijection.} \quad (21) $$

As $Y = X \setminus \Sigma_T$ is an open subset, there is an induced CS set structure $(Y, S)$. The condition (21) becomes

$$ \iota_0: \pi_0(Y \setminus \Sigma_S) \to \pi_0(Y) \text{ is a bijection.} $$

Without loss of generality, we may assume that $Y$ is connected. All strata of $(Y, S)$ are exceptional and none are of codimension 1 (cf. (13)). From Definition 1.4, we deduce that the link is a homology sphere of dimension $\ge 1$, hence the normality of the CS set $(Y, S)$. Finally, Lemma 5.3 implies that $Y \setminus \Sigma_S$ is connected and therefore property (21) is true. $\square$

**Proposition 5.5.** Let $(X, S, \bar{p})$ be a connected, normal perverse CS set. Then, any link $(L, \mathcal{L})$ is $\bar{p}$-connected; i.e., $\pi_0^{\bar{p}}(L, \mathcal{L}) = 0$.

**Proof.** Let $\varphi: \mathbb{R}^m \times \hat{c}L \to U$ be a conical chart. As $\mathbb{R}^m \times L \times ]0, 1[$ is an open subset of $X$, it is a CS set and any of its links are links of $X$. So, $\mathbb{R}^m \times L \times ]0, 1[$ is a connected, normal CS set. From Lemma 5.3, we deduce that $\mathbb{R}^m \times L \times ]0, 1[$, $1[ \setminus \Sigma = \mathbb{R}^m \times (L \setminus \Sigma_\mathcal{L}) \times ]0, 1[$ is connected. Any $\bar{p}$-allowable point of the CS set $\mathbb{R}^m \times L \times ]0, 1[$ can be replaced by a regular one (see Lemma 5.2) and any two regular points of $\mathbb{R}^m \times (L \setminus \Sigma_\mathcal{L}) \times ]0, 1[$ are connected by a regular path. So, $0 = \pi_0^{\bar{p}}(\mathbb{R}^m \times L \times ]0, 1[, \mathcal{I} \times \mathcal{L} \times \mathcal{I}) = \pi_0^{\bar{p}}(L, \mathcal{L})$, cf. Proposition 4.1. $\square$

# 6. INTERSECTION FUNDAMENTAL GROUP AND CS-COARSENINGS

We establish the existence of isomorphisms between the fundamental intersection groups of a CS-coarsening in two situations. The first concerns the case without exceptional stratum and is an extension of [CSAT25, Theorem 5.6] to general perversities. The second takes into account the existence of some specific exceptional strata within Thom-Mather's spaces.
**Proposition 6.1.** Let $(X, S, T, x_0)$ be a pointed CS-coarsening without exceptional stratum and $\bar{p}$ be a perversity on $S$ with $\bar{p} \le \bar{t}$. Then there is an isomorphism,

$$ \pi_1^{\bar{p}}(X, S, x_0) \xrightarrow{\sim} \pi_1^{\ell_*\bar{p}}(X, T, x_0). \qquad (22) $$

*Proof.* The surjectivity is a direct consequence of Lemma 5.1. For the injectivity, the proof done in [CSAT25, Theorem 5.6] for a codimensional perversity adapts immediately. The only modification is the replacement of $\bar{p}(\mathrm{codim} T)$ by $\bar{p}(T)$. $\square$

If there are exceptional strata, the result cannot be true, as shows [CSAT25, Example 5.3], recalled in Example 8.3. We give below a sufficient condition for keeping the isomorphism between intersection fundamental groups in the presence of exceptional strata. We do not need the whole structure of Thom-Mather stratified space, or smoothly stratified space of [ABP52], only the existence of neighborhoods of singular strata without control data.

A stratification on a CS set, $(X, S)$, is equivalent ([CT20, Proposition 1.5] or [Lur, Appendix A.5] for a global presentation of this point of view) to the data of a surjective continuous map $p_S: X \to S$, where $S$ is the poset of strata endowed with the Alexandrov topology $(V \subset S$ is open if $S \le S'$ and $S \in U$ implies $S' \in U)$.

Let $(X, S, T)$ be a CS-coarsening. We have a partition of $S$ in three parts, formed of the exceptional, fountain or source strata, respectively. We define $S_e$ as the quotient of $S$ by the equivalence relation generated by $S \sim S^t$ if $S$ exceptional. This provides us with a new filtered space $p_{S_e}: X \to S_e$ where the singular strata of $S_e$ are the singular strata of $S$ which are not exceptional in $(X, S, T)$. A priori, given Example 8.6, there is no reason why $(X, S_e)$ should be a CS set. In fact, it is true and can be deduced from [SA21, Proposition 3.10]. We provide in Proposition 6.3 an independant simple proof in the case of a pre-Thom-Mather space.

We have three CS-coarsenings, $(X, S, T)$, $(X, S, S_e)$, $(X, S_e, T)$, which give a decomposition of the identity map in

$$ (X, S) \xrightarrow{\stackrel{\ell_e}{\longrightarrow}} (X, S_e) \xrightarrow{\stackrel{\ell_\Sigma}{\longrightarrow}} (X, T). \qquad (23) $$

By construction, the following properties are satisfied.

i) A singular source stratum or a fountain stratum $S$ in $(X, S, T)$ is not modified in $(X, S_e)$, so $S = S^t$. Therefore $S$ becomes a source stratum in $(X, S, S_e)$ and $(X, S, S_e)$ has only exceptional and source strata,

ii) The CS-coarsening $(X, S_e, T)$ has no exceptional strata.

**Definition 6.2.** A pre-Thom-Mather space of dimension $n$ is a Hausdorff filtered space $(X, S)$ of dimension $n$, whose $i$-dimensional strata are $i$-dimensional topological manifolds for each $i$, and such that, for each singular stratum $S \in S$, there exist an open neighborhood $E$ of $S$, endowed with the induced filtration, a continuous map $\Pi: E \to S$, an open cover, $\mathfrak{U}$, of $S$, a filtered compact space, $(L, \mathcal{L})$, of (formal) dimension $n - \dim S - 1$, and, for each $U \in \mathfrak{U}$, a stratified homeomorphism

$$ \varphi_U: (\Pi^{-1}(U), S) \to (U \times \mathring{c}L, \mathcal{I} \times \mathring{c}\mathcal{L}), $$

verifying $\mathrm{pr}_1 \circ \varphi_U = \Pi$ and $\varphi_U(x) = (x, v)$ for any $x \in U$, with $v$ the apex of the cone. The map $\Pi$ is a conical neighborhood of $S$ and $\{\mathfrak{U}, (\varphi_U)_{U \in \mathfrak{U}}\}$ is a conical trivialization of $\Pi$. The filtered space $L$ is the link of $S$.

**Proposition 6.3.** Let $(X, S, T)$ be a CS-coarsening where $(X, S)$ is a pre-Thom-Mather space. Then, in the decomposition (23) of the identity map, the filtered space $(X, S_e)$ is a pre-Thom-Mather space. Moreover, any singular stratum in $S_e$ is a singular stratum in $S$ and one can choose the same conical neighborhood for $S$ in $(X, S)$ and in $(X, S_e)$.
**Proof.** We first show that the association $(X, S, \mathcal{T}) \mapsto (X, S_e)$ is hereditary. Let $Y$ be an open subset of $X$ and $Q \in S_Y$. Then $Q$ is a connected component of $S \cap Y$ with $S \in S$. As the restriction to $Y$ keeps the dimension of strata, we deduce that $Q$ is singular if and only if $S$ is, and that $Q$ is exceptional if and only if $S$ is. We deduce $(S_e)_Y = (S_Y)_e$ as expected.

Let $S \in S_e$ be a singular stratum. By construction, $S \in S$ is a singular source stratum of $(X, S, S_e)$. As $(X, S)$ is a pre-Thom-Mather space, there exist a conical neighborhood $\Pi: E \to S$ and a conical trivialization $\{\mathfrak{U}, (\varphi_U)_{U \in \mathfrak{U}}\}$, with a stratified homeomorphism

$$ \varphi_U : (\Pi^{-1}(U), S) \to (U \times \mathring{c}L, \mathcal{I} \times \mathring{c}\mathcal{L}). $$

As the association $(X, S, \mathcal{T}) \mapsto (X, S_e)$ is hereditary, we get a stratified homeomorphism

$$ \varphi_U : (\Pi^{-1}(U), S_e) \to (U \times \mathring{c}L, (\mathcal{I} \times \mathring{c}\mathcal{L})_e). $$

We have to determine $(\mathcal{I} \times \mathring{c}\mathcal{L})_e$. As $\varphi_U(S) = U \times \{\mathfrak{v}\}$, we reduce the question to $\Pi^{-1}(U) \setminus S$ and the study to $(\mathcal{I} \times \mathcal{L} \times \mathcal{I})_e$. The construction of $S_e$ consists of a quotient in the poset $S$ (or an amalgamation of strata in $X$), thus $(\mathcal{I} \times \mathcal{L} \times \mathcal{I})_e$ is of the form $(\mathcal{I} \times \mathcal{L}' \times \mathcal{I})$ where $\mathcal{L}'$ is a coarsening of $\mathcal{L}$. This gives the result. $\square$

**Remark 6.4.** If $(X, S)$ is a Thom-Mather space, it is (see [Fri20, Section 2.8.2] for details) a pseudomanifold. Therefore, $(L, \mathcal{L})$ is a pseudomanifold. The space $L$ is also endowed with a stratification $\mathcal{L}_{\mathcal{T}}$ induced by $\mathcal{T}$, giving a coarsening $(L, \mathcal{L}, \mathcal{L}_{\mathcal{T}})$. One can observe that the coarsening $\mathcal{L}'$ of the previous proof corresponds to its associated $\mathcal{L}_e$.

In reference to the decomposition (23) of the identity map, let us notice that, in the stratified map $\iota_{\Sigma}$, the two stratifications have the same regular set. In [CSAT25], we have proven that $\iota_{\Sigma}$ induces an isomorphism between intersection homotopy groups for GM-perversities. In Theorem A, we extend it to general perversities of MacPherson ([Mac91], Definition 3.1). The case $\iota_e$ was not considered in [CSAT25]; we study it in Theorem B. The next result is a step in this direction.

**Proposition 6.5.** Let $(X, S, \mathcal{T}, x_0)$ be a pointed CS-coarsening where $(X, S)$ is a connected normal pre-Thom-Mather space with a finite number of strata. Let $\bar{p}$ be a K-perversity of pushforward perversity $\iota_*\bar{p}$ and such that $\bar{p} \le \bar{t}$. If the cone of the link of any exceptional stratum verifies $\pi_1^{\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L}, x_0) = \{1\}$, then the identity map induces an isomorphism,

$$ \pi_1^{\bar{p}}(X, S, x_0) \xrightarrow{\sim} \pi_1^{\iota_*\bar{p}}(X, \mathcal{T}, x_0). \qquad (24) $$

**Proof.** Let us consider the decomposition recalled in (23),

$$ (X, S) \xrightarrow{\iota_e} (X, S_e) \xrightarrow{\iota_{\Sigma}} (X, \mathcal{T}). \qquad (25) $$

By Proposition 6.1 and Lemma 3.6, the simplicial map $\iota_{\Sigma}$ induces an isomorphism between the intersection fundamental groups. So we are reduced to the study of the CS-coarsening $(X, S, S_e)$ induced by $(X, S, \mathcal{T})$.

The proof is done by induction on the depth and the number of minimal singular strata in $S$. Let $S \in S$ be a minimal stratum with a conical neighborhood $\Pi: E \to S$ of associated link $(L, \mathcal{L}_S)$. The inductive step consists of an open cover of $X$ by $E$ and $X \setminus S$. Let us observe that $E \cap (X \setminus S) = E \setminus S$. The induction hypothesis can be applied to $X \setminus S$ and $E \setminus S$. Therefore, with the Van Kampen theorem of intersection fundamental groups ([CSAT25, Theorem 5.1]), the proof is reduced to the case of $E$ with the induced filtrations. The hypotheses of connectivity, required in [CSAT25, Theorem 5.1], are satisfied: for $E$, it is a consequence of the connectivity of $S$ and $\mathring{c}L$, for $X \setminus S$ it comes from $X \setminus \Sigma_S \subset X \setminus S$ and $\overline{X \setminus \Sigma_S} = X$.

By [Gaj96, Theorem 2.3], the map $\Pi$ is a locally trivial filtered fibration from which we deduce ([Gaj96, Theorem 2.2]) a long exact sequence in intersection homotopy groups.
(When the spaces are path connected, we omit the reference to a basepoint.) We already have informations on some groups in this sequence. First, as there is only one stratum in $S$, we have $\pi_k^q(S, \mathcal{I}) = \pi_k(S)$ for any perversity $\bar{q}$. We know from Proposition 5.5 that $\pi_0^{\bar{p}}(L, \mathcal{L}_S) = \pi_0^{\bar{p}}(L, \mathcal{L}_{S_e}) = 0$. The cone formula (17) being also verified for $\pi_0^{\bar{p}}$, we get $\pi_0^{\bar{p}}(\hat{c}L, \hat{c}\mathcal{L}_S) = \pi_0^{\bar{p}}(\hat{c}L, \hat{c}\mathcal{L}_{S_e}) = 0$. We have two possibilities for the singular stratum $S \in S$.

Suppose that $S$ is exceptional in $(X, S, S_e)$. By hypothesis, we have $\pi_1^{\bar{p}}(\hat{c}L, \hat{c}\mathcal{L}_S) = \{1\}$. Here, the total space of the locally trivial fibration $\hat{c}L \to E \to S$ has only one stratum in $S_e$; thus the corresponding long exact sequence of intersection homotopy groups is the long exact sequence of homotopy groups. So we have a diagram of short exact sequences,

$$ \begin{tikzcd} \pi_2(S) \arrow[r] \arrow[d, "\cong"] & \pi_1^{\bar{p}}(\hat{c}L, \hat{c}\mathcal{L}_S) \arrow[r] \arrow[d, "\cong"] & \pi_1^{\bar{p}}(E, S) \arrow[r] \arrow[d, "\cong"] & \pi_1(S) \arrow[r] \arrow[d, "\cong"] & 0 \\ \pi_2(S) \arrow[r] \arrow[d, "\cong"] & \pi_1(\hat{c}L) \arrow[d, "\psi_{\hat{c}L}"] & \pi_1(E) \arrow[d, "\psi_E"] & \pi_1(S) \arrow[r] \arrow[d, "\cong"] & 0 \end{tikzcd} $$

and an isomorphism $\pi_1^{\bar{p}}(E, S) \cong \pi_1(E)$.

Suppose that $S$ is not exceptional in $(X, S, S_e)$. Then $S$ is singular in $(X, S_e)$ and, by Proposition 6.3, we can choose $\Pi$ as conical neighborhood for $S$ and $S_e$. We thus have a diagram of exact sequences:

$$ \begin{tikzcd} \pi_2(S) \arrow[r] \arrow[d, "\cong"] & \pi_1^{\bar{p}}(\hat{c}L, \hat{c}\mathcal{L}_S) \arrow[r] \arrow[d, "\psi_{\hat{c}L}"] & \pi_1^{\bar{p}}(E, S) \arrow[r] \arrow[d, "\psi_E"] & \pi_1(S) \arrow[r] \arrow[d, "\cong"] & 0 \\ \pi_2(S) \arrow[r] \arrow[d, "\cong"] & \pi_1^{t_*\bar{p}}(\hat{c}L, \hat{c}\mathcal{L}_{S_e}) \arrow[r] \arrow[d, "\psi_{\hat{c}L}"] & \pi_1^{t_*\bar{p}}(E, S_e) \arrow[r] \arrow[d, "\psi_{S_e}"] & \pi_1(S) \arrow[r] \arrow[d, "\cong"] & 0. \end{tikzcd} $$

**Claim:** The map $\psi_{\hat{c}L}$ is an isomorphism, from which we deduce that $\psi_E: \pi_1^{\bar{p}}(E, S) \cong \pi_1^{\bar{p}}(E, S_e)$ is an isomorphism, as required.

From Definition 6.2 and Proposition 4.1, we have to prove that the homomorphism

$$ \iota_{\Pi^{-1}(U)}: \pi_1^{\bar{p}}(\Pi^{-1}(U), S) \to \pi_1^{t_*\bar{p}}(\Pi^{-1}(U), S_e) $$

is an isomorphism. Since $S^{t_e} = S$, the stratum $S$ is a source and we have $D\bar{p}(S) = \iota_* D\bar{p}(S) = D\iota_*\bar{p}(S)$, cf. Lemma 3.5. We use now the cone formula (17). If $D\bar{p}(S) = 0$, the domain and codomain of $\iota_{\Pi^{-1}(U)}$ are equal to $\{1\}$. If not, we have $\pi_1^{\bar{p}}(\Pi^{-1}(U), S) \cong \pi_1^{\bar{p}}(\Pi^{-1}(U)\setminus S, S)$ and a similar isomorphism with the stratification $S_e$. The result comes from the induction applied to

$$ \iota_{\Pi^{-1}(U)\setminus S}: \pi_1^{\bar{p}}(\Pi^{-1}(U)\setminus S, S) \to \pi_1^{t_*\bar{p}}(\Pi^{-1}(U)\setminus S, S_e). $$

□

**Example 6.6.** From Lemma 5.1 and the cone formula (17), we know that the hypothesis $\pi_1^{\bar{p}}(\hat{c}L, \hat{c}S) = \{1\}$ in Proposition 6.5 is a consequence of $\pi_1(L\setminus\Sigma_L) = \{1\}$. Yet, this last condition is stronger, as shows the following example.

Let $T$ be the pinched torus described in [GM80]. Its stratification $S$ has exactly one singular point and verifies $\pi_1(T\setminus\Sigma_T) = \mathbb{Z}$. We choose $\bar{p} = \bar{0}$. As we have already pointed out, the allowable 1-simplexes are the simplexes included in the regular part and an allowable 2-simplex can meet the singular set in dimension 0. So, the generator of $\pi_1(T\setminus\Sigma_T)$ is killed in $\pi_1^{\bar{0}}(T, S)$. In short, we have $\pi_1(T\setminus\Sigma_T) = \mathbb{Z}$ and $\pi_1^{\bar{0}}(\hat{c}T, \hat{c}S) = \{1\}$.
## 7. PROOFS OF THEOREMS A AND B

Let $(X, S, \mathcal{T})$ be a CS-coarsening endowed with a $K$-perversity $\bar{p}$. The identity map induces a simplicial map $\mathcal{G}(\iota): \mathcal{G}_{\bar{p}}(X, S) \to \mathcal{G}_{\iota, \bar{p}}(X, \mathcal{T})$. In this section, we prove that the latter induces a homology isomorphism $H_*(\mathcal{G}_p(X, S); \mathcal{G}(\iota)^*\mathcal{E}) \xrightarrow{\approx} H_*(\mathcal{G}_{\iota, \bar{p}}(X, \mathcal{T}); \mathcal{E})$, for any local system $\mathcal{E}$ on $\mathcal{G}_{\iota, \bar{p}}(X, \mathcal{T})$. We then deduce the proofs of Theorems A and B.

To achieve this objective, recall [Fri20, Theorem 5.1.4], see also [CSAT19, Theorem 5.1].

**Proposition 7.1.** Let $\mathcal{F}_X$ be the category whose objects are (stratified homeomorphic to) open subsets of a given CS set $(X, S)$ and whose morphisms are stratified homeomorphisms and inclusions. Let $\mathcal{Ab}_\star$ be the category of graded abelian groups. Let $F_*, G_* : \mathcal{F}_X \to \mathcal{Ab}$ be two functors and $\Phi: F_* \to G_*$ be a natural transformation satisfying the conditions listed below.

(i) The functors $F_*$ and $G_*$ admit exact Mayer-Vietoris sequences and the natural transformation $\Phi$ induces a commutative diagram between these sequences.

(ii) If $\{U_\alpha\}$ is a increasing collection of open subsets of $X$ and $\Phi: F_*(U_\alpha) \to G_*(U_\alpha)$ is an isomorphism for each $\alpha$, then $\Phi: F_*(\cup_\alpha U_\alpha) \to G_*(\cup_\alpha U_\alpha)$ is an isomorphism.

(iii) Let $(\varphi, V)$ be a conical chart of a singular point $x \in S$ with $S \in \mathcal{S}$. If $\Phi: F_*(V\setminus S) \to G_*(V\setminus S)$ is an isomorphism, then so is $\Phi: F_*(V) \to G_*(V)$.

(iv) If $U$ is an open subset of $X$ contained within a single stratum and homeomorphic to an Euclidean space, then $\Phi: F_*(U) \to G_*(U)$ is an isomorphism.

Then $\Phi: F_*(X) \to G_*(X)$ is an isomorphism.

This is the appropriate tool to demonstrate the existence of an isomorphism between homologies on CS sets, as below in Theorems 7.3 and 7.5. In the proof of these theorems, we also use a combination of homotopy and homology computations, where the following lemma will be essential. Weak equivalences can be detected from connected components, fundamental groups and isomorphisms of homology groups for any local coefficient system ([Qui67, Proposition II.3.4]). We use the next slight modification.

**Lemma 7.2 ([CSAT25, Lemma 3.8].)** Let $f: K \to K'$ be a simplicial map between Kan simplicial sets whose induced maps verify the following properties: $\pi_0(K) \cong \pi_0(K')$, $\pi_1(K, x) \cong \pi_1(K', f(x))$ for any $x \in K_0$, $H_j(K; f^*\mathcal{E}) \cong H_j(K'; \mathcal{E})$, for any local coefficient system $\mathcal{E}$ on $K'$, any $j \le \ell$ and $H_{\ell+1}(K; f^*\mathcal{E}) \to H_{\ell+1}(K'; \mathcal{E})$ is a surjection. Then, the map induced by $f$ between the homotopy groups is an isomorphism for $j \le \ell$ and a surjection for $j = \ell + 1$.

**Theorem 7.3.** Let $(X, S, \mathcal{T})$ be a CS-coarsening without exceptional strata, and let $\bar{p}$ be a $K$-perversity with $\bar{p} \le \bar{t}$, of pushforward perversity $\iota_*\bar{p}$. Then, for any local coefficient system $\mathcal{E}$ on $\mathcal{G}_{\iota, \bar{p}}(X, \mathcal{T})$, the identity map induces an isomorphism

$$ \mathcal{G}(\iota)_*: H_*(\mathcal{G}_{\bar{p}}(X, S); \iota_*\mathcal{E}) \xrightarrow{\approx} H_*(\mathcal{G}_{\iota, \bar{p}}(X, \mathcal{T}); \mathcal{E}). $$

*Proof.* According to (16), the map $\mathcal{G}(\iota)$ is well defined. From [SA24, Proposition 2.3], the identity map $(X, S) \to (X, \mathcal{T})$ can be decomposed in a sequence of simple CS-coarsenings (Definition 2.4). So, with Lemma 3.6, we can reduce the problem to a simple CS-coarsening where any point $x \in X$ has two conical charts, one in $(X, S)$ and the other in $(X, \mathcal{T})$, connected as detailed after the loc. cit. definition and recalled below.

The proof of Theorem 7.3 consists of the verifications of the hypotheses of Proposition 7.1. For properties (ii) and (iv), it is immediate. Condition (i) comes from the existence of a Mayer-Vietoris sequence in [CSAT25, Theorem 3.13]. So, we are reduced to the verification of (iii).
Let us consider a local coefficient system $\mathfrak{C}$ on $\mathscr{G}_{\iota_*\bar{p}}(X, \mathcal{T})$. We also have the local coefficient systems obtained by pullback of $\mathfrak{C}$, on $\mathscr{G}_{\bar{p}}(X, \mathcal{S})$ or on simplicial subsets. We do not mention the systems of coefficients in the rest of the proof.

Let $x \in S$ be a point in a singular stratum $S \in \mathcal{S}$. We may consider a small enough $\mathcal{S}$-conical chart containing $x$ as in Section 2. We distinguish two cases.

• The stratum $S$ is a source stratum, as described in Diagram (3) and recalled for the convenience of the reader,

$$ (\mathbb{R}^a \times \mathring{c}L, \mathcal{I} \times \mathring{c}\mathcal{L}) \xrightarrow{\psi} (W, \mathcal{S}) \qquad (26) $$
$$ (\mathbb{R}^a \times \mathring{c}L, \mathcal{I} \times \mathring{c}\mathcal{L}') \xrightarrow{\psi} (W, \mathcal{T}). $$

Using Proposition 4.1, Property (iii) of Proposition 7.1 is as follows:

> "If $\mathscr{G}(\iota)_* : H_*(\mathscr{G}_{\bar{p}}(L, \mathcal{L})) \xrightarrow{\cong} H_*(\mathscr{G}_{\iota_*\bar{p}}(L, \mathcal{L}'))$ is an isomorphism, then so is $\mathscr{G}(\iota)_* : H_*(\mathscr{G}_{\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L})) \xrightarrow{\cong} H_*(\mathscr{G}_{\iota_*\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L}')).$"

Recall from Lemma 3.5 the equalities $D_{\iota_*\bar{p}}(v) = \iota_*D\bar{p}(v) = D\bar{p}(v)$, where $v$ is the apex of the cone $\mathring{c}L$. From (18), we deduce isomorphisms and a surjection as follows:

$$ \iota_* : H_k(\mathscr{G}_{\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L})) \xrightarrow{\cong} H_k(\mathscr{G}_{\iota_*\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L})), \quad \text{if } k \le D\bar{p}(v), \text{ and} $$

$$ \iota_* : H_k(\mathscr{G}_{\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L})) \to H_k(\mathscr{G}_{\iota_*\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L})), \quad \text{if } k = D\bar{p}(v) + 1. $$

Let $x_0$ be any regular basepoint. As the conical chart has no exceptional stratum, Lemma 7.2, Propositions 5.4 and 6.1 imply isomorphisms and a surjection as

$$ \iota_* : \pi_k^{\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L}, x_0) \xrightarrow{\cong} \pi_k^{\iota_*\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L}', x_0), \quad \text{if } k \le D\bar{p}(v), \text{ and} $$

$$ \iota_* : \pi_k^{\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L}, x_0) \to \pi_k^{\iota_*\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L}', x_0), \quad \text{if } k = D\bar{p}(v) + 1. $$

On the other hand, we know from (17) that $\pi_{\ell}^{\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L}, x_0) = \pi_{\ell}^{\iota_*\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L}', x_0) = 0$, if $\ell > D\bar{p}(v)$. We conclude that the map

$$ \iota_* : \pi_*^{\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L}, x_0) \xrightarrow{\cong} \pi_*^{\iota_*\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L}', x_0) $$

is an isomorphism. So Lemma 7.2 gives the claim:

$$ \iota_* : H_*(\mathscr{G}_{\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L})) \xrightarrow{\cong} H_*(\mathscr{G}_{\iota_*\bar{p}}(\mathring{c}L, \mathring{c}\mathcal{L})). $$

• The stratum $S$ is a fountain stratum as described in the diagram (6) which is repeated here for the convenience of the reader,

$$ (\mathbb{R}^a \times (\mathring{c}\mathbb{S}^{b-1} \times \mathring{c}L), \mathcal{I} \times (\mathcal{I} \times \mathring{c}\mathcal{L})_{(u,v)}) \xrightarrow{\phi'} (W, \mathcal{S}) \qquad (27) $$
$$ ((\mathbb{R}^a \times \mathring{c}\mathbb{S}^{b-1}) \times \mathring{c}L, \mathcal{I} \times \mathcal{I} \times \mathring{c}\mathcal{L}) \xrightarrow{\psi'} (W, \mathcal{T}). $$

(Recall that $u$ is the apex of $\mathring{c}\mathbb{S}^{b-1}$ and $v$ that of $\mathring{c}L$.) Condition (iii) of Proposition 7.1 becomes the following claim, where the involved perversities are described in Remark 3.8.

“The map

$$ \iota : A := (\mathbb{R}^a \times \mathring{c}\mathbb{S}^{b-1} \times \mathring{c}L, (\mathcal{I} \times \mathring{c}\mathcal{L})_{(u,v)}) \to B := (\mathbb{R}^a \times \mathring{c}\mathbb{S}^{b-1} \times \mathring{c}L, \mathcal{I} \times \mathring{c}\mathcal{L}) \quad (28) $$

induces an isomorphism $\iota_* : H_k(\mathscr{G}_{\bar{p}}A) \to H_k(\mathscr{G}_{\iota_*\bar{p}}B).$

We now use a version of the cone formula (18) for a product of two Gajer spaces on cones, its proof being postponed to Lemma 7.4. There are isomorphisms and a surjection as follows:

$$ \iota_* : H_k(\mathscr{G}_{\bar{q}}A) \xrightarrow{\cong} H_k(\mathscr{G}_{\iota_*\bar{p}}B), \quad \text{if } k \le D\bar{p}(\{u, v\}), \text{ and} \quad (29) $$

$$ \iota_* : H_k(\mathscr{G}_{\bar{p}}A) \to H_k(\mathscr{G}_{\iota_*\bar{p}}B), \quad \text{if } k = D\bar{p}(\{u, v\}) + 1. $$
As the conical chart has no exceptional stratum, with Lemma 7.2, Proposition 5.4 and Proposition 6.1, we also conclude with isomorphisms and a surjection:

$$ \iota_* : \pi_k^{\bar{p}}(A, x_0) \xrightarrow{\cong} \pi_k^{\iota_* \bar{p}}(B, x_0), \quad \text{if } k \le D\bar{p}(\{\mathbf{u}, \mathbf{v}\}), \text{ and} $$

$$ \iota_* : \pi_k^{\bar{p}}(A, x_0) \to \pi_k^{\iota_* \bar{p}}(B, x_0), \quad \text{if } k = D\bar{p}(\{\mathbf{u}, \mathbf{v}\}) + 1, $$

for any basepoint $x_0 \in \mathbb{R}^a \times \check{\mathbb{S}}^{b-1} \times (L\setminus\Sigma)\times]0, 1[$. On the other hand, we have from (17): $\pi_{\ell}^{\bar{p}}(A, x_0) = 0$ if $\ell > D\bar{p}(\{\mathbf{u}, \mathbf{v}\})$ and $\pi_{\ell}^{\iota_*\bar{p}}(B, x_0) = 0$ if $\ell > D\bar{p}(\{\mathbf{v}\})$. So, since $D\bar{p}(\mathbf{v}) \le D\bar{p}(\{\mathbf{u}, \mathbf{v}\})$ (cf. Remark 3.8), we obtain $\pi_{\ell}^{\bar{p}}(A, x_0) = \pi_{\ell}^{\iota_*\bar{p}}(B, x_0) = 0$ if $\ell > D\bar{p}(\{\mathbf{u}, \mathbf{v}\})$ and the map

$$ \iota_* : \pi_{\star}^{\bar{p}}(A, x_0) \xrightarrow{\cong} \pi_{\star}^{\iota_* \bar{p}}(B, x_0) $$

is an isomorphism for any basepoint $x_0$. Finally Lemma 7.2 gives the claim. $\square$

<figure><img src="image_2.png" /></figure>

We prove the formula for two cones (29) used in the proof of Theorem 7.3.

**Lemma 7.4.** Let $(L, \mathcal{L})$ be a compact stratified space and $b > 0$ be an integer. We denote by $\mathbf{u}$ the apex of $\check{\mathbb{S}}^{b-1}$ and by $\mathbf{v}$ that of $\check{\mathcal{L}}$. Then, for any $K$-perversity $\bar{p}$ on $A$, the stratified map defined by the identity,

$$ \iota: A := (\mathbb{R}^a \times \check{\mathbb{S}}^{b-1} \times \check{\mathcal{L}}, \mathcal{I} \times (\mathcal{I} \times \check{\mathcal{L}})_{(\mathbf{u}, \mathbf{v})}) \to B := (\mathbb{R}^a \times \check{\mathbb{S}}^{b-1} \times \check{\mathcal{L}}, \mathcal{I} \times \mathcal{I} \times \check{\mathcal{L}}), \\ \text{induces} $$

$$ \left\{ \begin{array}{ll} \text{isomorphisms} & \iota_* : H_k(\mathcal{G}_{\bar{p}}A) \xrightarrow{\cong} H_k(\mathcal{G}_{\iota_*\bar{p}}B) & \text{if } k \le D\bar{p}(\{\mathbf{u}, \mathbf{v}\}), \text{ and} \\ \text{an epimorphism} & \iota_* : H_k(\mathcal{G}_{\bar{p}}A) \to H_k(\mathcal{G}_{\iota_*\bar{p}}B) & \text{if } k = D\bar{p}(\{\mathbf{u}, \mathbf{v}\}) + 1. \end{array} \right. $$

*Proof.* For sake of simplicity, we set $\bar{q} = \iota_*\bar{p}$. From (16), we know that $\iota$ induces a homomorphism $\iota_* : H_*(\mathcal{G}_{\bar{p}}A) \to H_*(\mathcal{G}_{\bar{q}}B)$. We consider the two stratified sets

$$ C := (\mathbb{R}^a \times (\check{\mathbb{S}}^{b-1} \times \check{\mathcal{L}}) \setminus \{(\mathbf{u}, \mathbf{v})\}, \mathcal{I} \times (\mathcal{I} \times \check{\mathcal{L}})_{(\mathbf{u}, \mathbf{v})}) = (\mathbb{R}^a \times (\check{\mathbb{S}}^{b-1} \times \check{\mathcal{L}}) \setminus \{(\mathbf{u}, \mathbf{v})\}, \mathcal{I} \times \mathcal{I} \times \check{\mathcal{L}}) \\ \text{and} $$

$$ D := (\mathbb{R}^a \times \check{\mathbb{S}}^{b-1} \times (\check{\mathcal{L}}\setminus\{\mathbf{v}\}), \mathcal{I} \times (\mathcal{I} \times \check{\mathcal{L}})_{(\mathbf{u}, \mathbf{v})}) = (\mathbb{R}^a \times \check{\mathbb{S}}^{b-1} \times (\check{\mathcal{L}}\setminus\{\mathbf{v}\}), \mathcal{I} \times \mathcal{I} \times \check{\mathcal{L}}). $$

The first step of the proof is the determination of $H_*(\mathcal{G}_{\bar{q}}B, \mathcal{G}_{\bar{q}}C)$. To do this, we first use the excision theorem ([CSAT25, Theorem 3.14]) of $Z = \mathbb{R}^a \times \{\mathbf{u}\} \times (\check{\mathcal{L}}\setminus\{\mathbf{v}\})$ relatively to the pair $(C, D)$, which gives:

$$ \begin{aligned} H_k(\mathcal{G}_{\bar{q}}C, \mathcal{G}_{\bar{q}}D) &\cong H_k(\mathcal{G}_{\bar{q}}(C\setminus Z), \mathcal{G}_{\bar{q}}(D\setminus Z)) \\ &= H_k(\mathcal{G}_{\bar{q}}((\check{\mathbb{S}}^{b-1}\setminus\{\mathbf{u}\}) \times \check{\mathcal{L}}), \mathcal{G}_{\bar{p}}((\check{\mathbb{S}}^{b-1}\setminus\{\mathbf{u}\}) \times (\check{\mathcal{L}}\setminus\{\mathbf{v}\}))). \end{aligned} $$

The stratified space $(\check{\mathbb{S}}^{b-1}\setminus\{\mathbf{u}\}, \mathcal{I})$ is stratified homeomorphic to $\mathbb{S}^{b-1}\times]0, 1[$ with the trivial filtration. From Proposition 4.1, we get

$$ H_k(\mathcal{G}_{\bar{q}}C, \mathcal{G}_{\bar{q}}D) \cong H_k(\mathcal{G}_{\bar{q}}B, \mathcal{G}_{\bar{q}}D) \oplus H_{k-b+1}(\mathcal{G}_{\bar{q}}(\check{\mathcal{L}}), \mathcal{G}_{\bar{q}}(\check{\mathcal{L}}\setminus\{\mathbf{v}\})). \quad (30) $$

Thus, for each $k \in \mathbb{N}$, the inclusion $C \hookrightarrow B$ induces an epimorphism $\nu_k : H_k(\mathcal{G}_{\bar{q}}C, \mathcal{G}_{\bar{q}}D) \to H_k(\mathcal{G}_{\bar{q}}B, \mathcal{G}_{\bar{q}}D)$. So, for any $k$, the long exact sequence of the triple $(\mathcal{G}_{\bar{q}}B, \mathcal{G}_{\bar{q}}C, \mathcal{G}_{\bar{q}}D)$, breaks down in short exacts sequences,

$$ 0 \to H_{k+1}(\mathcal{G}_{\bar{q}}B, \mathcal{G}_{\bar{q}}C) \to H_k(\mathcal{G}_{\bar{q}}C, \mathcal{G}_{\bar{q}}D) \xrightarrow{\nu_k} H_k(\mathcal{G}_{\bar{q}}B, \mathcal{G}_{\bar{q}}D) \to 0. \quad (31) $$

Given $k \le D\bar{p}(\{\mathbf{u}, \mathbf{v}\})$, we deduce from (14):

$$ k - b + 1 \le D\bar{p}(\{\mathbf{u}, \mathbf{v}\}) - b + 1 \le D\bar{q}(\mathbf{v}) + b - b + 1 = D\bar{q}(\{\mathbf{v}\}) + 1. $$

Thus, the cone formula (19) and the isomorphism (30) imply that $\nu_k$ is an isomorphism and therefore

$$ H_k(\mathcal{G}_{\bar{q}}B, \mathcal{G}_{\bar{q}}C) = 0, \text{ if } k \le D\bar{q}(\{\mathbf{u}, \mathbf{v}\}) + 1. \quad (32) $$
The homeomorphism $\iota: A \to B$ induces morphisms between the homology long exact sequences of the pairs $(\mathscr{G}_{\bar{p}}A, \mathscr{G}_{\bar{p}}C)$ and $(\mathscr{G}_{\bar{q}}B, \mathscr{G}_{\bar{q}}C)$. We denote them by $\iota_{\star}^{i}$, for $i = 1, 2, 3$, as follows:

$$ \dots \longrightarrow H_{k+1}(\mathscr{G}_{\bar{p}}A, \mathscr{G}_{\bar{p}}C) \longrightarrow H_k(\mathscr{G}_{\bar{p}}C) \longrightarrow H_k(\mathscr{G}_{\bar{p}}A) \longrightarrow H_k(\mathscr{G}_{\bar{p}}A, \mathscr{G}_{\bar{p}}C) \longrightarrow \dots \\ \Big|_{\iota_k^3} \Big| \Big|_{\iota_k^1} \Big| \Big|_{\iota_k^2} \Big| \Big|_{\iota_k^3} \\ \dots \longrightarrow H_{k+1}(\mathscr{G}_{\bar{q}}B, \mathscr{G}_{\bar{q}}C) \longrightarrow H_k(\mathscr{G}_{\bar{q}}C) \longrightarrow H_k(\mathscr{G}_{\bar{q}}B) \longrightarrow H_k(\mathscr{G}_{\bar{q}}B, \mathscr{G}_{\bar{p}}C) \longrightarrow \dots $$

The map $\iota_k^1$ is an isomorphism for each $k$ since $D\bar{p} = D\bar{q}$ outside $\{(\mathbf{u}, \mathbf{v})\}$. Applying (19) together with the stratified homeomorphism $h$ (4), we get $H_k(\mathscr{G}_{\bar{p}}A, \mathscr{G}_{\bar{p}}C) = 0$ for each $k \le D\bar{p}(\{\mathbf{u}, \mathbf{v}\}) + 1$. So, with the 5-lemma and (32), we conclude that $\iota_k^2: H_k(\mathscr{G}_{\bar{p}}A) \to H_k(\mathscr{G}_{\bar{p}}B)$ is an isomorphism for $k \le D\bar{p}(\{\mathbf{u}, \mathbf{v}\})$ and an epimorphism for $k = D\bar{p}(\{\mathbf{u}, \mathbf{v}\}) + 1$. $\square$

**Theorem 7.5.** Let $(X, S, T)$ be a CS-coarsening. Let $\bar{p}$ be a $K$-perversity of pushforward perversity $\iota_{\star}\bar{p}$, such that $\bar{p} \le \bar{t}$ and $\pi_1^*(\dot{c}L, \dot{c}S, x_0) = \{1\}$ for the link $L$ of any exceptional stratum, with $x_0$ regular. Then, for any local coefficient system $\mathfrak{E}$ on $\mathscr{G}_{\iota_{\star}\bar{p}}(X, T)$, the identity map induces an isomorphism

$$ \mathscr{G}(\iota)_{\star}: H_{\star}(\mathscr{G}_{\bar{p}}(X, S); \iota^{\star}\mathfrak{E}) \cong H_{\star}(\mathscr{G}_{\iota_{\star}\bar{p}}(X, T); \mathfrak{E}). $$

*Proof.* We follow the process of the proof of Theorem 7.3 using the decomposition in simple coarsenings established in [SA24, Proposition 2.3]. In particular, the proof reduces to the verification of the item (iii) of Proposition 7.1. Let $x \in S$ be a point in a singular stratum $S \in S$. We may consider a small enough $S$-conical charts containing $x$ and suppose that the stratum $S$ is exceptional in $(X, S, T)$, since the two other cases were dealt with in the proof of Theorem 7.3. Let us recall the local situation (2) in the diagram:

$$ \begin{tikzcd} (\mathbb{R}^a \times \mathring{c}\mathbb{S}^{b-1}, \mathcal{I} \times \mathring{c}\mathcal{I}) \arrow[r, "\phi"] & (W, S) \\ (\mathbb{R}^{a+b}, \mathcal{I}) \arrow[u,r, "f"'] \arrow[u,l, "\psi"] \arrow[r, "\iota"] & (W, T). \end{tikzcd} \quad (33) $$

Notice that $b \ge 2$ since $\bar{p}$ is a $K$-perversity (cf. Remark 3.4). Using Proposition 4.1, Property (iii) of Proposition 7.1 becomes the following claim

$$ \text{"$H_{\star}(\mathscr{G}_{\bar{p}}(\dot{c}\mathbb{S}^{b-1}, \dot{c}\mathcal{I})) \cong H_{\star}(\mathbb{R}^{b}, \mathcal{I})."} \quad (34) $$

The cone formula (17) gives

$$ \pi_k^{\bar{p}}(\dot{c}\mathbb{S}^{b-1}, \dot{c}\mathcal{I}) = \begin{cases} \pi_k(\mathbb{S}^{b-1}) & \text{if } k \le D\bar{p}(\mathbf{v}), \\ 0 & \text{if } k > D\bar{p}(\mathbf{v}). \end{cases} \quad (35) $$

The hypothesis $\pi_1^*(\dot{c}\mathbb{S}^{b-1}, \dot{c}\mathcal{I}) = 0$ implies that the homological sphere $\mathbb{S}^{b-1}$ is a homotopy sphere $\mathbb{S}^{b-1}$ if $1 \le D\bar{p}(\mathbf{v})$. From (13) applied to $D\bar{p}$, we have $D\bar{p}(\mathbf{v}) \le b-2$ and the equality (35) implies $\pi_k^*(\dot{c}\mathbb{S}^{b-1}, \dot{c}\mathcal{I}) = 0$ for any $k$ which implies (34). $\square$

*Proof of Theorem A.* We apply Lemma 7.2. The isomorphisms on $\pi_0$ and $\pi_1$ have been established in Propositions 5.4 and 6.1. The isomorphisms in homology are in Theorem 7.3. $\square$

*Proof of Theorem B.* It is a proof similar to that of Theorem A. The only modifications are the replacement of Proposition 6.1 by Proposition 6.5 and of Theorem 7.3 by Theorem 7.5. $\square$
## 8. REFINEMENT AND EXAMPLES

Let $(X, S, \mathcal{T})$ be a CS-coarsening. In this section, we use Theorems A and B to get isomorphisms on intersection homotoy groups, with a perversity $\bar{q}$ on $(X, \mathcal{T})$ and its pullback $\iota^*\bar{q}$ on $(X, S)$. We also show how the GM-perversities fit in our setting.

**Theorem 8.1 (Invariance by refinement).** Let $(X, S, \mathcal{T}, x_0)$ be a pointed CS-coarsening without exceptional strata, and let $\bar{q}$ be a perversity on $(X, \mathcal{T})$ with $\bar{q} \le \bar{t}$, of pullback perversity $\iota^*\bar{q}$. Then, the identity map induces an isomorphism

$$ \pi_{\star}^{\iota^*\bar{q}}(X, S, x_0) \cong \pi_{\star}^{\bar{q}}(X, \mathcal{T}, x_0). $$

*Proof.* From Lemma 3.7, we know that $\iota^*\bar{q}$ is a K-perversity and $\iota^*\bar{q} \le \bar{t}$. So, the perversity $\bar{p} := \iota^*\bar{q}$ satisfies the hypotheses of Theorem A. The result now follows from $\iota_*\bar{p} = \iota_*\iota^*\bar{q} = \bar{q}$, as was noticed after Definition 3.2. $\square$

With the same proof, we get also a refinement theorem in the case of a Thom-Mather space with some exceptional strata.

**Theorem 8.2.** Let $(X, S)$ be a normal connected Thom Mather space with a finite number of strata and $(X, S, \mathcal{T})$ be a CS-coarsening without 1-exceptional strata. Let $\bar{q}$ be a perversity on $(X, \mathcal{T})$ with $\bar{q} \le \bar{t}$, of pullback perversity $\iota^*\bar{q}$, such that $\pi_1^{\iota^*\bar{q}}(\hat{c}L, \hat{c}S, x_0) = \{1\}$ for the link $L$ of any exceptional stratum with $x_0$ regular. Then, the identity map induces an isomorphism

$$ \pi_{\star}^{\iota^*\bar{q}}(X, S, x_0) \cong \pi_{\star}^{\bar{q}}(X, \mathcal{T}, x_0). $$

If a stratum $S$ is exceptional, we have $\bar{q} = 0$ in $W\backslash S$, where $W$ is a small enough neighborhood of $S$. Thus, $\iota^*\bar{q} = 0$ on $W\backslash S$ and therefore on the link $L$. In short, the hypothesis on the link of exceptional strata can also be stated as $\pi_1^0(\hat{c}L, \hat{c}S, x_0) = \{1\}$.

Below, we recall [CSAT25, Example 4.3] which justifies the hypothesis made on the exceptional strata.

**Example 8.3.** Let $\Sigma^2 P$ be the double suspension of the Poincaré sphere. This suspension being homeomorphic to the sphere $S^5$, we get a CS-coarsening $(S^5, S, \mathcal{T})$, where $\mathcal{T}$ is the trivial stratification with only one regular stratum. The stratification $S$ has one regular stratum and four exceptional strata $S_0, S_1, S_2, S_3$, with $\dim S_0 = \dim S_1 = 0$ and $\dim S_2 = \dim S_3 = 1$. Let us define a K-perversity $\bar{p}$ on $S$ by $D\bar{p} = \bar{1}$, so which verifies $\bar{p} \le \bar{t}$. By definition of the pushforward perversity, we have $\iota_*\bar{p} = \bar{0}$. The computation made in [CSAT25] gives $\pi_1^{\bar{p}}(S^5, S) \ne \{1\}$ and $\pi_1^{\iota_*\bar{p}}(S^5, \mathcal{T}) = \{1\}$.

Invariance of intersection homology was established by Goresky and MacPherson in the PL setting ([GM80]) and after in the topological setting ([GM83]). They use codimensional perversities with a growing property; we recall their definition.

**Definition 8.4.** A GM-perversity is a codimensional perversity $\bar{p}$ verifying $\bar{p}(0) = \bar{p}(1) = \bar{p}(2) = 0$ and, for any $1 \le k$,

$$ \bar{p}(k) \le \bar{p}(k+1) \le \bar{p}(k) + 1. \qquad (36) $$

We remind from Definition 3.1 that the top perversity $\bar{t}$ is defined by $\bar{t}(k) = k - 2$ if $k > 0$. Also, any codimensional perversity $\bar{p}$ verifies $\bar{p}(0) = 0$,

In [CSAT25], we have shown the topological invariance of the intersection homotopy groups for GM-perversities and CS sets without exceptional strata. We prove below that Theorem A implies this result, in the slight more general paradigm of the perversities used by H. King ([Kin85]).
**Corollary 8.5.** Let $\bar{p}$ be a codimensional perversity verifying $\bar{p} \le \bar{t}$ and the growing property (36). Let $(X, S, \mathcal{T}, x_0)$ be a pointed CS-coarsening without exceptional strata, then the identity map induces the isomorphism

$$ \iota_*: \pi_*^{\bar{p}}(X, S, x_0) \cong \pi_*^{\bar{p}}(X, \mathcal{T}, x_0). $$

*Proof.* We denote also $\bar{p}$ the perversity on $(X, S)$ induced by $\bar{p}$; i.e., we set $\bar{p}(S) = \bar{p}(\mathrm{codim} S)$. We claim that $\bar{p}$ is a $K$-perversity for the CS-coarsening $(X, S, \mathcal{T})$. First, let $(S, Q)$ be a couple of strata of $S$ such that $S \le Q$ and $S^t = Q^t$. For the property (K1) of Definition 3.3, we have to check:

$$ \bar{p}(\mathrm{codim} Q) \le \bar{p}(\mathrm{codim} S) \le \bar{p}(\mathrm{codim} Q) + \bar{t}(\mathrm{codim} S) - \bar{t}(\mathrm{codim} Q). $$

This is clear if $S, Q$ are regular strata. If they are both singular, this comes from (36). By hypothesis, there is no exceptional strata.

For the second property, we consider a couple of strata $(S, Q)$ of $S$ such that $\dim S = \dim Q$ and $S^t = T^t$. The axiom (K2) is trivially satisfied since we have

$$ \bar{p}(S) = \bar{p}(\mathrm{codim} S) = \bar{p}(\mathrm{codim} Q) = \bar{p}(Q). $$

If $S \in S$ is a singular stratum, the inequalities

$$ \bar{p}(S) = \bar{p}(\mathrm{codim} S) \le \bar{t}(\mathrm{codim} S) = \mathrm{codim} S - 2 = \bar{t}(S). $$

give $\bar{p} \le \bar{t}$. Finally, let us compute $\iota_* \bar{p}$. Let $T \in \mathcal{T}$ and $S \in S$ be a source stratum of $T$, then from Lemma 3.5 i), we deduce

$$ \iota_* \bar{p}(T) = \bar{p}(S) = \bar{p}(\mathrm{codim} S) = \bar{p}(\mathrm{codim} T) = \bar{p}(T). $$

We have set $\iota_* \bar{p} = \bar{p}$. The hypotheses being verified, we can apply Theorem A and get the result. □

Notice that Corollary 8.5 and Theorems 7.3 and 7.5 also imply a topological invariance of the homology of the involved Gajer spaces.

We present an example where the notion of CS set is not preserved, neither by refinement nor by coarsening.

**Example 8.6.** We give a chain of coarsenings

$$ (X, S) \xrightarrow{\iota_1} (X, \mathcal{R}) \xrightarrow{\iota_2} (X, \mathcal{T}) $$

whose extremities are CS sets and the middle is merely a filtered space. Let $T^2 \subset \mathbb{R}^3$ be the torus. Its cone, $\hat{c}T^2$ is a closed subset of $\mathbb{R}^3 \times ]-1, 1[$. We define $X = \mathbb{R}^3 \times ]-1, 1[ \times \mathbb{R}$, choose two different points $\{P, Q\}$ of $T^2$ and denote by $v$ the apex of the cone. We define three filtrations on the space $X$ as follows.

* $S: X_0 = \{(v, 0)\} \subset X_1 = \hat{c}\{P, Q\} \times \{0\} \subset X_3 = \hat{c}\mathbb{T} \times \{0\} \subset X_4 = X$.

* $\mathcal{R}: X_1 = \hat{c}\{P, Q\} \times \{0\} \subset X_3 = \hat{c}\mathbb{T} \times \{0\} \subset X_4 = X$.

* $\mathcal{T}: X_4 = X$.

Notice that $S$ is the filtration $\mathcal{R}_{(v,0)}$ as defined in Example 2.5. The figure below illustrates the subspace $X_3$ with $S$ and $\mathcal{R}$ stratifications.

<figure><img src="image_3.png" /></figure>
The space $X$ is homeomorphic to $\mathbb{R}^5 \cong \mathring{c}S^4$. The stratification $\mathcal{S}$ of $X$ is the conic filtration where $S^4$ has for only singular stratum the torus $T^2$ viewed as a subspace of $S^3 \subset S^4$. It is therefore a CS set.

If $(X, \mathcal{R})$ were a CS set, then $X_1 \subset X_3$ would be a CS set (this follows directly from the definition of a CS set). In that case, any two points in $X_1$ would have the same homological link ([Fri20, Lemma 6.3.23]). This is not the case, since the link of $v \in X_1$ is $T^2$, while the link of $P \in X_1$ is $S^2$.

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LABORATOIRE DE MATHÉMATIQUES DE LENS, EA 2462, UNIVERSITÉ D'ARTOIS, SP18, RUE JEAN SOUVRAZ, 62307 LENS CEDEX, FRANCE

Email address: martin.saraleguiaranguren@univ-artois.fr

DÉPARTEMENT DE MATHÉMATIQUES, UMR-CNRS 8524, UNIVERSITÉ DE LILLE, 59655 VILLENEUVE D'ASCQ CEDEX, FRANCE

Email address: Daniel.Tanre@univ-lille.fr