Intersection Homotopy Groups

• Intersection homotopy groups are defined for filtered spaces with perversity functions, using Gajer’s simplicial set of p-full simplices to capture stratified topological invariants.
• The construction employs full facewise compatibility conditions that guarantee the Kan property, enabling a functorial and computational framework analogous to classical homotopy theory.
• Key structural results include a Van Kampen theorem analogue and a Hurewicz comparison that relate these groups to intersection homology, ensuring invariance under stratified maps.

Searching arXiv for recent and foundational papers on intersection homotopy groups and related constructions. Search query: "intersection homotopy groups Gajer perversity CS sets" Search query: "site:arxiv.org intersection homotopy groups Gajer simplicial set" Intersection homotopy groups are homotopy groups attached to a filtered space together with a perversity, designed as a homotopy-theoretic counterpart to Goresky–MacPherson intersection homology. Their modern definition uses the homotopy groups of a Kan simplicial set introduced by P. Gajer, built from simplices satisfying perversity-controlled allowability conditions on all faces. The construction addresses a basic obstacle: Goresky–MacPherson intersection homology is not, in general, the homology of a space, so there is no preferred ambient space whose ordinary homotopy groups could simply be declared to be the “intersection homotopy groups” (Chataur et al., 2022).

1. Conceptual origin and ambient setting

The starting point is a filtered space $X$ with strata, singular part $X_{n-1}$, and regular part $X\setminus X_{n-1}$. A perversity is a function on the strata,

$$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$

For a singular stratum $S$, the top perversity is

$t(S)=\operatorname{codim}S-2,$

and the complementary perversity $Dp$ is defined by $Dp+p=t$ (Chataur et al., 2022).

This framework is modeled on the role perversities already play in intersection homology. The distinctive issue is that allowability conditions are imposed relative to the singular strata, and these conditions do not automatically behave well under ordinary simplicial operations. As a result, the definition of intersection homotopy groups is not obtained by taking homotopy groups of the topological space $X$ itself, except in special cases. The principal setting for the theory is that of Siebenmann CS sets, that is, locally conical stratified spaces; later refinements also treat Thom–Mather and pre-Thom–Mather hypotheses (Chataur et al., 2022).

A common misconception is to identify intersection homotopy groups with ordinary homotopy groups of a singular space endowed with a stratification. The theory is more rigid: the groups depend on the pair $(X,p)$, not on $X_{n-1}$0 alone, unless additional invariance results apply. Another misconception is that intersection homology should automatically determine a space up to homotopy. The construction exists precisely because no such preferred realizing space is available in general (Chataur et al., 2022).

2. Definition through Gajer’s simplicial set

For a singular simplex $X_{n-1}$1, the $X_{n-1}$2-allowability condition is

$X_{n-1}$3

for every singular stratum $X_{n-1}$4. Allowability alone is insufficient for a simplicial-set construction, because it is not stable under passage to faces. The remedy is to require full facewise compatibility: a simplex is $X_{n-1}$5-full if $X_{n-1}$6 and all its iterated faces are $X_{n-1}$7-allowable (Chataur et al., 2022).

The set of all $X_{n-1}$8-full simplices forms a simplicial set, denoted in the literature by $X_{n-1}$9 and also by $X\setminus X_{n-1}$0. A basic structural result is that this simplicial set satisfies the Kan condition. For a pointed perverse space $X\setminus X_{n-1}$1, with $X\setminus X_{n-1}$2 a regular point, the intersection homotopy groups are then defined by

$X\setminus X_{n-1}$3

(Chataur et al., 2022).

This definition is functorial for stratified maps $X\setminus X_{n-1}$4 satisfying $X\setminus X_{n-1}$5. Stratified homotopies induce simplicial homotopies on the Gajer spaces. Consequently, stratified homotopy equivalent perverse spaces have equivalent Gajer spaces and therefore the same intersection homotopy groups (Chataur et al., 2022).

The later coarsening-and-refinement theory keeps the same simplicial model but extends the class of perversities. In that setting, general perversities are defined on the poset of strata rather than only as functions of codimension, and the identity map under a coarsening induces a canonical simplicial map

$X\setminus X_{n-1}$6

once the inequality $X\setminus X_{n-1}$7 is available (Saralegi-Aranguren et al., 5 Aug 2025).

3. Fundamental structural theorems

The basic low-dimensional theorem is a Van Kampen theorem for the intersection fundamental group. If $X\setminus X_{n-1}$8 is a perverse space with $X\setminus X_{n-1}$9, and $$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$0 is an open cover such that $$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$1, $$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$2, and $$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$3 are path-connected, then the canonical homomorphism

$$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$4

is surjective, and its kernel is the normal subgroup generated by the elements

$$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$5

where the $$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$6 are induced by the inclusions $$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$7 (Chataur et al., 2022).

This formal parallel with the classical Van Kampen theorem is significant because it shows that $$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$8 is not merely an ad hoc invariant of a simplicial set. It behaves as a fundamental group should behave with respect to open coverings, provided the perversity constraints are respected.

At the opposite end of the theory lies a recovery theorem. If $$p:\mathcal S_X\to \mathbb Z\cup\{+\infty\}, \qquad p=0 \text{ on regular strata.}$$9 is a connected normal Thom–Mather space with finitely many strata and $S$0 is the top perversity, then

$S$1

Thus, for top perversity in this setting, intersection homotopy groups coincide with the ordinary homotopy groups of the underlying topological space (Chataur et al., 2022).

A related cone formula appears in the later literature. For the open cone $S$2 on a compact filtered space $S$3, with apex $S$4,

$S$5

This provides a local computational tool compatible with the conical nature of CS sets (Saralegi-Aranguren et al., 5 Aug 2025).

4. Hurewicz comparison with intersection homology

The central comparison with Goresky–MacPherson theory begins from the inclusion of chain complexes

$S$6

which induces

$S$7

Composing the classical Hurewicz map for the simplicial set $S$8 with $S$9 yields the $t(S)=\operatorname{codim}S-2,$0-intersection Hurewicz homomorphism

$t(S)=\operatorname{codim}S-2,$1

(Chataur et al., 2022).

For any perverse space, the comparison is already exact in low degrees: $t(S)=\operatorname{codim}S-2,$2 Accordingly, for a $t(S)=\operatorname{codim}S-2,$3-connected perverse CS set, the degree-one Hurewicz map identifies $t(S)=\operatorname{codim}S-2,$4 with the abelianization of $t(S)=\operatorname{codim}S-2,$5 (Chataur et al., 2022).

In higher degrees, the theory requires hypotheses on links. If $t(S)=\operatorname{codim}S-2,$6 and every link $t(S)=\operatorname{codim}S-2,$7 of $t(S)=\operatorname{codim}S-2,$8 satisfies

$t(S)=\operatorname{codim}S-2,$9

then

$Dp$0

is an isomorphism for $Dp$1 and a surjection for $Dp$2. The proof proceeds by comparing the homology of $Dp$3 with intersection homology and then applying the classical Hurewicz theorem to the Kan simplicial set $Dp$4 (Chataur et al., 2022).

These results clarify the status of intersection homotopy groups. They are not defined as an auxiliary reformulation of intersection homology, but they become tightly linked to intersection homology under explicit connectivity assumptions. A plausible implication is that the link conditions play the same structural role here that local conical control plays elsewhere in the subject: they govern when perversity-controlled homotopy can be detected homologically.

5. Invariance, intrinsic stratification, coarsenings, and refinements

Topological invariance is subtle. A CS set $Dp$5 admits an intrinsic stratification $Dp$6, and the identity map $Dp$7 is stratified. The crucial condition is that $Dp$8 and $Dp$9 have the same regular part: $Dp+p=t$0 Equivalently, no singular stratum of $Dp+p=t$1 becomes regular in the coarsened structure. Under this hypothesis, if $Dp+p=t$2 is a CS set without codimension $Dp+p=t$3 strata and $Dp+p=t$4 is a GM-perversity, then

$Dp+p=t$5

(Chataur et al., 2022).

The 2025 extension recasts this as a theory of coarsenings and refinements of CS structures on the same underlying space. A coarsening $Dp+p=t$6 has strata of $Dp+p=t$7 that are unions of strata of $Dp+p=t$8. It introduces source strata, fountain strata, and exceptional strata; an exceptional stratum is a singular stratum for $Dp+p=t$9 that becomes regular in $X$0. To control perversities under such changes, the paper defines $X$1-perversities, characterized by growth conditions analogous to the classical GM inequalities but formulated on the stratum poset (Saralegi-Aranguren et al., 5 Aug 2025).

If there are no exceptional strata and $X$2, then the identity induces an isomorphism

$X$3

More generally, if $X$4 is a normal connected Thom–Mather space with finitely many strata, $X$5 is a $X$6-perversity with $X$7, and the link $X$8 of every exceptional stratum satisfies

$X$9

then the same invariance conclusion holds (Saralegi-Aranguren et al., 5 Aug 2025).

The same paper derives refinement invariance by pullback of perversities: $(X,p)$0 under the corresponding hypotheses (Saralegi-Aranguren et al., 5 Aug 2025).

The limitations are explicit. The double suspension of the Poincaré sphere gives a space homeomorphic to $(X,p)$1 but with a nontrivial singular stratification; for a suitable GM-perversity $(X,p)$2, one has

$(X,p)$3

and similarly for the double suspension. Since $(X,p)$4 topologically, this shows that $(X,p)$5 is not a topological invariant without the “same regular part” hypothesis (Chataur et al., 2022).

6. Related meanings of “intersection” in adjacent literatures

The phrase “intersection homotopy groups” also appears, or is naturally suggested, in several neighboring contexts, but these uses are distinct from the perversity-based theory.

In geometric group theory and low-dimensional topology, a recurrent formula expresses higher homotopy groups as quotients of subgroup intersections by symmetric commutator subgroups. For a cofibrant $(X,p)$6-partition and suitable normal subgroups $(X,p)$7,

$(X,p)$8

The same mechanism is applied to link groups, braid groups, and surface groups, including formulas recovering $(X,p)$9 from special pure braids or from normal subgroups of punctured-surface groups. Here the term does not denote a separate perversity-sensitive invariant; rather, higher homotopy is realized algebraically as the difference between an intersection subgroup and the symmetric commutator subgroup (Li et al., 2010).

In finite-group combinatorics, intersection data on proper nontrivial subgroups gives rise to an intersection graph $X_{n-1}$00 and an intersection complex $X_{n-1}$01. The graph is the $X_{n-1}$02-skeleton of the complex, and $X_{n-1}$03 is homotopy equivalent to the order complex of the poset of proper nontrivial subgroups. Consequently, the two complexes have the same homotopy groups after choosing compatible basepoints. If the domination number of $X_{n-1}$04 is $X_{n-1}$05, then $X_{n-1}$06 is contractible. This literature studies homotopy invariants attached to subgroup-intersection complexes, but it does not define a separate object called “intersection homotopy groups” in the stratified-topology sense (Kayacan, 2016).

In the algebraic study of complete intersections, the homotopy obstruction set $X_{n-1}$07 is defined from polynomial homotopies of points on an affine quadric $X_{n-1}$08. It detects when a surjection $X_{n-1}$09 lifts to a surjection $X_{n-1}$10, acquires a natural abelian group structure under stable-range hypotheses, and is closely related to Euler class groups. The paper explicitly interprets $X_{n-1}$11 as an algebraic analogue of an “intersection homotopy group,” but this is again a different notion from the homotopy groups $X_{n-1}$12 of a Gajer space (Mandal et al., 2016).

Taken together, these usages show that the phrase has a narrow technical meaning in stratified topology and a wider family of analogical meanings elsewhere. In the narrow sense, intersection homotopy groups are the homotopy groups of Gajer’s simplicial set of $X_{n-1}$13-full simplices. In the wider sense, the phrase points to a recurrent mathematical pattern: homotopy is extracted from data governed by intersections—of strata, of subgroups, or of ideals—after imposing an additional structure that isolates the genuinely higher-dimensional obstruction.