From samples to persistent stratified homotopy types

Abstract

The natural occurrence of singular spaces in applications has led to recent investigations on performing topological data analysis (TDA) in a stratified framework. In many applications, there is no a priori information on what points should be regarded as singular or regular. For this purpose we describe a fully implementable process that provably approximates the stratification for a large class of two-strata Whitney stratified spaces from sufficiently close non-stratified samples. Additionally, in this work, we establish a notion of persistent stratified homotopy type obtained from a sample with two strata. In analogy to the non-stratified applications in TDA which rely on a series of convenient properties of (persistent) homotopy types of sufficiently regular spaces, we show that our persistent stratified homotopy type behaves much like its non-stratified counterpart and exhibits many properties (such as stability, and inference results) necessary for an application in TDA. In total, our results combine to a sampling theorem guaranteeing the (approximate) inference of (persistent) stratified homotopy types of sufficiently regular two-strata Whitney stratified spaces.

Appendices

Appendix A Some details on abstract homotopy theory

Remark A.1

There are some subtleties to be considered, which come down to the order in which one passes to the persistent and homotopical perspective. We emphasize that by $\textrm{ho}{\textbf {T}}^{I}$, for some indexing category I, we mean the localization of the functor category at pointwise weak equivalences, and not the functor category $(\textrm{ho}{\textbf {T}})^{I}$, obtained by localizing at weak equivalences first. The universal property of the localization induces a canonical functor

$$\begin{aligned} \textrm{ho}{\textbf {T}}^{I} \rightarrow (\textrm{ho}{\textbf {T}})^{I}. \end{aligned}$$

This functor is essentially never an equivalence of categories. For example, for ${\textbf {T}} = {{\textbf {Top}}}$ with the usual class of weak equivalences, the notion of isomorphism on the left-hand side is fine enough to compute homotopy limits and colimits. This is not the case on the right-hand side (see for example Hirschhorn (2003), for an introduction to the theory). Generally, the functor will be neither essentially surjective nor fully faithful. Essential surjectivity, for example, comes down to whether or not a homotopy commutative diagram is equivalent to an actual commutative diagram (see Dwyer and Kan (1984) for a detailed discussion.)

To see that faithfulness is generally not the case, consider replacing ${\mathbb {R}}_+$ by $I = \{ 0 < 1\}$, and taking ${\textbf {T}} = {{\textbf {Top}}}$, $D = \{* \rightarrow S^1 \}$ and $D' = \{* \rightarrow X\}$, for some pointed space X. Both objects may be considered as pointed spaces. Then, the hom-objects from D to $D'$ in $\textrm{ho}{\textbf {T}}^{I}$ are the homotopy groups of X. In $(\textrm{ho}{\textbf {T}})^{I}$, however, the hom-object is given by free homotopy classes from $S^1$ to X, i.e. by the abelianization of the homotopy group of X.

In the special case where $I = {\mathbb {R}}_+$ and ${\textbf {T}} = {{\textbf {Top}}}$ this leaves, a priori, an ambivalence by what one means by a persistent homotopy type. Given a persistent space, i.e. an object in ${{\textbf {Top}}}^{{\mathbb {R}}_+}$, one can either consider its isomorphism class in $\textrm{ho}( {{\textbf {Top}}}^{{\mathbb {R}}_+})$ or in $(\textrm{ho}{{\textbf {Top}}})^{\mathbb R_+}$. We argue that the former is the conceptually better notion since properties P(1) to Item P3 may already be stated on this level. At the same time, due to the comparison functor between the two categories, results obtained in $\textrm{ho}( {{\textbf {Top}}}^{{\mathbb {R}}_+})$ are generally stronger than results in $(\textrm{ho}{{\textbf {Top}}})^{{\mathbb {R}}_+}$.

However, one should note that when passing to the algebraic world by applying homology index-wise, both perspectives agree. Finally, we may add that for most applications the difference is negligible. This is a consequence of Lemma A.2 which, among other things, implies, as long as one restricts to persistent objects which are tame in the sense that their homotopy type only changes at finitely many points, then the functor

(A1)

induces a bijection on isomorphism classes. In particular, there is no difference in the resulting notion of persistent homotopy type.

Lemma A.2

Let ${\mathcal {M}}$ be a (simplicial) model category and I be any small indexing category. Then,

$$\begin{aligned} F: \textrm{ho}{\mathcal {M}}^{I} \rightarrow (\textrm{ho}{\mathcal {M}})^{I} \end{aligned}$$

reflects isomorphisms. Furthermore, let I be a finite, totally ordered poset. Then F is essentially surjective and full. In particular, two objects in $\textrm{ho}{\mathcal {M}}^{I}$ are isomorphic, if and only if their images under F are isomorphic.

Proof

Ultimately, this comes down to the fact that homotopy coherent diagrams of the particularly simple shapes involved are easy to understand. The proof requires a series of standard arguments in the theory of model categories. See Hirschhorn (2003) for a comprehensive overview. To see that the functor characterizes isomorphisms, note that a morphism in a functor category is an isomorphism, if and only if is so pointwise. Since a morphism descends to an isomorphism in the homotopy category, if and only if it is a weak equivalence, and weak equivalences in ${\mathcal {M}}^{I}$ (equipped with any of the usual model structures) are defined pointwise, this shows that a morphism in ${\mathcal {M}}^{I}$ is an isomorphism in $\textrm{ho}({\mathcal {M}}^{I})$ if and only if its image under F is an isomorphism.

The next statement holds in general. However, we show only the case of a simplicial model category, since all model categories relevant in this paper fulfill this property and the availability of a canonical cylinder object makes the proof somewhat more digestible. Essential surjectivity is immediate, as functors D defined on a totally ordered set $I \cong [n]=\{ 0,..., n\}$ are entirely determined by their values valued on $D(i \le i+1)$ and conversely any sequence of morphisms $X_i \rightarrow X_{i+1}$, uniquely determines a functor. Thus, (up to an isomorphism in the right-hand side category) being a functor with values in the homotopy category $\textrm{ho}{\mathcal {M}} ^{I}$ is equivalent to being a functor with values in ${\mathcal {M}}^{I}$.

Now, to see fullness, consider objects D and $D'$ on the left-hand side. Without loss of generality, we may assume that D is a cofibrant and $D'$ a fibrant object with respect to the injective model structure (which exists since I is a Reedy category in the obvious fashion. See Hirschhorn (2003) for an introduction to Reedy model structures.) In particular, morphisms in the homotopy categories between D and $D'$ and between $D_i, D_i'$ are given by (simplicial) homotopy classes. We now proceed to show fullness by induction over n. The case $n = 0$ is trivial. Now let $f: D \rightarrow D'$ be a morphism in $(\textrm{ho}{\mathcal {M}})^{I}$, i.e. we are given a homotopy commutative diagram

(A2)

by inductive assumption, we can assume that up to $D_n$ the diagram is actually commutative. It remains to show, that there exists a morphism $g_{n+1}$, simplicially homotopic to $f_{n+1}$, such that the right-hand square commutes on the nose. Let $H: D_n \rightarrow D'^{\Delta ^1}_{n+1}$ be the adjoint to the simplicial homotopy from the right down to the down right composition. We may instead solve the induced lifting problem

(A3)

such a lift exists, since, by assumption, the left-hand side map is a cofibration and the right-hand side map is a fibration. Thus, we have shown fullness. $\square$

Lemma A.3

Let $\mathcal {M}$ be a relative category (i.e. a category equipped with a notion of weak equivalence). Let $U \subset \Omega \times {\mathbb {R}}_+$ be a subset containing $\Omega \times \{0 \}$. Let $D \in \mathcal {M}^{U}$, be such that

$$\begin{aligned}D(v, 0) \rightarrow D(v, \alpha )\end{aligned}$$

is a weak equivalence, for all $(v, \alpha ) \in U$. Let $D\mid _{\Omega \times \{0\}}$ be homotopically constant of value $M \in {\mathcal {M}}$. Then D is also homotopically constant of value M.

Proof

This follows from the fact that $\Omega$ is initial in U. Let $i:\Omega \times \{ 0\} \hookrightarrow U$ be the inclusion. Note that, in this specific scenario

$$\begin{aligned} i_*(D')(v, \alpha ) = D'_{(v, 0)} \end{aligned}$$

since any slice involved in the right Kan-extension have a terminal object of the form $(v,0)$. In particular, this means that $i_*$ preserves weak equivalences between all objects. Furthermore, by assumption, this equality implies that the natural transformation

$$\begin{aligned} i_* D\mid _{(\Omega , 0)} \rightarrow D \end{aligned}$$

is a weak equivalence. Now, by assumption, $D\mid _{(\Omega , 0)}$ is weakly equivalent to some constant functor C in $\mathcal {M}^{\Omega \times \{0 \}}$. In particular, this implies that there is a zigzag of weak equivalences

(A4)

applying $i_*$ and using the fact that it preserves weak equivalences between all objects, we thus obtain a zigzag of weak equivalences

(A5)

which induces an isomorphism in $\textrm{ho}{\mathcal {M}}^{U}$. Finally, note that if C is constant of value $M \in {\mathcal {M}}$, then so is $i_*C$. $\square$

Appendix B Results on definable and Whitney stratified spaces

1.1 B.1 Definable sets can be thickened

The following lemma seems folklore knowledge to some degree. We provide it here for the sake of completeness. It seems to us that, with some extra technical effort, methods used in Chazal and Lieutier (2005) may even be used to obtain strongly stratified map** cylinder neighborhoods. However, the following result suffices for our purposes.

Lemma B.1

Let ${X}\subset Y \subset {\mathbb {R}}^N$ be definable with respect to some o-minimal structure and X compact. Then, there exists a $\varepsilon >0$ such for $0< \alpha < \varepsilon$ the following holds:

• 1. ${X}\hookrightarrow {X}_{\alpha } \cap Y$ is a strong deformation retract.

• 2. There is a homeomorphism $({X}_{\alpha } \cap Y) {\setminus } X \cong \textrm{d}_{{X}}^{-1}{(\frac{\alpha }{2})} \times (0, \alpha ]$, such that the diagram

  

  commutes.

Furthermore, if $Y= {\mathbb {R}}^N$, then $\varepsilon$ may be taken to be the weak feature size of ${X}$ as in Chazal and Lieutier (2005, Definition 3.1).

Proof

The statement on the homemomorphism type of the complements is an immediate application of Hardt’s theorem for definable sets together with the fact that $\textrm{d}_{X}$ is definable (see e.g. van den Dries (1998)). One may then use the isotopies induced by flows used for example in Chazal and Lieutier (2005) to extend this homeomorphism to the case where $Y = {\mathbb {R}}^N$ and $\varepsilon$ is the weak feature size. To see that the latter is positive, note that the argument for positivity of weak feature sizes of semialgebraic sets in Fu (1985, Remark 5.3) also applies to the definable case. Finally, we need to see that the inclusion is a strong deformation retraction. Note that by the triangulability of definable sets (see for example (van den Dries 1998, Theorem 2.9)), ${\mathbb {R}}^N$ may be equipped with a triangulation compatible with X and Y. In particular, by subdividing if necessary, X has arbitrarily small map** cylinder neighborhoods in Y, given by piecewise linear regular neighborhoods. Furthermore, this means that $X \hookrightarrow X_{\alpha } \cap Y$ is a cofibration. Thus, it suffices to show that $X \hookrightarrow X_{\alpha } \cap Y$ is a homotopy equivalence. Now, for $\alpha< \alpha ' < \varepsilon$, with $\varepsilon$ such that 2 holds. Then, we have inclusions

$$\begin{aligned} X \hookrightarrow X_{\alpha } \cap Y \hookrightarrow N \hookrightarrow X_{\alpha '} \cap Y, \end{aligned}$$

where N and $N'$ are regular neighborhoods with respect to the piecewise linear structure induced by the triangulation. By the open cylinder structure (assumption 2) of the set $(X_{\alpha '} \cap Y) \setminus X$, the inclusion $X_{\alpha } \cap Y \hookrightarrow X_{\alpha '} \cap Y$ is a homotopy equivalence. The same holds for the inclusion $X \hookrightarrow N$. It follows by the two-out-of-six property of homotopy equivalences, that all maps are homotopy equivalences. $\square$

1.2 B.2 Proof of Proposition 3.41

Proof of Proposition 3.41

The map $\beta$ is clearly continuous on ${S}_q \times {S}_p$. The condition on $\beta$ is thus equivalent to the extension by 0 to $\Delta _{{S}_p}$ being continuous. Indeed, by continuity of $\textbf{d}(-,-)$, this extension condition immediately implies condition (b). For the converse, as $\beta \ge 0$, it suffices to show upper semi-continuity. This is the content of Proposition B.2.

Proposition B.2

Let $W= (X, s: X \rightarrow P)$ be a Whitney stratified space. Then, the restriction of $\beta$ to $W_{\ge p} \times W_{p} \rightarrow {\mathbb {R}}$ is upper semi-continuous.

Proof

$\beta$ is clearly continuous on the strata of $W\times W$. Now, suppose $(x_n, y_n) \in W_{\ge p} \times W_{p}$ is a sequence converging to a point $(x,y) \in W_{p'} \times W_{p}$, for some $p' \ge p$. Then, for sufficiently large $n \in {\mathbb {N}}$, we have $s(x_n) \ge p'$. To show upper semi-continuity, we may thus without loss of generality assume that $x_n$ lies in the same stratum $W_{q}$. We show that any subsequence of $(x_n,y_n)$ has a further subsequence (all named the same by abuse of notation), for which $\beta (x_n,y_n)$ converges to a value lesser or equal then $\beta (x,y)$. By compactness of Grassmannians, we may first restrict to a subsequence such that $\textrm{T}_{x_n}(W_q)$ and $l(x_n,y_n)$ converge to $\tau$ and l respectively. By Whitney’s condition (a) Whitney (1965a, 1965b)—which by Mather (2012) follows from condition (b)—we have $\textrm{T}_{x}(W_{p'}) \subset \tau$. Summarizing, this gives:

$$\begin{aligned} \lim \beta (x_n,y_n) = \textbf{d}(l,\tau )&\le \textbf{d}(l,\textrm{T}_{x}(W_{p'})). \end{aligned}$$

Now, in case when $x \ne y$, the last expression equals $\beta (x,y)$ by definition. In the case when $x=y$ then, by condition (b), $l \subset \tau$. Thus, again, we have

$$\begin{aligned} \lim \beta (x_n,y_n) = \textbf{d}(l,\tau ) = 0 = \beta (y,y) \end{aligned}$$

finishing the proof. $\square$

1.3 B.3 A normal bundle version of $\beta$

Furthermore, we are going to make use of the following fiberwise version of $\beta$.

Construction B.3

Again, in the framework of Construction 3.40, assume that $W= ({X}, s:{X}\rightarrow P)$ is a Whitney stratified space, with $W_p$ compact. Take N to be a standard tubular neighborhood of $W_p$ in ${\mathbb {R}}^N$ with retraction $r: N \rightarrow W_p$. Note that by Whitney’s condition (a), for N sufficiently small, $r_{\mid W_q}$ is a submersion for $q \ge p$. In particular, by Nocera and Volpe (2021, Lemma 2.1) the fiber of

$$\begin{aligned} W^{y}:=(r)_{\mid N \cap W^{\ge p}}^{-1}(y) \end{aligned}$$

is a Whitney stratified space over $\{q \in P\mid q \ge p \}$ with the p-stratum given by $\{y\}$. Furthermore, we have

$$\begin{aligned} \textrm{T}_{x}(W_q) \cap \nu _{r(x)}(W_p) = \textrm{T}_{x}(W^{r(x)}_q),\end{aligned}$$

where $\nu _{r(x)}(W_p)$ denotes the normal space of $W_p$ at r(x). In particular, the dimension of these spaces is constant, and they vary continuously in x. Then, consider the following function:

$$\begin{aligned} {{\tilde{\beta }}}_{p}(-): N \cap W^{\ge p} \rightarrow {\mathbb {R}}{} & {} {\left\{ \begin{array}{ll} x &{}\mapsto \textbf{d}(l(x,r(x)),\textrm{T}_{x}(W^{r(x)}_{s(x)})) \text {, for } s(x) >p\\ x &{} \mapsto 0 \text {, for } s(x) = p. \end{array}\right. } \end{aligned}$$

Noting that $l(x,r(x)) \in \nu _{r(x)}(W_{p})$, by an analogous argument to the proof of Proposition 3.41, one obtains that ${{\tilde{\beta }}}_{p}(-)$ is continuous on $W_{q} \cup W_{p}$. Note that if we restrict ${{\tilde{\beta }}}_{p}(-)$ to $W^{y}$, then we obtain the function $\beta (-,y)$ associated to $W^{y}$. Let us denote this $\beta _{y}$. In particular, by compactness of $W_{p}$, we obtain that the functions $\beta _{y}$ can be globally bounded by any $\delta >0$, for N sufficiently small.

1.4 B.4 Definability of $\beta$

Proposition B.4

Let ${S}= ({X}, s:{X}\rightarrow P)$ be as in Construction 3.40. Then, if ${X}\subset {\mathbb {R}}^N$ is definable, then so is $\beta$.

Proof

As all the strata of ${X}\times {X}$ are again definable, it suffices to show that $\beta$ is definable on the strata of ${X}\times {X}$. Furthermore, as $\beta$ is 0 along $\Delta _{{X}}$, it suffices to show definability away from the diagonal. Here $\beta$ is equivalently given by

$$\begin{aligned} \beta (x,y) = \inf _{v \in \textrm{T}_{x}({X}_{s(x)})}{ \vert \vert \frac{x-y}{\vert \vert x-y \vert \vert } -v \vert \vert }. \end{aligned}$$

It follows from the fact that for $q \in P$, $\textrm{T}_{}({X}_q) \subset {\mathbb {R}}^N\times {\mathbb {R}}^N$ is definable (see Coste (2000) and Lemma B.5) that this defines a definable function ${X}_q \times {X}_p \rightarrow \mathbb R$. $\square$

1.5 B.5 Proof of Proposition 4.20

We begin by proving a series of technical lemmas.

Lemma B.5

Consider two definable maps $f: X \rightarrow {\mathbb {R}}$, $\pi : X \rightarrow Y$ such that f is bounded from above on every fiber of $\pi$. Then the map

$$\begin{aligned} g: Y&\rightarrow {\mathbb {R}} \\ y&\mapsto \sup _{x \in \pi ^{-1}(y)}f(x) \end{aligned}$$

is again definable.

Proof

This is immediate, if one interprets the graph of g in terms of a formula being expressible with respect to the o-minimal structure. $\square$

Lemma B.6

Let $X \rightarrow \{ p < q\}$ be a stratified metric space and Y a first countable, locally compact Hausdorff space. Let $\pi : X \rightarrow Y$ be a proper map, such that both the fibers of $\pi$, as well as the fibers of $\pi _{\mid X_{p}}$ vary continuously in the Hausdorff distance. Let $f:X \rightarrow {\mathbb {R}}$ be upper semi-continuous and continuous on the strata. Then,

$$\begin{aligned} g: Y&\rightarrow {\mathbb {R}} \\ y&\mapsto \sup _{x \in \pi ^{-1}(y)}f(x) \end{aligned}$$

is continuous.

Proof

Note first that as the fibers of $\pi$ are compact and f is upper semi continuous, it takes its maximum on every fiber. Now, let $y_n \rightarrow y$ be a convergent sequence in Y. We show that any of its subsequences $y'_n$, has a further subsequence ${\tilde{y}}_n \rightarrow y$, with

$$\begin{aligned} \sup _{x \in \pi ^{-1}({{\tilde{y}}}_n)}f(x) \rightarrow \sup _{x \in \pi ^{-1}(y)}f(x).\end{aligned}$$

Let $x'_n \in \pi ^{-1}(y_n)$ for all n such that $f(x'_n) = \sup _{x \in \pi ^{-1}(y'_n)}f(x)$. As Y is locally compact and $\pi$ is proper, $x'_n$ has a convergent subsequence ${{\tilde{x}}}_n \rightarrow {{\tilde{x}}}$. Define ${{\tilde{y}}}_n:=\pi ({{\tilde{x}}}_n)$. Since the fibers of $\pi$ vary continuously and ${{\tilde{y}}}_n \rightarrow y$, we also have ${{\tilde{x}}} \in \pi ^{-1}(y)$. Thus, we have

$$\begin{aligned} \limsup \sup _{x \in \pi ^{-1}({{\tilde{y}}}_n)}f(x) = \limsup f( {{\tilde{x}}}_n) \le f({{\tilde{x}}}) \le g(y). \end{aligned}$$

It remains to see the converse inequality for a subsequence of ${{\tilde{y}}}_n$. Let ${\hat{x}} \in \pi ^{-1}(y)$ be such that $f({\hat{x}}) = \sup _{x \in \pi ^{-1}(y)}f(x)$. By assumption we can find a sequence $x''_n$ with $x''_n \in \pi ^{-1}({{\tilde{y}}}_n)$ converging to ${\hat{x}}$. If ${\hat{x}} \in X_p$, then $x''_n$ can be taken to be in $X_p$, as $\pi ^{-1}({{\tilde{y}}}_n) \cap X_p$ converges to $\pi ^{-1}(y) \cap X_p$. If ${\hat{x}} \in X_q$, then, as the latter is open, $x''_n$ ultimately lies in $X_q$. Hence, by continuity of f on the strata, we have

$$\begin{aligned} g(y) = f({\hat{x}}) = \lim f(x''_n) = \liminf {f(x''_n)} \le \liminf \sup _{x \in \pi ^{-1}({{\tilde{y}}}_n)}f(x). \end{aligned}$$

$\square$

As a consequence of the prior two lemmas we obtain:

Lemma B.7

If $W$ is a definably Whitney stratified over $P=\{ p <q\}$. Then the map

$$\begin{aligned} {\hat{\beta }}: W_p \times {\mathbb {R}}_{\ge 0}&\rightarrow {\mathbb {R}} \\ (y,d)&\mapsto \sup _{\vert \vert x-y \vert \vert =d, x\in W}\beta (x,y) \end{aligned}$$

is continuous in a neighborhood of $W_p \times \{0\}$, definable and vanishes on $W_p \times \{0\}.$

Proof

Definability follows immediately from Lemma B.5. consider the map

$$\begin{aligned} B: W\times W_p \rightarrow W_p \times {\mathbb {R}}_{\ge 0} (x,y) \mapsto (y, \vert \vert x-y \vert \vert ). \end{aligned}$$

Over $W_p \times {\mathbb {R}}_{>0}$ it is given by submersion on each stratum of $W\times W_p$. In particular, by Thom’s first isotopy lemma (Mather 2012, Proposition 11.1) it is a fiber bundle with fibers $\partial \textrm{B}_{d}(y)$ at (y, d) over ${\mathbb {R}}_{>0}$. In particular, the fibers of B vary continuously over $W_p \times {\mathbb {R}}_{>0}$. Additionally, for $(y_n, d_n) \rightarrow (y,0)$ the fiber converges to the point y. Hence, B fulfills the requirements of Lemma B.6. Furthermore, $\beta : W\times W_p \rightarrow {\mathbb {R}}$ also fulfills the requirements of Lemma B.6, showing the continuity of ${\hat{\beta }}$. Lastly, ${\hat{\beta }}$ vanishes on $W_p \times {\mathbb {R}}_{\le 0}$ by definition of $\beta$. $\square$

We now have all the tools available to obtain a proof of Proposition 4.20,

Proof of Proposition 4.20

We conduct this proof for the case of $P= \{ p < q \}$ and $K = W_p$ (with notation as in Definition 4.18). The general case follows analogously by working strata-wise and then passing to maxima. By Lemma B.7 for d small enough, the function ${\hat{\beta }}: W_p \times {\mathbb {R}}_{\ge 0} \rightarrow {\mathbb {R}}$ fulfills the requirements of Lojasiewicz’ theorem for (polynomially bounded) o-minimal structures (Loi 2016). Hence, we find $\hat{\phi }: {\mathbb {R}}_{\ge 0} \rightarrow {\mathbb {R}}_{\ge 0}$ to be a definable and monotonous bijection such that on $W_p \times [0,d]$ we have

$$\begin{aligned}{\hat{\phi }}( {\hat{\beta }}(y,t) ) \le t. \end{aligned}$$

If the relevant o-minimal structure is polynomially bounded, then there exist $n >0$, such that

$$\begin{aligned} t^{n} \le {\hat{\phi }}(t) \end{aligned}$$

for $t \in [0,d']$. Hence, we obtain

$$\begin{aligned} {\hat{\beta }}(y,t)^n&\le {\hat{\phi }}( {\hat{\beta }}(y,t) ) \le t. \\ \implies {\hat{\beta }}(y,t)&\le t^{\alpha } \end{aligned}$$

for $t \in [0,d]$, $\alpha = \frac{1}{n}$ and $d:=\phi ^{-1}(d')$. $\square$

1.6 B.6 Proof of Lemma 4.2

Proof of Lemma 4.2

The first result is immediate from the local conical structure of ${X}$. The second is immediate from the definition of a homology stratification, as clearly ${X}- {S}_p$ is a homology manifold. For the final result, note first that by the local conical structure, having local homology isomorphic to $\text {H}_{\bullet }({\mathbb {R}}^q;0)$ is an open condition on ${S}_p$. In particular, since this condition holds on all of ${X}- {S}_p$ it is an open condition on all of ${X}$. Thus, $s: {X}\rightarrow \{ p < q \}$ as defined in the statement is actually a stratification of ${X}$. To see that this is indeed a homology stratification we need to see that the local isomorphism condition is fulfilled. By construction, we have ${X}- {S}_p \subset {\tilde{s}}^{-1}\{q\}$. Within ${S}_p \cap {\tilde{s}}^{-1}\{p\}$ the local isomorphism condition again holds by the local conical structure of ${X}$. Thus, it remains to consider the case where $x \in {S}_p$, and $\textrm{H}_{\bullet }({X};x) \cong \text {H}_{\bullet }({\mathbb {R}}^q;0)$. We need to show that, for $U_x \cong {\mathbb {R}}^{q-p-1} \times \mathring{C}(L_{x})$, an open neighborhood of x, the natural map

$$\begin{aligned} \textrm{H}_{\bullet }({X};x) \cong \textrm{H}_{\bullet }(W, W-U_x) \rightarrow \textrm{H}_{\bullet }({X};y) \end{aligned}$$

is an isomorphism, for all $y \in U_x$. The only nontrivial degree in this case is $q=\dim {W}$. By an application of the Künneth formula $L_x$ is again an orientable manifold. Hence, up to suspension, from this perspective, the claim reduces to the fact that if $L_x$ is an orientable, closed manifold. Then, under the natural isomorphism

$$\begin{aligned} \textrm{H}_{\bullet }(CL_x, L_x) \cong \tilde{\textrm{H}}_{\bullet -1}(L_x) \end{aligned}$$

the fundamental class of $L_x$ induces a fundamental class of $CL_x$. $\square$