1. Introduction
The (seemingly) simple question of how to organize the fibers of a map has led to significant advancement in mathematics and related disciplines. Recently, this blossoming has been most apparent in topological data analysis (TDA). Indeed, the case where and is simply a function already motivates persistence theory and the use of tools such as Reeb graphs (the higher dimensional analogues of which are called Reeb spaces).
In the present, we connect with the work of Edelsbrunner, Harer, and Patel [14]
on Reeb spaces for a piecewise linear (PL) vector valued function (we refer to a map with the specific hypotheses imposed as an EHP map). In particular, we develop simple tools that allow us to refine the stratification of the Reeb space previously given. In contrast to
ibid, our stratification depends entirely on the Jacobi set, and we give an explicit description of how this stratification also extends to both the domain and codomain of the piecewise linear function in question.Let be a EHP map and let denote the corresponding Reeb space.

There is a stratification of such that each stratum is connected; and

With respect to this filtration the Stein factorization is a map of stratified spaces.
Morally, the stratification of is a refinement of that constructed in [14]. This is not literally true as we give a slightly different definition of critical simplices. We make an explicit comparison between resulting Jacobi sets in Appendix A. As the stratification of and are obtained by extending stratifications of Jacobi sets to the ambient space, our stratification can differ slightly from that of Edelsbrunner, Harer, and Patel.
A convenient consequence of the generality of our constructions is that they apply in a variety of settings beyond the EHP setup. In particular, we obtain wellbehaved stratifications on the codomain of higher Morse functions. For Morse 2functions we prove the following.
Let be a Morse 2function. There is a stratification of such that the topology of preimages is locally constant on top/open strata.
This note is but the first of several works on applying stratified and homotopical techniques to the theory and application of persistence modules. Indeed, the perspective of persistence modules as (constructible) cosheaves on stratified spaces, [9, 10], drives this work. One motivator for the present work is the observation that subposets of the stratifying poset of a space are equivalent to multiparameter filtrations of the space. Recent work in multiparameter persistence often advocates for analyzing such filtrations by focusing on onedimensional ‘slices’ through the parameter space (e.g., [18, 17]), and these slices are equivalent to linear subposets of the stratifying poset.
To wit, one consequence of Theorem B is that to any Morse 2function, there is an associated persistence module (over ). This construction explicitly extends the case of an ordinary Morse function , which is a classical ingress to persistence theory. This example and related higher dimensional analogues will be explained in detail in forthcoming work.
It is our hope that by bringing the tools of higher () categories and homotopical algebra to the setting of persistence theory and TDA, new discriminating invariants and computational techniques will grow.
Outline
In stratspaces, we present the necessary background on stratified spaces. In particular, we introduce the (connected) ambient stratification and prove several fundamental properties. reeb and higher contain our main theorems, related to Reeb spaces and higher Morse functions, respectively. Appendix A discusses three different notions of criticality for functions on combinatorial manifolds including the one used in the bulk of the paper and that from [14].
Acknowledgements
AS is supported by the National Science Foundation under NIH/NSF DMS 1664858. The authors thank David Ayala for discussion and shared insight.
2. Stratified Spaces
Stratified spaces have been used throughout topology for over 60 years with major foundational contributions due to Thom, Mather, Goresky and MacPherson. The paradigm we use is developed extensively in the work of Ayala, Francis, and Tanaka [2] and first appeared in Appendix A of Lurie [19].
2.1. Stratified Recollections
2.1.1. Poset Topologies and Operations
A poset is a pair consisting of

a set , and

a relation on
such that defines a partial order on . I.e., for all , the following hold

(reflexivity) .

(transitivity) if and , then .

(antisymmetry) if and , then .
The upward closed topology defines a functor, .
Note that for , the topology is rarely Hausdorff. Two points in are only contained in disjoint open sets when they are not comparable in the poset structure. Then if is Hausdorff, it is equivalent to the discrete topology, which is certainly not true for general .
Let and be posets. We can explicitly describe the product poset as follows. The underlying set is , and we declare if and .
This construction is categorical.
The category has (finite) products.
Let be the standard total order on . Then admits three partial orders

Lexicographic order: if and only if , or and ;

Product order: if and only if and ;

Closure of direct product: if and only if and , or and .
Only the first partial order is a total order. As the name indicates, the product order is the categorical product.
Let be a poset. The left cone on is a new poset, , that is defined by adjoining a new minimum element to . More rigorously,
Similarly, we can define the right cone on ,
which adjoins a new maximum to .
Note that , , and more generally, . Let , then we have isomorphisms of posets
2.1.2. The category of stratified spaces
A stratified topological space is a triple consisting of

a paracompact, Hausdorff topological space, ,

a poset , equipped with the upward closed topology, and

a continuous map .
Given a stratified topological space , and any , the stratum, , is defined as
Any paracompact, Hausdorff space defines a stratified topological space via . We call this the trivial stratification.
is stratified over , where , , and are mapped to , , and , respectively. We often find it convenient to indicate a stratified space pictorially, by drawing the strata. For example, below is a depiction of the stratified space :
A map of stratified topological spaces to is a pair of continuous maps making the following diagram commute.
A continuous map is an open embedding if both and are open (topological) embeddings.
We will let denote the category of stratified topological spaces with morphisms given by continuous maps. The category has many nice properties, e.g., it has finite products.
A refinement of a stratified space is a stratified space and a map of stratified spaces such that is the identity, and is a continuous surjection; diagrammatically we have the following.
2.2. Native stratifications
It turns out that certain classes of topological spaces have canonical stratifications, we refer to such stratifications as native. For concreteness, we focus our attention to simplicial complexes.
Let be a simplicial complex. The native stratification of is defined by , where is the poset such that, for simplices , we have the relation if and only if is a face of . Suppose that is the lowest dimensional simplex containing . Then .
Let be a simplicial complex consisting of a single edge endowed with the subspace topology. Define the map , where for and for . Then the triple does not define a stratification of , since any open neighborhood disjoint from has a preimage that is not open (i.e., is not continuous).
notstrat is turned into an example if we instead consider the map , and we map to the maximum element of .
We note a couple key properties of the native stratification.
Let be a simplicial complex. Then is a refinement of the coarsest stratification of such that each stratum is a connected manifold.
Let be a simplicial complex. There is a map of posets such that the composite is the skeletal filtration.
The following is immediate from Proposition 2.1.1 and the definition of simplicial map.
Let be a simplicial map. Then there is a map of posets that extends to a functor .
Note that the native stratification makes sense in other categories: complexes, CWcomplexes, simplicial objects, etc. One notable caution is that composition with a “dimension” map need not result in the corresponding skeletal filtration, but rather only a collapsed filtration e.g., consider the CW structure on consisting of one 0cell and one 2cell.
2.3. Ambient stratifications
We now consider how an embedded stratified space determines a stratification of the ambient space. The example of a simplicial complex—equipped with its native stratification—embedded in some Euclidean space is an illustrative example to track throughout.
Suppose that is a stratified space and is a topological embedding into the paracompact, Hausdorff space , e.g., . (We will conflate with its image in .) To extend the stratification of to a stratification of compatible with the usual topology on , we can append a final element to the poset, , and map the complement, , to this element. This is exactly the result of taking the right cone on . We will denote the resulting map .
Note that we often want the strata to be connected. In this case we introduce a new stratifying poset , which is a refinement of .
Let be a stratified space, a proper topological embedding, and . Define the poset, , as the set , subject to the following generating relations:

The relations of ;

For and , if and only if , i.e., the stratum is in the closure of the connected component indexed by .
There is an obvious extension of the map , and we call this stratification the connected ambient stratification.
Let be a stratified space, a proper topological embedding, The maps and define stratified spaces that restrict to the stratification on . Moreover, by collapsing the maximal elements in the poset, the connected ambient stratification refines the conical extension of the stratification:
Define a category as follows:

Objects: an object is a triple , where is a stratified space, is a paracompact, Hausdorff space, and is a proper (topological) embedding.

Morphisms: a morphism is a map such that

The image of under is contained in ;

The restriction is a stratified map; and

The preimage of is contained in , i.e., .

We will at times condense notation and simply denote an object by . Note that is well defined as the identity is clearly a morphism and composition is inherited from composition in spaces. The conditions on are equivalent to asking that the following is a pullback diagram
This rephrasing makes composition apparent as pasting of pullback squares.
The connected ambient stratification defines a functor . In particular, given a morphism , we have a commutative diagram (in spaces)
Proof.
Lemma 2.3 shows that we have a welldefined functor at the level of objects. As the map is continuous, it takes path components into path components, so we obtain a map at the level of morphisms. Functoriality follows in a straightforward manner from composition in . ∎
2.4. Combinatorial Manifolds
Let us briefly recall some combinatorial and piecewiselinear (PL) topology. A standard reference is [22] or Section 3.9 of [23]. (In the latter reference, a combinatorial manifold is called a triangulated PL manifold.)
A combinatorial manifold, , is a PL manifold equipped with a compatible triangulation, i.e., a PL homeomorphism for some simplicial complex .
A PL manifold is a topological manifold equipped with an equivalence class of triangulations, while the combinatorial structure picks out a particular triangulation. It is no surprise that any PL manifold admits a structure of a combinatorial manifold, though the condition is still stronger than being a triangulated topological manifold. A classical theorem of Whitehead is that any smooth manifold admits a unique PL structure, so a triangulated smooth manifold has a unique structure as a combinatorial manifold.
The standard PL disk is given by the cube and its boundary is the standard PL sphere. In actuality, any convex polyhedron of dimension presents the standard PL disk as they are all PL homeomorphic and similarly for PL spheres.
Let and be topological spaces. The join of and , denoted , is given by , where
Morally, the join of spaces is obtained by adjoining all line segments connecting the two spaces (this is literally true for polyhedral spaces embedded in Euclidean space). The join of simplicial complexes is again a simplicial complex, the join of PL manifolds is again PL, and the join of combinatorial manifolds is again a combinatorial manifold. Let be a simplicial complex.

For a simplex, the star of , is the set of simplices in that contain as a face.

For a simplex, the link of , , is the closure of the star of set minus the union of stars of the faces of , i.e., .

For a simplex and , the link of , , is given by
A simplicial complex is a PL manifold if and only if the link of each simplex is PL homeomorphic to a PL sphere.
For , let be a PL disk, a PL sphere, and any PL manifold. Assuming the factors are disjoint, we have the following PL homeomorphisms:
3. Reeb Spaces
Reeb graphs have long been an object of interest, and, roughly, are graphic summaries of level sets of real valued functions.The Reeb graph is formed by identifying points that lie in the same connected component of a level set. First introduced in [21] for use in Morse theory, they have since been widely used in applications (see [5] for a survey) and have been studied for their mathematical properties (e.g., [3, 11, 8]) as well as in terms of efficient computation (e.g., [13, 12] etc.) Reeb spaces generalize the notion of Reeb graphs to the case of an arbitrary map between spaces.
Let be a continuous map of spaces. The Reeb space is the quotient space defined by the relation: for , whenever and if and are in the same connected component of .
Let be the canonical quotient map. It is often useful to visualize the relationship between these spaces in diagramatic form:
[column sep=small] X [rr, ”f”] [dr, ”q”’] & & Y
& _f [ur, ”g”’] &
where is the Stein factorization of , the unique continuous map making the diagram commute.
Let be the Reeb space corresponding to the map . If is normal, then is Hausdorff.
Proof.
Since is normal, it suffices to show that is an open subset of . Let , meaning and do not lie in the same connected component of a level set of . Note that the level set componenets containing and , which we denote and , respectively, are disjoint, and as they are connected components, both are closed in . Since and are disjoint closed subsets of a normal space, we can find disjoint open sets containing them, which we denote and , respectively. Then is an open set containing . We claim that . Suppose, towards a contradiction, that . Then and would be in the same connected component of a level set of while simultaneously being in and , a contradiction. Since we have found an open neighborhood around an arbitrary point of , we know is closed in , and thus, is Hausdorff. ∎
As our future applications are based in the realm of computational geometry, we consider the following hypotheses, a subset of those in [14]:
[The EHP Hypotheses] Let be a triangulated PL manifold and be a function that, restricted to each simplex, is affine. We further require that is generic, so, in particular, no structural properties of the image of can be altered by arbitrarily small changes in the vertex set of .
Note that the condition of normality needed for the above lemma is met by , so the associated Reeb space is Hausdorff. Additionally, we observe that the condition of genericity of implies that restricted to some simplex for is injective, i.e., the image of this simplex is again an simplex.
Note that for a point in the interior of a simplex, , the link is naturally presented as a join of spheres where and . If is the triangulation of a smooth manifold, then the first factor in the join can be identified with the unit sphere in tangent bundle to at , while the second factor is the unit normal sphere at . This observation motivates the following definitions.
Let satisfy , and suppose that is dimensional.

A point is regular if there exists a PL homeomorphism such that
respects the join decomposition and is a PL homeomorphism of spheres on the second factor.

The Jacobi set of is the collection of critical points, i.e., those points that are not regular. The Jacobi set of the map is denoted .
In the case that is a Morse function on a smooth manifold, is exactly the critical points of .
Let be an icosohedron in , and let be the standard height function for a direction that satisfies the EHP hypothesis (namely, this means all vertices of have a unique height with respect to ). This is illustrated in icos. We see that
is the join of two copies of , and the image is indeed a PL homeomorphism onto the second factor. Thus, we see that is regular. Instead, consider . Although this link is also the join of two copies of , since is maximal, the image of . Thus, we see that is critical.
Let satisfy . The Jacobi set is an embedded simplicial complex in of dimension at most .
Proof.
As the function is affine restricted to any given simplex, , if is critical than for any , is also critical, i.e., criticality is constant on simplices. Also, by genericity of , if is a critical simplex, then .
To complete the proof it is sufficient (and necessary) to show that criticality is a closed condition or, equivalently, that regularity is an open condition. Let be a codimension one face, , and . We claim that if is regular, then is regular. Indeed, we have
There is a PL homeomorphism that is the identity on the last factor of the join and such that . Now if is regular via the decomposition , then the resulting decomposition of proves that is regular. By induction on codimension, if the simplex is critical, then any point in its closure is critical as well. ∎
Lemma 2.3 then determines two stratifications of : one determined by and a refinement thereof given by .
In [14] the authors define critical simplices using reduced homology. We discuss this definition (and another) of the Jacobi set in Appendix A.
3.1. The stratification of
Next, we will show that can determine similar stratifications of .
Since contains only and lower dimensional simplices and since is generic, we see that takes simplices to simplices, hence is a collection of simplices in . Even so, there is no guarantee that these simplices will not intersect in , which precludes us from immediately endowing with a simplicial structure. Nonetheless, we have the following.
If is an EHP map, then admits a refinement into simplices so that, for where , whenever a point and , we have as subsets of .
The proof of this proposition is not exceedingly difficult, but is a bit tedious so we have relegated it to Appendix B. In what follows, we may abuse notation and use to mean taken with this refinement. Then, we can define the following.
Let be an EHP map, and suppose that has been refined into simplices as described in Jsimp. We define the poset , where, for , we have if and only if as subsets of , and stratify via the map , where, if is the lowest dimensional simplex containing , then .
Given the stratification as above, we can stratify via via connectedstrat, so that each stratum is a connected manifold.
3.2. The stratification of
Since is a quotient of , we would like to use the universal property to stratify . However, the stratifying map is not constant on the fibers of the quotient map . (Nor does the map —in general—satisfy the conditions under which is functorial.)
However, we have the composition of continuous maps
which equips with a stratification; we will denote the stratifying map by . Our stratification of is exactly built so that it’s obvious that the Stein factorization is a stratified map.
The Stein factorization is a map of stratified spaces, i.e., the following commutes in the category of topological spaces
If If the EHP map is injective on the skeleton of , then and are stratified maps.
Proof.
If is injective on the skeleton of , then none of the images of simplices of will intersect transversely (i.e., has a simplicial complex structure), so . Furthermore, we have meaning that restricting and to elements of defines surjections onto . ∎
3.3. Comparing to the EHP stratification
A main result of [14] is defining the so called “canonical stratification” of Reeb spaces, which we will refer hereafter as the EHP stratification. The EHP stratification is described using an algorithmic construction. First, is triangulated by coarsening a decomposition of into prisms created by preimages of the affine hulls of images of simplices. This triangulation is inherited by , and the EHP stratification of is the skeletal stratification of this triangulation. As noted in [14] a precise version of this algorithm is limited to , since a Boolean subroutine used to determine when two triangulated spaces of dimension are homeomorphic is undecidable for .
Although not explicitly considered in [14], the triangulation (and implied skeletal filtration) of and the EHP stratification of induce a stratification of that causes the Stein factorization to be a map of stratified spaces (similar to steinstrat). One of the more obvious differences between the EHP stratification and our stratification in reebstrat is that, by construction, each stratum of is connected, whereas this is not true in general for a stratum of the EHP stratification.
4. Stratifying Targets of (Higher) Morse Functions
In this section, we extend the methods of reeb to stratify the target of more general Morse functions. In the present section we do not assume . Our main results are in targetstrat and focus specifically Morse 2functions, but we additionally discuss how Morse nfunctions could be treated similarly in generalizations. To build up to higher Morse functions, we start in the warmup setting of Morse 1functions.
4.1. Morse 1Functions
Consider the upright height function on the torus (upright).
There are four critical points and we will stratify via the stable manifolds with respect to gradient flow of the function (so we flow down), i.e., for a critical point , the corresponding stratum, , is given by
where is the gradient flow of .
Let be a Morse function on a compact smooth manifold. The stable stratification of is determined by the stratifying map , where is the poset of critical values with and are related if there is a flow line from to .
The following is just a rephrasing of standard results in Morse theory, see [20].
The stable stratification is well defined, i.e., the triple defines a stratified space. Moreover, this stable stratification is Euclidean, i.e., each stratum is diffeomorphic to for some .
One should note that this stratification is rather wild. In particular, it is often desirable that an intersection of the closure of strata is contained in a stratum of lower dimension; this is violated in the upright torus example as the intersection of the closures of the strata corresponding to the critical points of index 1 is the just the upper index 1 critical point. The situation can be remedied by restricting to the case that the map is Morse–Smale.
A Morse function is Morse–Smale if for each pair of critical points such that , the the unstable manifold of intersects the stable manifold of transversely.
An illustrative example of a MorseSmale function is the height function on the tilted torus (tilted).
[Lurie A.5.5] Let be a stratified space and a point. The space is conically stratified at if there exists an open neighborhood, , of and a stratified homeomorphism where is a topological space and is space stratified by .
Note that any stratified space of [2] is conical.
Let be a closed smooth manifold and a Morse–Smale function. The stable stratification of is conical.
Proof.
Indeed this statement is Proposition 2 of Laudenbach [6]. The key idea is that when is Morse–Smale and the gradient vector field admits a normal form, then the closure of is a submanifold with at worst conical singularities. The “normal form” (or “Special Morse” in ibid.) is that the vector field is the gradient of with respect to local coordinates as in the Morse Lemma. This condition is satisfied by construction in the current setting. ∎
In the case that is MorseSmale, one can prove moreover that the stable stratification is conically smooth. Conical smoothness is a desirable condition that has significant implications for the behavior of links and the exit/entrance path category [2, 19, 10]. In the case , there is a canonical identification of a link with the corresponding moduli space of gradient flows, see [7, 1].
4.2. Morse 2functions
Recently Morse 2functions have been a critical tool in low dimensional topology, especially in dimension 4. The work of Gay and Kirby is of particular note and [15] is a good introduction to the relevant ideas.
A smooth proper function is a Morse 2function if it is locally a generic Morse homotopy. Equivalently, is smooth whose only singularities are folds and cusps.
That is locally a generic Morse homotopy means that each point admits a neighborhood, , such that for a smooth manifold of dimension . Further, restricted to , where is a 1parameter family of Morse functions. Finally, the family is generic, so the only permitted singularities occur at the transverse crossing of critical values or at birth/death singularities. If is compact, then there are only finitely many crossings and birth/death singularities in the singular locus . We illustrate these phenomena in the example below, see also Section 2 of [15].
Suppose that we have a family of diffeomorphisms such as the one indicated at the bottom of morse2. Then, for every , let be the standard height function. Since is smooth, we can define a Morse2 function where . See morse2 for an illustration of this example.
It follows from general Cerf/Stability Theory that Morse 2functions are generic in the space of smooth maps [16]. Moreover, following Section 2 of [4], we can assume our Morse 2functions are compatible with coordinate projections of .
In the space of Morse 2functions, , those functions, , that satisfy

[label=()]

The composite , is Morse; and

The composite , restricted to the fibers of is Morse;
constitue an open, dense subspace.
[s2 continued] Consider the map where records the height of point under , a diffeomorphism of as in s2. First, we observe that that the composite is simply the identity function on , and is therefore Morse. Next, we observe that the composite restricted to (some fiber of ) is equivalent to the height function on , and is therefore also Morse. Thus, satisfies both conditions of cerf.
[A Morse Homotopy NonExample] Consider the map where records the height of under a diffeomorphism that rotates in such a way that is a ‘horizontal torus’ (like pushing the upright torus in upright back to lay flat) for some . Then does not satisfy the conditions of cerf, since restricted to the fiber has a degenerate critical point (both the maximum and minimum of are entire circles rather than single points), and is not Morse. As our primary interest moving forward is the stratification of the target of a Morse 2function, we defer stratifying the domain to the following two (open) tasks. Let be a Morse 2function.

Describe as a stratified manifold such that stratification is obtained as a refinement of that determined by the horizontal Morse function , e.g., by using restricted to the fibers of the horizontal function;

Define an analog of the MorseSmale condition for Morse 2functions such that the stratification of is conical/conically smooth.
4.2.1. Remark on gradient flow
A key tool in Morse theory is that of gradient like vector fields. Of course for a vector valued function with target at least two dimensional, , the analogue of the gradient, the Jacobian, does not define a vector field. However, by choosing a horizontal distribution we can still define parallel transport along paths. Our presentation is a brief summary of Sections 4.1 and 4.2 of [4].
Let be smooth. A connection for is a subset that is a pointwise orthogonal complement to with respect to some metric on . (This is a generalization of an Ehresmann connection in the case that is a fiber bundle.) Given a path , we would like obtain a lift such that for all . Moreover, we want the lift to be unique in order to define parallel transport, monodromy, etc. There are some technical difficulties that arise in this most general setting (significantly parallel transport may only be partially defined), but for paths of regular values we obtain familiar results.
[[4]] Let be smooth, a connection for and a path such that for each is a regular value of . Then,

There is a well defined parallel transport map along , ;

The map is a diffeomorphism;

The map is independent, up to isotopy, of the choice of and depends only on the homotopy class of relative to its endpoints within the connected component of , where is the singular locus of .
4.3. The stratification of the target
Given a Morse function , there is an associated persistence module/cosheaf over stratified by critical values, see [10]. We now describe the target stratification determined by a Morse 2function . (As noted above, we defer the full discussion of the associated persistence module to a follow up paper.) The idea is quite simple: if is the singular locus of , we equip it with a simplicial structure and we consider the stratification determined by . The difficulty is in showing that this process is welldefined and unique up to equivalence.
4.3.1. Simplicial Structure
Let be the singular locus of a Morse 2function . We describe a recipe for replacing by an embedded simplicial complex, , of dimension 1. The output is such that there is a well defined native stratification . We build up in the following way;

0simplex at each crossing;

0simplex at each birthdeath;

0simplex at each vertical tangency;

appropriate 1simplices using segments of (parametrized by arc length).
[Uniqueness] The stratification is the coarsest stratification of such that points of vertical tangency are strata and such that each stratum is a connected manifold.
4.3.2. The stratification
For a Morse 2function, equip with the stratification. This stratification is quite wellbehaved as the following proposition indicates.
Let be a Morse 2function. The topology of preimages only changes when crossing the singular locus .
Proof.
Let be in the same connected component. Since each stratum is a connected manifold, there is a (smooth) path with and . By construction, is a regular value for all . Let be a connection for (the space of such is nonempty), and the associated parallel transport operator. Proposition 4.2.1 (b) implies that is a diffeomorphism. ∎
4.4. Generalizations
In [15], Morse 2functions can have as codomain any (oriented) surface . The key idea is that any point admits a neighborhood diffeomorphic to such that restricted to the preimage of this neighborhood the function is a generic Morse homotopy. Note that at this generality there is no global notion of “index,” but rather only locally defined indices for the Morse homotopy.
One could give a definition of Morse function for higher by iterating the definitions of Gay and Kirby [15] or by simply fixing singularity types as in [16]. Again stratifying the domain of such a function is subtle, but the technique described above readily extends to stratify the target of any (reasonable) Morse function.
Appendix A The Jacobi Set and Critical Simplices
Let satisfy and the dimension of be . There are (at least) three definitions for when a simplex of dimension is critical. (From before, this dimension is the critical dimension.) To begin, let us recall the definition from above; in this appendix only, we will call this notion of critical Lcritical. (Also, recall that being regular is an open condition and constant on simplices.)
A dimensional simplex is Lcritical if it is not regular, where

: The point is regular if there exists a PL homeomorphism such that
respects the join decomposition and is a PL homeomorphism of spheres on the second factor; or

: The simplex is regular if there exists a point in the interior of that is regular.
Under , a map has a welldefined derivative as explained in §3.10 of [23]. Inspired by smooth geometry one could make the following definition.
A dimensional simplex is Dcritical if

: The differential at the point , is not surjective; or

: There is a point for which is not surjective.
In [14], the authors give a homological definition of criticality. To begin, let be a unit vector and define a function as . Let be a simplex and a unit normal to , then determines upper and lower subcomplexes of the link .
The simplex is Hcritical if its upper link has a nonvanishing reduced Betti number (with respect to coefficients).
The Hcritical condition is actually symmetric in upper versus lower link as one can see by considering the unit vector .
All three of these notions of criticality are distinct. The Hcritical condition has the advantage that it can algorithmically be implemented and checked by machine. In the absence of PL homology spheres and acyclic complexes that are not collapsible, Hcriticality would agree with Lcriticality; of course, such spaces do exist even in low dimension, e.g., the Poincaré sphere in dimension 3 or Bing’s house in dimension 2.
The following illustrates a complex that has no Dcritical vertices, but a whole slew of Hcritical vertices. Triangulate by a large number of points. Consider the two sphere, . Now define a function on that sends the two cone points to and randomly assigns to the vertices on the (equatorial) ; extend this function linearly to obtain a PL map . At any vertex, the differential will be surjective, yet many of the equatorial points will be Hcritical as the upper link will be a disjoint union of intervals.
Appendix B Proof of Proposition 3.1
In this appendix, we prove Jsimp, which is a result of the following lemma.
Let be a generic PL map from a finite simplicial complex comprised entirely of simplices. Then is a disjoint set of dimensional convex regions, and for every , we have .
Proof.
Since is generic and PL, the image of any simplex of is again an simplex. Since only consists of simplices, is exactly the intersections of these simplices in (not including the shared faces of simplices that were also adjacent in ). Any nontransverse intersection would be removable by an arbitrary perturbation of some vertex of , violating the genericity of . Similarly, no triple of simplex images can intersect, since an arbitrary perturbation of a vertex of one of the simplices involved could change the intersection to three twoway intersections. Thus, intersections of images of simplices are pairwise transverse intersections, and therefore are convex with dimension . Observe that since intersections are only pairwise, we have . If the collection of such intersections were not disjoint, this would imply an intersection three simplex images, which is not possible by the reasoning above. ∎
Recall the following proposition;
Jsimp If is an EHP map, then admits a refinement into simplices so that, for where , whenever a point and , we have as subsets of .
This is a corollary of intersections. Since points of with more than one preimage form dimensional convex polytopes, we can triangulate these intersection regions so that is a collection of simplices. Although there are many ways to triangulate a convex polytope (and the method of triangulation does not effect our results), a simple explicit triangulation is to place a Steiner point at the barycenter of each dimensional face to startriangulate each dimensional face (whenever the face is not already an simplex), proceeding from .
Note that although this corollary does not generally imply is a simplicial complex, it does maintain the property that every simplex of this refinement is either disjoint from, or a subset/superset of every other simplex in the refinement, and thus is a natural candidate for stratification by containment (see cP).
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