The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, Gonz{á}lez, and Rudyak with the aim of constructing a cellular model of the configuration space of a sphere. Although the original aim was not achieved in the project, the notion of cellular stratified spaces turns out to be useful, at least, in the study of configuration spaces of graphs. In particular, the notion of totally normal cellular stratified spaces was used successfully in a joint work with the former students of the author arXiv:1312.7368 to study the homotopy type of configuration spaces of graphs with a small number of vertices. Roughly speaking, totally normal cellular stratified spaces correspond to acyclic categories in the same way regular cell complexes correspond to posets. In this paper, we extend this correspondence by replacing cells by stellar cells and acyclic categories by topological acyclic categories.
This is a sequel to [1106.3772], in which a systematic study of cellular stratified spaces and related concepts was initiated. In this paper, we study important operations on cellular and stellar stratified spaces, including taking subspaces, subdivisions, and products. We also introduce a procedure which extends the classical duality for simplicial complexes. The results in this paper will be used to construct "Salvetti-type" models for configuration spaces.
The notion of regular cell complexes plays a central role in topological combinatorics because of its close relationship with posets. A generalization, called totally normal cellular stratified spaces, was introduced by the third author by relaxing two conditions; face posets are replaced by acyclic categories and cells with incomplete boundaries are allowed. The aim of this article is to demonstrate the usefulness of totally normal cellular stratified spaces by constructing a combinatorial model for the configuration space of graphs. As an application, we obtain a simpler proof of Ghrist's th…
The notion of cellular stratified spaces was introduced in a joint work of the author with Basabe, González, and Rudyak [1009.1851] with the aim of constructing a cellular model of the configuration space of a sphere. In particular, it was shown that the classifying space (order complex) of the face poset of a totally normal regular cellular stratified space $X$ can be embedded in $X$ as a strong deformation retract. Here we elaborate on this idea and develop the theory of cellular stratified spaces. We introduce the notion of cylindrically normal cellular stratified spaces and associate a top…
We develop a theory of conically smooth stratified spaces and their smooth moduli, including a notion of classifying maps for tangential structures. We characterize continuous space-valued sheaves on these conically smooth stratified spaces in terms of tangential data, and we similarly characterize 1-excisive invariants of stratified spaces. These results are based on the existence of open handlebody decompositions for conically smooth stratified spaces, an inverse function theorem, a tubular neighborhood theorem, an isotopy extension theorem, and functorial resolutions of singularities to smo…
In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those model structures together, we construct a cofibrantly generated model structure on the category of all stratified spaces. In both model categories, weak-equivalences are characterized by stratified homotopy groups.
We correct some oversights in the paper "A spectral sequence for stratified spaces and configuration spaces of points" by the second named author. In particular we explain that an additional hypothesis should be added to Theorem 4.15 in said paper.
Let $P$ be a poset. We define a new homotopy theory of suitably nice $P$-stratified topological spaces with equivalences on strata and links inverted. We show that the exit-path construction of MacPherson, Treumann, and Lurie defines an equivalence from our homotopy theory of $P$-stratified topological spaces to the $\infty$-category of $\infty$-categories with a conservative functor to $P$. This proves a stratified form of Grothendieck's homotopy hypothesis, verifying a conjecture of Ayala-Francis-Rozenblyum. Our homotopy theory of stratified spaces has the added benefit of capturing all exam…
We exhibit a Quillen equivalence between two model categories encoding the homotopy theory of stratified spaces : the model category of filtered simplicial sets, and that of filtered spaces. Additionally, we introduce a new class of filtered spaces, that of vertical filtered CW-complexes, providing a nice model for the homotopy category of stratified spaces.
In previous work, the first author defined homotopy theories for stratified spaces from a simplicial and a topological perspective. In both frameworks stratified weak-equivalences are detected by suitable generalizations of homotopy links. These two frameworks are connected through a stratified version of the classical adjunction between the realization and the functor of singular simplices. Using a modified version of this adjunction, the first author showed that over a fixed poset of strata the two homotopy theories were equivalent. Building on this result we now show that the unmodified adj…
In this master thesis, we extend results from classical simple homotopy theory to the world of stratified homotopy theory. To obtain a well-established framework to work in, we prove a series of results on two model categories of simplicial sets and topological spaces, both equipped with a notion of filtration, introduced by Sylvain Douteau in his PHD thesis. In particular, we show that there is a fully faithful embedding of homotopy categories from the (finite) filtered simplicial into the filtered topological setting. We also use these results to characterize the morphisms in the topological…
In this article we consider the homotopy theory of stratified spaces through a simplicial point of view. We first consider a model category of filtered simplicial sets over some fixed poset $P$, and show that it is a simplicial combinatorial model category. We then define a generalization of the homotopy groups for any fibrant filtered simplicial set $X$ : the filtered homotopy groups $sπ_n(X)$. They are diagrams of groups built from the homotopy groups of the different pieces of $X$. We then show that the weak equivalences are exactly the morphisms that induce isomorphisms on those filtered h…
An IP-space is a pseudomanifold whose defining local properties imply that its middle perversity global intersection homology groups satisfy Poincaré duality integrally. We show that the symmetric signature induces a map of Quinn spectra from IP bordism to the symmetric $L$-spectrum of $\Z$, which is, up to weak equivalence, an $E_\infty$ ring map. Using this map, we construct a fundamental $L$-homology class for IP-spaces, and as a consequence we prove the stratified Novikov conjecture for IP-spaces.
A stratified space is a topological space together with a decomposition into strata corresponding to different types of singularities. Examples of such spaces appear everywhere in topology and geometry. The study of stratified spaces involves invariants such as intersection cohomology which are only invariants under stratification-preserving homotopies. In this thesis, we study the homotopy theory of stratified spaces with respect to those stratified homotopies. To do so, we construct model categories for stratified spaces and we introduce new invariants to characterize them, the filtered homo…
We apply the theory of directed topology developed by Grandis [9, 10] to the study of stratified spaces by describing several ways in which a stratification or a stratification with orientations on the strata can be used to produce a related directed space structure. This description provides a setting for the constructions of state-sum TQFTs with defects of [5, 8], which we extend to a similar construction of a Dijkgraaf- Witten type TQFT in the case where the defects (lower dimensional strata) are not sources or targets, but sources on one side and targets on the other, according to an orien…
The goal of this article is to extend a theorem of Lurie \[ \mathsf{Sh}_A (X) = \mathsf{Fun}(\mathsf{Exit}_A (X), \mathsf{S}) \] representing constructible sheaves with values in $ \mathsf{S} $, the $ \infty $-category of spaces, on a stratified space $ X $ with poset of strata $ A $, as functors from the exit paths $ \infty $-category $ \mathsf{Exit}_A (X) $ to $ \mathsf{S} $. Lurie's representation theorem works provided $ A $ satisfy the ascending chain condition. This typically rules out infinite dimensional examples of stratified space. Building on it and with the help of a stratified hom…
The fundamental groupoid of a locally 0 and 1-connected space classifies covering spaces, or equivalently local systems. When the space is topologically stratified Treumann, based on unpublished ideas of MacPherson, constructed an `exit category' (in the terminology of this paper, the `fundamental category') which classifies constructible sheaves, equivalently stratified etale covers. This paper generalises this construction to homotopically stratified sets, in addition showing that the fundamental category dually classifies constructible cosheaves, equivalently stratified branched covers. The…
In 1920s R. L. Moore introduced \emph{upper semicontinuous} and \emph{lower semicontinuous} decompositions in studying decomposition spaces. Upper semicontinuous decompositions were studied very well by himself and later by R.H. Bing in 1950s. In this paper we consider lower semicontinuous decompositions $\mathcal D$ of a topological space $X$ such that the decomposition spaces $X/\mathcal D$ are Alexandroff spaces. If the associated proset (preordered set) of the decomposition space $X/\mathcal D$ is a poset, then the decomposition map $π:X \to X/\mathcal D$ is \emph{a continuous map from the…
We derive spectral sequences for the intersection homology of stratified fibrations and approximate tubular neighborhoods in manifold stratified spaces. These neighborhoods include regular neighborhoods in PL stratified spaces.
In this thesis we define the notion of a locally stratified space. Locally stratified spaces are particular kinds of streams and d-spaces which are locally modelled on stratified spaces. We construct a locally presentable and cartesian closed category of locally stratified spaces that admits an adjunction with the category of simplicial sets. Moreover, we show that the full subcategory spanned by locally stratified spaces whose associated simplicial set is a quasicategory has the structure of a category with fibrant objects. We define the fundamental category of a locally stratified space and …
Recovering homological features of spaces from samples has become one of the central themes of topological data analysis, leading to many successful applications. Many of the results in this area focus on global homological features of a subset of a Euclidean space. In this case, homology recovery predicates on imposing well understood geometric conditions on the underlying set. Typically, these conditions guarantee that small enough neighborhoods of the set have the same homology as the set itself. Existing work on recovering local homological features of a space from samples employs similar …
We construct an explicit equivariant cellular decomposition of the $(4n-1)$-sphere with respect to binary polyhedral groups, and describe the associated cellular homology chain complex. As a corollary of the binary octahedral case, we deduce an $\mathfrak{S}_3$-equivariant decomposition of the flag manifold of $SL_3(\mathbb{R})$.
In a previous work we proved the uniqueness and functoriality of primary unfoldings on simple Thom-Mather spaces, which is a functor to the category of smooth manifolds. In this article we extend these results for any stratified Thom-Mather pseudomanifold with arbitary finite length, through a new kind of intermediate desingularizations, the unbendings, which coincide with primary unfoldings in the simple case.
We study bordism groups and bordism homology theories based on pseudomanifolds and stratified pseudomanifolds. The main seam of the paper demonstrates that when we uses classes of spaces determined by local link properties, the stratified and unstratified bordism theories are identical; this includes the known examples of pseudomanifold bordism theories, such as bordism of Witt spaces and IP spaces. Along the way, we relate the stratified and unstratified points of view for describing various (stratified) pseudomanifold classes.
We study the GKM theory for a equivariant stratified space having orbifold structures in tis successive quotients. Then, we introduce the notion of an \emph{almost simple polytope}, as well as a \emph{divisive toric variety} generalizing the concept of a divisive weighted projective space. We employ the GKM theory to compute the generalized equivariant cohomology theories of toric varieties associated to almost simple polytopes and divisive toric varieties.
We show that intersection homology extends Poincare duality to manifold homotopically stratified spaces (satisfying mild restrictions). This includes showing that, on such spaces, the sheaf of singular intersection chains is quasi-isomorphic to the Deligne sheaf.
A poset-stratified space is a pair $(S, S \xrightarrow πP)$ of a topological space $S$ and a continuous map $π: S \to P$ with a poset $P$ considered as a topological space with its associated Alexandroff topology. In this paper we show that one can impose such a poset-stratified space structure on the homotopy set $[X, Y]$ of homotopy classes of continuous maps by considering a canonical but non-trivial order (preorder) on it, namely we can capture the homotopy set $[X, Y]$ as an object of the category of poset-stratified spaces. The order we consider is related to the notion of \emph{dependen…
The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincaré duality holds between the ordinary rational cohomology groups of the cell-complexes associated to complementary perversities. The cohomology of these intersection spaces defines a cohomology theory HI for singular spaces, which is not isomorphic to intersection cohomology IH. Mirror symmetry tends to interchange IH and HI. The theory IH can be tied to type IIA string theory, while HI can be tied to IIB theory. For pseudomanifolds with …
We prove the existence of a model structure on the category of stratified simplicial sets whose fibrant objects are precisely $n$-complicial sets, which are a proposed model for $(\infty,n)$-categories, based on previous work of Verity and Riehl. We then construct a Quillen equivalent model based on simplicial presheaves over a category that can facilitate the comparison with other established models.
This work forms a foundational study of factorization homology, or topological chiral homology, at the generality of stratified spaces with tangential structures. Examples of such factorization homology theories include intersection homology, compactly supported stratified mapping spaces, and Hochschild homology with coefficients. Our main theorem characterizes factorization homology theories by a generalization of the Eilenberg--Steenrod axioms; it can also be viewed as an analogue of the Baez--Dolan cobordism hypothesis formulated for the observables, rather than state spaces, of a topologic…
In this paper, we go into the study of the 2-category SSS_Σof Σ-constructible stacks. The notions of constructible stack was introduced by D. Treumann. It is a natural generalization of constructible sheaf. D. Treumann has also introduced the exit-path 2-category, which is a stratified version of the fundamental 2-groupoid and he has showed that these two 2-categories are equivalent. Our approach is different. We show the 2-equivalence between SSS_Σand a 2-category whose objects are combinatoric data of 2-representations, functors of 2-representations and isomorphisms of functors.
By a theorem of A.Björner, for every interval $[u,v]$ in the Bruhat order of a Coxeter group $W$, there exists a stratified space whose strata are labeled by the elements of $[u,v]$, adjacency is described by the Bruhat order, and each closed stratum (resp., the boundary of each stratum) has the homology of a ball (resp., of a sphere). Answering a question posed by Björner, we suggest a natural geometric realization of these stratified spaces for a Weyl group $W$ of a semisimple Lie group $G$, and prove its validity in the case of the symmetric group. Our stratified spaces arise as links in th…
Intersection homology is a topological invariant which detects finer information in a space than ordinary homology. Using ideas from classical simple homotopy theory, we construct local combinatorial transformations on simplicial complexes under which intersection homology remains invariant. In particular, we obtain the notions of stratified formal deformations and stratified spines of a complex, leading to reductions of complexes prior to computation of intersection homology. We implemented the algorithmic execution of such transformations, as well as the calculation of intersection homology,…
This article is devoted to the study of smooth desingularization, which are customary employed in the definition of De Rham Intersection Cohomology with differential forms. In this paper we work with the category of Thom-Mather simple spaces. We construct a functor which sends each Thom-Mather simple space into a smooth manifold called its primary unfolding. Hence we prove that the primary unfoldings are unique up Thom-Mather isomorphisms.
We show that conically smooth stratified spaces embed fully faithfully into $\infty$-categories. This articulates a stratified generalization of the homotopy hypothesis proposed by Grothendieck. As such, each $\infty$-category defines a stack on conically smooth stratified spaces, and we identify the descent conditions it satisfies. These include $\mathbb{R}^1$-invariance and descent for open covers and blow-ups, analogous to sheaves for the h-topology in $\mathbb{A}^1$-homotopy theory. In this way, we identify $\infty$-categories as striation sheaves, which are those sheaves on conically smoo…
We construct a spectral sequence associated to a stratified space, which computes the compactly supported cohomology groups of an open stratum in terms of the compactly supported cohomology groups of closed strata and the reduced cohomology groups of the poset of strata. Several familiar spectral sequences arise as special cases. The construction is sheaf-theoretic and works both for topological spaces and for the étale cohomology of algebraic varieties. As an application we prove a very general representation stability theorem for configuration spaces of points.
We extend the existing idea of "cellular poset", introduce a collection of "cellular methods" for the computation of homology of intersection lattice of a complicated subspace arrangement, and for the computation of multiplicative structure induced by intersection. As an application, we give a presentation of cohomology ring of orbit configuration space of standard action by cellular methods and a spectral sequence associated with Grothendieck fibration of poset.
For any abelian compact Lie group $G$, we introduce a family of $G$-stratified pseudomanifolds, whose main feature is the preservation of the orbit spaces in the category of stratified pseudomanifolds. Which generalize a previous definition given by R. Popper. We also find a sufficient condition for the existence of equivariant unfoldings, so we have Intersection Cohomology with differential forms, as defined in \cite{S}. Moreover, if $G$ act on a manifold $M$, we find a equivariant unfolding of $M$ which induces a canonical unfolding on the $k$-orbits space for every closed subgroup $K$ of $G…
The flag vector contains all the face incidence data of a polytope, and in the poset setting, the chain enumerative data. It is a classical result due to Bayer and Klapper that for face lattices of polytopes, and more generally, Eulerian graded posets, the flag vector can be written as a cd-index, a non-commutative polynomial which removes all the linear redundancies among the flag vector entries. This result holds for regular CW complexes. We relax the regularity condition to show the cd-index exists for Whitney stratified manifolds by extending the notion of a graded poset to that of a quasi…
In this paper we complete the description of the $B\mathbb{Z} /p$-cellularization of the classifying spaces of all finite groups, for all primes $p$. The techniques are based in a careful analysis of the $p$-fusion structure of the groups involved -with special attention to their strongly closed subgroups- and Chachólski's description of the $A$-cellular approximation.
On a smoothly stratified space, we identify intersection cohomology of any given perversity with an associated weighted $L^2$ cohomology for iterated fibred cusp metrics on the smooth stratum. In particular given a Witt space, we identify the $L^2$ cohomology of iterated fibred cusp metrics with the middle perversity intersection cohomology of the corresponding stratified space.
Let $M$ denote a two-dimensional Moore space (so $H_2(M; \Z) = 0$), with fundamental group $G$. The $M$-cellular spaces are those one can build from $M$ by using wedges, push-outs, and telescopes (and hence all pointed homotopy colimits). The question we address here is to characterize the class of $M$-cellular spaces by means of algebraic properties derived from the group $G$. We show that the cellular type of the fundamental group and homological information does not suffice, and one is forced to study a certain universal extension.
Many data-rich industries are interested in the efficient discovery and modelling of structures underlying large data sets, as it allows for the fast triage and dimension reduction of large volumes of data embedded in high dimensional spaces. The modelling of these underlying structures is also beneficial for the creation of simulated data that better represents real data. In particular, for systems testing in cases where the use of real data streams might prove impractical or otherwise undesirable. We seek to discover and model the structure by combining methods from topological data analysis…
