Local Cohomology and Stratification

Foundations of Computational Mathematics https://doi.org/10.1007/s10208-019-09424-0 Local Cohomology and Stratification Vidit Nanda1 Received: 14 August 2017 / Revised: 9 March 2019 / Accepted: 29 April 2019 © The Author(s) 2019 Abstract We outline an algorithm to recover the canonical (or, coarsest) stratification of a given finite-dimensional regular CW complex into cohomology manifolds, each of which is a union of cells. The construction proceeds by iteratively localizing the poset of cells about a family of subposets; these subposets are in turn determined by a collection of cosheaves which capture variations in cohomology of cellular neighborhoods across the underlying complex. The result is a nested sequence of categories, each containing all the cells as its set of objects, with the property that two cells are isomorphic in the last category if and only if they lie in the same canonical stratum. The entire process is amenable to efficient distributed computation. Keywords Canonical stratification · Local cohomology Mathematics Subject Classification 52S60 · 55N30 · 18E35 1 Introduction Setting aside technicalities for the moment, an n-dimensional stratification of a given topological space X is a nested sequence of closed subspaces ∅ = X−1 ⊂ X0 ⊂ · · · ⊂ Xn = X where each successive difference Xd − Xd−1 resembles a (possibly empty) ddimensional manifold whose connected components are the d-dimensional strata. The details which have been suppressed are designed to guarantee the uniformity of small neighborhoods around points in a given stratum. Stratified spaces are of fundamental importance in any branch of mathematics which seriously confronts singularities. Communicated by Herbert Edelsbrunnner. B Vidit Nanda 1 University of Oxford, Oxford, UK 123 Foundations of Computational Mathematics Every topological manifold admits a straightforward stratification into its connected components; less trivially, the following spaces all admit, and are often analyzed using, stratifications: finite CW complexes, quotients of properly discontinuous Lie group actions on smooth manifolds, (semi)algebraic varieties and (sub)analytic sets. The two-dimensional singular space below—let us call it Y—is built by pinching a torus along a meridian and attaching a disk across an equator: Any regular CW structure, such as the illustrated decomposition into little squares, constitutes a stratification of Y where d-dimensional strata are precisely the d-cells; passing to a subdivision further refines this stratification in the sense that every new cell is entirely contained in the interior of an old cell. On the other hand, one can discover a much coarser stratification by examining the topology of small neighborhoods around points of Y. Up to homeomorphism, these fall into three different classes depending on whether the central point is at the pinch, on the singular equator, or on one of the manifold-like two-dimensional regions: The neighborhoods above deformation retract onto their central vertices and are therefore contractible; however, their one-point compactifications (obtained by collapsing their boundaries to points) are new stratified spaces with potentially interesting topology. The compactified neighborhood around the pinch point is homeomorphic to two 2-spheres joined at their north and south poles with a spanning disk across the middle. The compactified neighborhood of any point in the singular equator resembles a 2-sphere whose interior has been partitioned into two by a disk. And finally, the compactified neighborhood around any point in either of the two-dimensional regions is homeomorphic to an ordinary 2-sphere: 123 Foundations of Computational Mathematics Our main result here involves algorithmically recovering the coarsest stratification of a finite regular CW complex where all strata are unions of cells. This is called the canonical stratification of the complex; its existence and uniqueness for a special class of spaces (called pseudomanifolds) plays a central role in Goresky and MacPherson’s proof of the topological invariance of intersection homology [14, Sec 4]. Our argument, much like theirs, has an intuitive geometric core but invokes algebraic and categorical machinery. For the purposes of this introductory section, we focus on geometry and ask: Given a finite cellulation of Y, how might one identify the canonical strata and determine which cells lie in each canonical stratum, as shown below? (Since the pinch point has a different compactified neighborhood than a generic point on the singular equator, it must constitute a separate stratum.) In light of the discussion above, one hopes to recover canonical strata by clustering together cells whenever they exhibit similar compactified neighborhoods. But already in this example, we encounter two significant difficulties: First, the compactified neighborhoods do not distinguish cells in the two 2-strata from each other. The second difficulty is somewhat subtler—although the compactified neighborhoods of cells in the 0-stratum and 1-stratum are not homeomorphic, both are homotopy-equivalent to a wedge of two 2-spheres. Therefore, they cannot be distinguished by weaker, more computable topological invariants such as cohomology. We tackle the first problem by constructing a complex of cosheaves which encodes how local topology varies across cells, and we bypass the second problem by working exclusively in the category of cohomologically stratified spaces. Consider the task of determining the canonical stratification of a finite-dimensional regular CW complex X into R-cohomology manifolds, where R is a fixed non-trivial commutative ring with unity. In this case, our complex of cosheaves assumes a particularly appealing form: It is a functor L • : Fc(X) → Ch(R) from the poset of cells in X (where x > y denotes that y is a face of x, or equivalently, that x is a co-face of y) to the category of lower-bounded R-module cochain complexes. To each cell x of X, it assigns a cochain complex L • (x): L 0 (x) βx0 L 1 (x) βx1 L 2 (x) βx2 ··· , where L d (x) is the free R-module generated by all d-dimensional co-faces of x— in particular, the module L d (x) is trivial for d < dim x and d > dim X. The codifferentials βx• are inherited verbatim from incidence degrees among cells of X taking values in R. Thus, L • (x) computes the (reduced) cohomology of a compactified small 123 Foundations of Computational Mathematics open neighborhood around x in X. Given another cell y satisfying x ≥ y, there is an inclusion map L • (x ≥ y) : L • (x) → L • (y) arising from the fact that co-faces of x are also co-faces of y. Here is a simple version of our main result. (The full statement has been recorded in Theorem 5.10.) Theorem 1.1 There is a category S obtained from Fc(X) by formally inverting a particular subset of those face relations x ≥ y for which L • (x ≥ y) induces isomorphisms on cohomology; two cells l (...truncated)