Related papers: Stratified Morse Theory in Arrangements
The method of intersection spaces associates cell-complexes depending on a perversity to certain types of stratified pseudomanifolds in such a way that Poincar\'e duality holds between the ordinary rational cohomology groups of the…
An arrangement is a finite set of hyperplanes in a finite dimensional complex affine space. A complex rank one local system on the arrangement complement is determined by a set of complex weights for the hyperplanes. We study the…
We define a notion of Morse function and establish Morse theory-like theorems over offsets of any compact set in a Euclidean space at regular values of their distance function. Using non-smooth analysis and tools from geometric measure…
Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of a complex hyperplane arrangement has the homotopy type of a CW complex in…
In recent years, Benson, Iyengar and Krause have developed a theory of stratification for compactly generated triangulated categories with an action of a graded commutative Noetherian ring. Stratification implies a classification of…
The aim of this short note is to develop a (co)homology theory for topological spaces together with the specialisation preorder. A known way to construct such a (co)homology is to define a partial order on the topological space starting…
We develop the theory of arrangements of spheres. Consider a finite collection of codimension-$1$ subspheres in a positive-dimensional sphere. There are two posets associated with this collection: the poset of faces and the poset of…
We study the Gauss-Manin connection for the moduli space of an arrangement of complex hyperplanes in the cohomology of a nonresonant complex rank one local system. Aomoto and Kita determined this Gauss-Manin connection for arrangements in…
In this paper, we study $k$-parabolic arrangements, a generalization of the $k$-equal arrangement for any finite real reflection group. When $k=2$, these arrangements correspond to the well-studied Coxeter arrangements. We construct a cell…
We prove several combinatorial results on path algebras over discrete structures related to directed graphs. These results are motivated by Morse theory on a manifold with boundary and, more generally, by Floer theory on a configuration…
The Lefschetz hyperplane section theorem asserts that an affine variety is homotopy equivalent to a space obtained from its generic hyperplane section by attaching some cells. The purpose of this paper is to describe attaching maps of these…
Homotopy connectedness theorems for complex submanifolds of homogeneous spaces (sometimes referred to as theorems of Barth-Lefshetz type) have been established by a number of authors. Morse Theory on the space of paths lead to an elegant…
We use derived localization of the bar and nerve constructions to provide simple proofs of a number of results in algebraic topology. This includes a recent generalization of Adams' cobar-construction to the non-simply connected case, and a…
Let $\textbf{H} = ((H, F^{\bullet}), L)$ be a polarized variation of Hodge structure on a smooth quasi-projective variety $U.$ By M. Saito's theory of mixed Hodge modules, the variation of Hodge structure $\textbf{H}$ can be viewed as a…
We provide a new approach to studying the moduli space of curves via Morse theory and hyperbolic geometry, by introducing a family of Morse functions on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$…
The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the…
We compute the cohomology ring of the complement of a toric arrangement with integer coefficients and investigate its dependency from the arrangement's combinatorial data. To this end, we study a morphism of spectral sequences associated to…
We introduce toric arrangements, essentially finite families of codimension 1 subtori of a torus or of their cosets, as a periodic generalization of hyperplane arrangements, compute cohomology of the complement of such an arrangement and…
Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic…
We give a short topological proof of coherence for categorified non-symmetric operads by using the fact that the diagrams involved form the 1-skeleton of simply connected CW complexes. We also obtain a "one-step" topological proof of Mac…