Mathematics

We define the notion of an approximate triangulation for a manifold M embedded in euclidean space. The basic idea is to build a nested family of simplicial complexes whose vertices lie in M and use persistent homology to find a complex in the family whose homology agrees with that of M . Our key examples are various Grassmann manifolds G k ( R n ) .

We construct a dga to computing the cohomology of ordered configuration spaces on an algebraic variety with vanishing Euler characteristic. It follows that the k -th Betti number of Conf(C,n) ( C is an elliptic curve) grows as a polynomial of degree exactly 2k−2 . We also compute H k (Conf(C,n)) for k≤5 and arbitrary n .

We give explicit formulas for the asymptotic Betti numbers of the unordered configuration spaces of an arbitrary finite graph over an arbitrary field.

Toric topology assigns to each n -dimensional combinatorial simple convex polytope P with m facets an (m+n) -dimensional moment-angle manifold Z P with an action of a compact torus T m such that Z P / T m is a convex polytope of combinatorial type P . A simple n -polytope is called B -rigid, if any isomorphism of graded rings H ∗ ( Z P ,Z)= H ∗ ( Z Q ,Z) for a simple n -polytope Q implies that P and Q are combinatorially equivalent. An ideal almost Pogorelov polytope is a combinatorial 3 -polytope obtained by cutting off all the ideal vertices of an ideal right-angled polytope in the Lobachevsky (hyperbolic) space L 3 . These polytopes are exactly the polytopes obtained from any, not necessarily simple, convex 3 -polytopes by cutting off all the vertices followed by cutting off all the "old" edges. The boundary of the dual polytope is the barycentric subdivision of the boundary of the old polytope (and also of its dual polytope). We prove that any ideal almost Pogorelov polytope is B -rigid. This produces three cohomologically rigid families of manifolds over ideal almost Pogorelov manifolds: moment-angle manifolds, canonical 6 -dimensional quasitoric manifolds and canonical 3 -dimensional small covers, which are "pullbacks from the linear model".

Toric topology assigns to each n -dimensional combinatorial simple convex polytope P with m facets an (m+n) -dimensional moment-angle manifold Z P with an action of a compact torus T m such that Z P / T m is a convex polytope of combinatorial type P . We study the notion of B -rigidity. A property of a polytope P is called B -rigid, if any isomorphism of graded rings H ∗ ( Z P ,Z)= H ∗ ( Z Q ,Z) for a simple n -polytope Q implies that it also has this property. We study families of 3 -dimensional polytopes defined by their cyclic k -edge-connectivity. These families include flag polytopes and Pogorelov polytopes, that is polytopes realizable as bounded right-angled polytopes in Lobachevsky space L 3 . Pogorelov polytopes include fullerenes -- simple polytopes with only pentagonal and hexagonal faces. It is known that the properties to be flag and Pogorelov polytope are B -rigid. We focus on almost Pogorelov polytopes, which are strongly cyclically 4 -edge-connected polytopes. They correspond to right-angled polytopes of finite volume in L 3 . There is a subfamily of ideal almost Pogorelov polytopes corresponding to ideal right-angled polytopes. We prove that the properties to be an almost Pogorelov polytope and an ideal almost Pogorelov polytope are B -rigid. As a corollary we obtain that 3 -dimensional associahedron A s 3 and permutohedron P e 3 are B -rigid. We generalize methods known for Pogorelov polytopes. We obtain results on B -rigidity of subsets in H ∗ ( Z P ,Z) and prove an analog of the so-called separable circuit condition (SCC). As an example we consider the ring H ∗ ( Z A s 3 ,Z) .

In this paper, we study how basis-independent partial matchings induced by morphisms between persistence modules (also called ladder modules) can be defined. Besides, we extend the notion of basis-independent partial matchings to the situation of a pair of morphisms with same target persistence module. The relation with the state-of-the-art methods is also given. Apart form the basis-independent property, another important property that makes our partial matchings different to the state-of-the-art ones is their linearity with respect to ladder modules.

In a 2012 note in Comptes Rendus Math{é}matique, the author did try to answer a question of Jean-Pierre Serre; it has recently been announced that the scope of that answer needs an adjustment, and the details of this adjustment are given in the present paper. The original question is the following. Consider the ring of integers O in an imaginary quadratic number field, and the Borel-Serre compactification of the quotient of hyperbolic 3-space by SL 2 (O). Consider the map α induced on homology when attaching the boundary into the Borel-Serre compactification. How can one determine the kernel of α (in degree 1) ? Serre used a global topological argument and obtained the rank of the kernel of α . He added the question what submodule precisely this kernel is.

Let p be prime. We prove that, for n odd, the p -torsion part of π q ( S n ) has cardinality at most p 2 1 p−1 (q−n+3−2p) , and hence has rank at most 2 1 p−1 (q−n+3−2p) . For p=2 these results also hold for n even. The best bounds proven in the existing literature are p 2 q−n+1 and 2 q−n+1 respectively, both due to Hans-Werner Henn. The main point of our result is therefore that the bound grows more slowly for larger primes. As a corollary of work of Henn, we obtain a similar result for the homotopy groups of a broader class of spaces.

In this paper, we generalize the tools that were introduced in [Dar19b] in order to study the Andreadakis problem for subgroups of IAn. In particular, we study the behaviour of the Andreadakis problem when we add inner automorphisms to a subgroup of IAn. We notably use this to show that the Andreadakis equality holds for the pure braid group on n strands modulo its center acting on the free group on n-1 generators , that is, for the (pure, based) mapping class group of the n-punctured sphere acting on its fundamental group.

The hierarchy poset and branch point poset for a data set both admit a calculus of least upper bounds. A method involving upper bounds is used to show that the map of branch points associated to the inclusion of data sets is a controlled homotopy equivalence, where the control is expressed by an upper bound relation that is constrained by Hausdorff distance.