arXiv subjects
Publications and source records attributed to Davide Lofano.
Sphere recognition is known to be undecidable in dimensions five and beyond, and no polynomial time method is known in dimensions three and four. Here we report on positive and negative computational results with the goal to explore the limits of sphere recognition from a practical point of view. An important ingredient are randomly constructed discrete Morse functions.
We construct a series HMT$(n)$ of $2$-dimensional simplicial complexes with torsion $H_1($HMT$(n))=(\mathbb{Z}_2)^{{k}\choose{1}} \times (\mathbb{Z}_4)^{{k}\choose{2}} \times \cdots \times (\mathbb{Z}_{2^k})^{{k}\choose{k}}$, $|H_1($HMT$(n))|=|$det(H$(n))|=n^{n/2} \in Θ(2^{n \log n})$, where the construction is based on the Hadamard matrices H$(n)$ for $n\geq 2$ a power of $2$, i.e., $n=2^k, \ k \geq 1$. The examples have linearly many vertices, their face vector is $f(HMT(n))=(5n-1,3n^2+9n-6,3n^2+4n-4)$. Our explicit series with torsion growth in $Θ(2^{n \log n})$ is constructed in quadratic time $Θ(n^{2})$ and improves a previous construction by Speyer with torsion growth in $Θ(2^{n})$, narrowing the gap to the highest possible asymptotic torsion growth in $Θ(2^{n^2})$ proved by Kalai via a probabilistic argument.
We implement an algorithm RSHT (Random Simple-Homotopy) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions. For triangulated d-manifolds with d < 7, we show that RSHT reduces to (random) bistellar flips. Among the many examples on which we test RSHT, we describe an explicit 15-vertex triangulation of the Abalone, and more generally, (14k+1)-vertex triangulations of Bing's houses with k rooms, which all can be deformed to a point using only six pure elementary expansions.
We construct a new class of maximal acyclic matchings on the Salvetti complex of a locally finite hyperplane arrangement. Using discrete Morse theory, we then obtain an explicit proof of the minimality of the complement. Our construction provides interesting insights also in the well-studied case of finite arrangements, and gives a nice geometric description of the Betti numbers of the complement. In particular, we solve a conjecture of Drton and Klivans on the characteristic polynomial of finite reflection arrangements. The minimal complex is compatible with restrictions, and this allows us to prove the isomorphism of Brieskorn's Lemma by a simple bijection of the critical cells. Finally, in the case of line arrangements, we describe the algebraic Morse complex which computes the homology with coefficients in an abelian local system.
In general a contractible complex need not be collapsible. Moreover, there exist complexes which are collapsible but even so admit a collapsing sequence where one "gets stuck", that is one can choose the collapses in such a way that one arrives at a nontrivial complex which admits no collapsing moves. Here we examine this phenomenon in the case of a simplex. In particular we characterize all values of $n$ and $d$ so that the n-simplex may collapse to a d-complex from which no further collapses are possible. Equivalently and in the language of high-dimensional generalizations of trees, we construct hypertrees that are anticollapsible, but not collapsible. Furthermore we examine anticollapsibility in random simplicial complexes.