arXiv subjects

Mauricio Gomez Lopez

Publications and source records attributed to Mauricio Gomez Lopez.

4 recordsLinked to original sources

arXiv2025

Realizing the Tutte polynomial as a cut-and-paste K-theoretic invariant

Cut-and-paste $K$-theory is a new variant of higher algebraic $K$-theory that has proven to be useful in problems involving decompositions of combinatorial and geometric objects, e.g., scissors congruence of polyhedra and reconstruction problems in graph theory. In this paper, we show that this novel machinery can also be used in the study of matroids. Specifically, via the $K$-theory of categories with covering families developed by Bohmann-Gerhardt-Malkiewich-Merling-Zakharevich, we realize the Tutte polynomial map of Brylawski (also known as the universal Tutte-Grothendieck invariant for matroids) as the $K_0$-homomorphism induced by a map of $K$-theory spectra.

math.KT↗

arXiv2024

The homotopy type of the PL cobordism category. I

In this paper, we introduce a bordism category $\mathcal{C}_d^{PL}$ whose objects are bundles of closed $(d-1)$-dimensional piecewise linear manifolds and whose morphisms are bundles of $d$-dimensional piecewise linear cobordisms. In the main theorem of this article, we show that the classifying space $B\mathcal{C}_d^{PL}$ is weak homotopy equivalent to an infinite loop space. We regard $\mathcal{C}_d^{PL}$ as the piecewise linear analogue of the category of smooth cobordisms which has been studied extensively in connection to the Madsen-Weiss Theorem, and the main result of this paper is a first step towards obtaining Madsen-Weiss type results in the context of piecewise linear topology.

math.GT↗

arXiv2024

The homotopy type of the PL cobordism category. II

In this article, we prove the PL analogue of the theorem of Galatius, Madsen, Tillmann, and Weiss which describes the homotopy type of the smooth cobordism category. More specifically, we introduce the PL Madsen-Tillmann spectrum $\mathbf{MT}PL(d)$ and prove that there is a weak homotopy equivalence of the form $B\mathsf{Cob}^{\mathrm{PL}}_d \simeq Ω^{\infty-1}\mathbf{MT}PL(d)$. We also discuss how to adjust the methods of this paper to obtain the topological version of our main result.

math.AT↗

arXiv2022

The homotopy type of the topological cobordism category

We define a cobordism category of topological manifolds and prove that if $d \neq 4$ its classifying space is weakly equivalent to $Ω^{\infty -1} MTTop(d)$, where $MTTop(d)$ is the Thom spectrum of the inverse of the canonical bundle over $BTop(d)$. We also give versions with tangential structures and boundary. The proof uses smoothing theory and excision in the tangential structure to reduce the statement to the computation of the homotopy type of smooth cobordism categories due to Galatius-Madsen-Tillman-Weiss.

math.AT↗