arXiv subjects

Andrés Carnero Bravo

Publications and source records attributed to Andrés Carnero Bravo.

8 recordsLinked to original sources

arXiv2026

Total cut complexes and their duals

We study the total $d$-cut complexes and their Alexander duals. We give some results about the connectivity in general and in terms of the grith of the graph. For $d\geq3$, the homotopy type of these complexes is calculated for: $p$th power of a cycle with at least $(2r)d$ vertices where $p\leq r$; the $r$th power of a cycle with at least $2rd-(r-1)$ vertices where $r\geq3$; and the $2$th power of a cycle with at least $3d$ vertices. These calculations solve a conjecture of Bayer, Denker, Milutinović, Rowlands, Sundaram and Xue. The homotopy type of the $2$-total cut complex for any $r$th power of a cycle with $r\geq3$ also is calculated, solving a conjecture of Chauhan, Shukla and Vinayak. We also study the complexes of cartesian products of paths and of cartesian products of complete graphs for the total $2$-cut complex.

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arXiv2026

Independence complexes of generalized Mycielskian graphs

We show that the homotopy type of the independence complex of the generalized Mycielskian of a graph $G$ is determined by the homotopy type of the independence complex of $G$ and the homotopy type of the independence complex of the Kronecker double cover of $G$. As an application we calculate the homotopy type for paths, cycles and the categorical product of two complete graphs.

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arXiv2025

On independence complexes of graph products

We study the independence complexes of graph products where at least one factor is a path. We also analyze the complexes of their induced subgraphs. We determine the homotopy type of the independence complex of the graphs $P_n\times P_m$, $P_n\boxtimes P_2$, $P_n\boxtimes P_3$ and $P_n\boxtimes P_4$. We also focus in the independence complexes of the induced subgraphs of $P_n\times P_3$, $P_n\boxtimes P_2$, $P_n\boxtimes P_3$, $P_n\boxtimes P_4$ and some lexicographic products $G\circ H$.

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arXiv2025

Left and right Bousfield localization on lattices

The key information of a model category structure on a poset is encoded in a transfer system, which is a combinatorial gadget, originally introduced to investigate homotopy coherence structures in equivariant homotopy theory. We describe how a transfer system associated with in a model structure on a lattice is affected by left and right Bousfield localization and provide a minimal generating system of morphisms which are responsible for the change in model structure. This leads to new concrete insights into the behavior of model categories on posets in general.

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arXiv2025

Polyhedral joins and graph complexes

We give a decomposition of the suspension of a polyhedral join in terms of the polyhedral smash product of the suspension of the family of pairs, and study some cases in which the formula can be desuspended, particularly for polyhedral joins over independence complexes of graphs. We also give some lower bounds for the connectivity of polyhedral joins. We use these results to study the homotopy type of the forest filtration for some lexicographic products of graphs.

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arXiv2025

The Forest Filtration of a Graph

Given a graph $G$, we define a filtration of simplicial complexes associated to $G$, $\mathcal{F}_0(G)\subseteq\mathcal{F}_1(G)\subseteq\cdots\subseteq\mathcal{F}_\infty(G)$ where the first complex is the independence complex and the last the complex is formed by the acyclic sets of vertices. We prove some properties of this filtration and we calculate the homotopy type for various families of graphs. We give an upper bound for the decycling number and generalizations of this parameter using the dimensions of the rational cohomology groups of these complexes. We also derive an upper bound for the Fibonacci numbers of ternary graphs.

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arXiv2023

Homotopy type of the independence complex of some categorical products of graphs

It was conjectured by Goyal, Shukla and Singh that the independence complex of the categorical product $K_2\times K_3\times K_n$ has the homotopy type of a wedge of $(n-1)(3n-2)$ spheres of dimension $3$. Here we prove this conjecture by calculating the homotopy type of the independence complex of the graphs $C_{3r}\times K_n$ and $K_2\times K_m\times K_n$. For $C_m \times K_n$ when $m$ is not a multiple of $3$, we calculate the homotopy type for $m = 4, 5$ and show that for other values it has to have the homotopy type of a wedge of spheres of at most $2$ consecutive dimensions and maybe some Moore spaces.

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arXiv2023

Homotopy type through homology groups

We show that if a complex has free finitely generated reduced homology groups for two consecutive dimensions and trivial homology for all other dimensions, then it must have the homotopy type of a wedge of spheres of two consecutive dimensions. We also show other pairs of dimensions for which the last result can be generalized.

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