Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2407.03832

Homology of graph burnings

Yuri Muranov·Anna Muranova
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we study graph burnings using methods of algebraic topology. We prove that the time function of a burning is a graph map to a path graph. Afterwards, we define a category whose objects are graph burnings and morphisms are graph maps which commute with the time functions of the burnings. In this category we study relations between burnings of different graphs and, in particular, between burnings of a graph and its subgraphs. For every graph, we define a simplicial complex, arising from the set of all the burnings, which we call a configuration space of the burnings. Further, simplicial structure of the configuration space gives burning homology of the graph. We describe properties of the configuration space and the burning homology theory. In particular, we prove that the one-dimensional skeleton of the configuration space of a graph $G$ coincides with the complement graph of $G$. The results are illustrated with numerous examples.

Explore related subjects

math.ATmath.COmath.CT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yuri Muranov, Anna Muranova. 2024-07-04. Homology of graph burnings. https://doi.org/10.1016/j.topol.2025.109486

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

The fifth algebraic transfer in generic degrees and validation of a localized Kameko's conjecture

This paper develops our previous works concerning the classical Peterson hit problem for the polynomial algebra on five variables over the mod--2 Steenrod algebra $\mathscr A$ in a generic family of degrees, together with applications to the fifth Singer algebraic transfer and a localized variation of Kameko's conjecture. As a topological illustration of the usefulness of the Steenrod algebra, we prove that $\mathbb{C}P^4/\mathbb{C}P^2$ and $\mathbb{S}^6\vee \mathbb{S}^8$ are not homotopy equivalent by showing that their mod--2 cohomologies are not isomorphic as $\mathscr A$-modules, and we further determine the homotopy type of the quotient $\mathbb{C}P^n/\mathbb{C}P^{\,n-2}$ for all $n\ge 3$. For the generic degrees under consideration, we determine the relevant cohit spaces and describe the associated $GL(5,\mathbb F_2)$-module structure. As a consequence, the fifth algebraic transfer is an isomorphism in an explicit infinite family of internal degrees. These results were independently verified by implementations in \texttt{SageMath} and \texttt{OSCAR}. We also study a localized form of Kameko's conjecture concerning the dimensions of the indecomposables $\mathbb F_2\otimes_{\mathscr A}\mathbb F_2[x_1,\ldots,x_m]$ relative to parameter vectors, and prove that this conjecture holds for all $m\ge 1$ in certain degrees.

math.AT↗
arXiv2026

A rigidity theorem for $π_*$-étale $\mathbb E_k$-algebras

We prove a rigidity theorem for $π_*$-étale $\mathbb E_k$-algebras over an $\mathbb E_{k+1}$-ring spectrum: the category of $π_*$-étale extensions of an $\mathbb E_k$-algebra is identified with the ordinary category of étale Dirac algebras over its graded homotopy Dirac ring. The proof develops a relative Goerss-Hopkins type obstruction theory in synthetic spectra, including an $I$-complete version. As an application, the completed obstruction theory constructs the $I_n$-complete $\mathbb E_3$-$MU_{(p)}$-algebra realization of the Lubin-Tate theory, hence an $\mathbb E_4$-orientation $MU_{(p)}\to E_n$.

math.AT↗
arXiv2026

Automated proofs of unstable Adams differentials

We present a computer-based approach to computing differentials in the unstable Adams spectral sequence by systematically applying the unstable Leibniz rule and naturality with respect to maps in the EHP sequence. We record our results in tables of upper and lower bounds on the orders of 2-primary unstable homotopy groups of spheres through the unstable 50-stem. We provide examples of proofs for several differentials and give a guide to interpreting the associated unstable Adams charts and flow chart diagrams for differential proofs.

math.AT↗