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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., Comment: 22 pages, 10 figures

The singular cubical homology theory for the category of quivers or digraphs can be constructed similarly to the classical singular homology theory for topological spaces. The case of digraphs and quivers differs from the topological case due to the possibility of using a large number of non-isomorphic line digraphs that correspond to the unit interval in algebraic topology. In this paper we introduce several different notions of quiver realizations of a cubical set and we describe relations between them. We also define various singular cubical homology theories on quivers and digraphs. Moreover, using quiver realizations of cubical sets we define a collection of path homology theories on the category of cubical sets and we describe their properties., Comment: 20 pages

We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy invariance of the introduced path homology groups. We provide examples of computation of these homology groups., Comment: 19 pages

We introduce a homotopy theory of digraphs (directed graphs) and prove its basic properties, including the relations to the homology theory of digraphs constructed by the authors in previous papers. In particular, we prove the homotopy invariance of homologies of digraphs and the relation between the fundamental group of the digraph and its first homology group. The category of (undirected) graphs can be identified by a natural way with a full subcategory of digraphs. Thus we obtain also consistent homology and homotopy theories for graphs. Note that the homotopy theory for graphs coincides with the one constructed in the paper of Babson et., Comment: 39 pages, 15 figures

Let $M^3$ be a 3-dimensional manifold with fundamental group $\pi_1(M)$ which contains a quaternion subgroup $Q$ of order 8. In 1979 Cappell and Shaneson constructed a nontrivial normal map $ f\colon M^3\times T^2\to M^3\times S^2$ which cannot be detected by simply connected surgery obstructions along submanifolds of codimension 0, 1, or 2, but it can be detected by the codimension 3 Kervaire-Arf invariant. The proof of non-triviality of $\sigma(f)\in L_5(\pi_1(M))$ is based on consideration of a Browder-Livesay filtration of a manifold $X$ with $\pi_1(X)\cong \pi_1(M)$. For a Browder-Livesay pair $Y^{n-1}\subset X^n$, the restriction of a normal map to the submanifold $Y$ is given by a partial multivalued map $\Gamma\colon L_n(\pi_1(X))\to L_{n-1}(\pi_1(Y))$, and the Browder-Livesay filtration provides an iteration $\Gamma^n$. This map is a basic step in the definition of the iterated Browder-Livesay invariants which give obstructions to realization of surgery obstructions by normal maps of closed manifolds. In the present paper we prove that $\Gamma^3(\sigma(f))=0$ for any Browder-Livesay filtration of a manifold $X^{4k+1}$ with $\pi_1(X)\cong Q$. We compute splitting obstruction groups for various inclusions $\rho\to Q$ of index 2, describe natural maps in the braids of exact sequences, and make more precise several results about surgery obstruction groups of the group $Q$.

In this paper we introduce a path complex that can be regarded as a generalization of the notion of a simplicial complex. The main motivation for considering path complexes comes from directed graphs(digraphs). We obtain a new notion of the path homology and cohomology of a digraph., Comment: 88 pages, 37 figures