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In Crew and Spirkl (2020), the authors generalize Stanley's chromatic symmetric function (Stanley, 1995) to vertex-weighted graphs. In this paper we find a categorification of their new invariant extending the definition of chromatic symmetric homology to vertex-weighted graphs. We prove the existence of a deletion–contraction long exact sequence for chromatic symmetric homology which gives a useful computational tool and allow us to answer two questions left open in Chandler et al. (2019). In particular, we prove that, for a graph G with n vertices, the maximal index with nonzero homology is not greater that n − 1. Moreover, we show that the homology is non-trivial for all the indices between the minimum and the maximum with this property. [ABSTRACT FROM AUTHOR]

This article is concerned with the Rabinowitz Floer homology of negative line bundles. We construct a refined version of Rabinowitz Floer homology and study its properties. In particular, we build a Gysin-type long exact sequence for this new invariant and discuss an application to the orderability problem for prequantization spaces. We also construct a short exact sequence for the ordinary Rabinowitz Floer homology and provide computational results. [ABSTRACT FROM AUTHOR]

We introduce a graded homology theory for graded étale groupoids. We prove that the graded homology of a strongly graded ample groupoid is isomorphic to the homology of its ε -th component with ε the identity of the grade group. For Z -graded groupoids, we establish an exact sequence relating the graded zeroth-homology to non-graded one. Specialising to the arbitrary graph groupoids, we prove that the graded zeroth homology group with constant coefficients Z is isomorphic to the graded Grothendieck group of the associated Leavitt path algebra. To do this, we consider the diagonal algebra of the Leavitt path algebra of the covering graph of the original graph and construct the group isomorphism directly. Considering the trivial grading, our result extends Matui's on zeroth homology of finite graphs with no sinks (shifts of finite type) to all arbitrary graphs. We use our results to show that graded zeroth-homology group is a complete invariant for eventual conjugacy of shift of finite types and could be the unifying invariant for the analytic and the algebraic graph algebras. [ABSTRACT FROM AUTHOR]

Let G be a group, let H be a subgroup of G and let Or(G) be the orbit category. In this paper we extend the definition of the relative group homology the- ories of the pair (G, H) defined by Adamson and Takasu to have coefficients in an Or(G)-module. There is a canonical comparison homomorphism defined by Cisneros- Molina and Arciniega-Nevrez from Takasu's theory to Adamson's theory. We give a necessary and sufficient condition on the subgroup H for which the comparison homomorphism is an isomorphism for all coefficients. We also use the Lück-Wiermann construction to introduce a long exact sequence for Adamson homology. Finally, we provide some examples of explicit computations for the comparison homomorphism. [ABSTRACT FROM AUTHOR]