Subdivisional spaces and graph braid groups
Abstract
We study the problem of computing the homology of the configuration spaces of a finite cell complex $X$. We proceed by viewing $X$, together with its subdivisions, as a subdivisional spacea kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we decompose $X$ and show that the homology of the configuration spaces of $X$ is computed by the derived tensor product of the Morse complexes of the pieces of the decomposition, an analogue of the monoidal excision property of factorization homology. Applying this theory to the configuration spaces of a graph, we recover a cellular chain model due to Świątkowski. Our method of deriving this model enhances it with various convenient functorialities, exact sequences, and module structures, which we exploit in numerous computations, old and new.
 Publication:

arXiv eprints
 Pub Date:
 August 2017
 arXiv:
 arXiv:1708.02351
 Bibcode:
 2017arXiv170802351A
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Geometric Topology
 EPrint:
 71 pages, 15 figures. Typo fixed. May differ slightly from version published in Documenta Mathematica