Subdivisional Spaces and Graph Braid Groups

  • Byung Hee An

    Center for Geometry and Physics, Institute for Basic Science, Pohang 37673, Republic of Korea
  • Gabriel C. Drummond-Cole

    Center for Geometry and Physics, Institute for Basic Science, Pohang 37673, Republic of Korea
  • Ben Knudsen

    Mathematics Department, Northeastern University, Boston 02115, USA
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Abstract

We study the problem of computing the homology of the configuration spaces of a finite cell complex . We proceed by viewing , together with its subdivisions, as a subdivisional space -- a kind of diagram object in a category of cell complexes. After developing a version of Morse theory for subdivisional spaces, we decompose and show that the homology of the configuration spaces of 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 Swiatkowski. 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.

Cite this article

Byung Hee An, Gabriel C. Drummond-Cole, Ben Knudsen, Subdivisional Spaces and Graph Braid Groups. Doc. Math. 24 (2019), pp. 1513–1583

DOI 10.4171/DM/709