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 \textit{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 \(X\) 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. \par 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.

20F36, 55R80, 55U05, 05C10

configuration spaces, cell complexes, subdivisional spaces, graphs, braid groups

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

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

Mathematics Department, Northeastern University, Boston 02115, USA