Euler characteristics of finite homotopy colimits

In this note, we provide a calculation of the Euler characteristic of a finite homotopy colimit of finite cell complexes, which depends only on the Euler characteristics of each space and resembles Mobius inversion. Versions of the result are known when the colimit is indexed by categories with various finiteness conditions, but the behavior is more uniform when we index by a finite quasicategory instead. The formula simultaneously generalizes the additive formula for Euler characteristic of a homotopy pushout and the multiplicative formula for Euler characteristic of a fiber bundle.

Keywords

Euler characteristic, quasicategory, Mobius function

2010 Mathematics Subject Classification

55M99, 55U10

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The author was supported by an NSF Postdoctoral Fellowship under grant 1803089.

Received 5 September 2019

Received revised 19 September 2019

Accepted 1 October 2019

Published 6 May 2020