𝖥𝖨-hyperhomology and ordered configuration spaces

Miller, Jeremy; Wilson, Jennifer

doi:10.1090/proc/14772

$\mathsf {FI}$-hyperhomology and ordered configuration spaces
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by Jeremy Miller and Jennifer C. H. Wilson

Proc. Amer. Math. Soc. 148 (2020), 993-1002

DOI: https://doi.org/10.1090/proc/14772

Published electronically: September 20, 2019

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Abstract:

Using a result of Gan–Li on $\mathsf {FI}$-hyperhomology and a semisimplicial resolution of configuration spaces due to Randal-Williams, we establish an improved representation stability stable range for configuration spaces of distinct ordered points in a manifold. Our bounds on generation degree improve the best known stability slope by a factor of 5/2 in the most general case. We adapt this result of Gan–Li to apply beyond stability arguments involving highly connected simplicial complexes, and our methods suggest that their result may be widely applicable to improving most stability ranges for $\mathsf {FI}$-modules in the current representation stability literature.

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