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$ \mathsf{FI}$-hyperhomology and ordered configuration spaces


Authors: Jeremy Miller and Jennifer C. H. Wilson
Journal: Proc. Amer. Math. Soc. 148 (2020), 993-1002
MSC (2010): Primary 18A25, 18G40, 55R40, 55R80, 55U10
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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Additional Information

Jeremy Miller
Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907
Email: jeremykmiller@purdue.edu

Jennifer C. H. Wilson
Affiliation: Department of Mathematics, East Hall 2074, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109
Email: jchw@umich.edu

DOI: https://doi.org/10.1090/proc/14772
Received by editor(s): March 11, 2019
Received by editor(s) in revised form: July 8, 2019
Published electronically: September 20, 2019
Additional Notes: The first author was supported in part by NSF grant DMS-1709726.
Communicated by: Mark Behrens
Article copyright: © Copyright 2019 American Mathematical Society