Secondary homological stability for unordered configuration spaces

Himes, Zachary

doi:10.1090/tran/9015

Secondary homological stability for unordered configuration spaces
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by Zachary Himes;

Trans. Amer. Math. Soc. 377 (2024), 3173-3241

DOI: https://doi.org/10.1090/tran/9015

Published electronically: March 20, 2024

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Abstract:

Secondary homological stability is a recently discovered stability pattern for a sequence of spaces exhibiting homological stability and it holds outside the range where the homology stabilizes. We prove secondary homological stability for the homology of the unordered configuration spaces of a connected manifold. The main difficulty is the case that the manifold is closed because there are no obvious maps inducing stability and the homology eventually is periodic instead of stable. We resolve this issue by constructing a new chain-level stabilization map for configuration spaces.

References

Byung Hee An, Gabriel C. Drummond-Cole, and Ben Knudsen, Subdivisional spaces and graph braid groups, Doc. Math. 24 (2019), 1513–1583. MR 4033833, DOI 10.4171/dm/709
David Ayala and John Francis, Factorization homology of topological manifolds, J. Topol. 8 (2015), no. 4, 1045–1084. MR 3431668, DOI 10.1112/jtopol/jtv028
Martin Bendersky and Sam Gitler, The cohomology of certain function spaces, Trans. Amer. Math. Soc. 326 (1991), no. 1, 423–440. MR 1010881, DOI 10.1090/S0002-9947-1991-1010881-8
Martin Bendersky and Jeremy Miller, Localization and homological stability of configuration spaces, Q. J. Math. 65 (2014), no. 3, 807–815. MR 3261967, DOI 10.1093/qmath/hat039
Christin Bibby and Nir Gadish, A generating function approach to new representation stability phenomena in orbit configuration spaces, Trans. Amer. Math. Soc. Ser. B 10 (2023), 241–287. MR 4546785, DOI 10.1090/btran/130
C.-F. Bödigheimer, F. Cohen, and L. Taylor, On the homology of configuration spaces, Topology 28 (1989), no. 1, 111–123. MR 991102, DOI 10.1016/0040-9383(89)90035-9
Federico Cantero and Martin Palmer, On homological stability for configuration spaces on closed background manifolds, Doc. Math. 20 (2015), 753–805. MR 3398727, DOI 10.4171/dm/505
Gunnar Carlsson and R. James Milgram, Stable homotopy and iterated loop spaces, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 505–583. MR 1361898, DOI 10.1016/B978-044481779-2/50014-6
Thomas Church, Homological stability for configuration spaces of manifolds, Invent. Math. 188 (2012), no. 2, 465–504. MR 2909770, DOI 10.1007/s00222-011-0353-4
Frederick R. Cohen, Thomas J. Lada, and J. Peter May, The homology of iterated loop spaces, Lecture Notes in Mathematics, Vol. 533, Springer-Verlag, Berlin-New York, 1976. MR 436146, DOI 10.1007/BFb0080464
Daniel Dugger and Daniel C. Isaksen, Topological hypercovers and $\Bbb A^1$-realizations, Math. Z. 246 (2004), no. 4, 667–689. MR 2045835, DOI 10.1007/s00209-003-0607-y
Johannes Ebert and Oscar Randal-Williams, Semisimplicial spaces, Algebr. Geom. Topol. 19 (2019), no. 4, 2099–2150. MR 3995026, DOI 10.2140/agt.2019.19.2099
Edward Fadell and James Van Buskirk, The braid groups of $E^{2}$ and $S^{2}$, Duke Math. J. 29 (1962), 243–257. MR 141128
Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847, DOI 10.1007/978-1-4613-0105-9
Søren Galatius, Alexander Kupers, and Oscar Randal-Williams, $E_2$-cells and mapping class groups, Publ. Math. Inst. Hautes Études Sci. 130 (2019), 1–61. MR 4028513, DOI 10.1007/s10240-019-00107-8
Søren Galatius, Alexander Kupers, and Oscar Randal-Williams, Cellular $E_k$-algebras, arXiv:1805.07184 [math.AT].
Søren Galatius and Oscar Randal-Williams, Stable moduli spaces of high-dimensional manifolds, Acta Math. 212 (2014), no. 2, 257–377. MR 3207759, DOI 10.1007/s11511-014-0112-7
Søren Galatius and Oscar Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds. I, J. Amer. Math. Soc. 31 (2018), no. 1, 215–264. MR 3718454, DOI 10.1090/jams/884
Allen Hatcher, A short exposition of the Madsen-Weiss theorem, arXiv:1103.5223 [math.GT].
Richard Hepworth, On the edge of the stable range, Math. Ann. 377 (2020), no. 1-2, 123–181. MR 4099630, DOI 10.1007/s00208-020-01955-0
Quoc P. Ho, Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras, arXiv:2004.00252 [math.AT].
Birger Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986. MR 842190, DOI 10.1007/978-3-642-82783-9
Ben Knudsen, Betti numbers and stability for configuration spaces via factorization homology, Algebr. Geom. Topol. 17 (2017), no. 5, 3137–3187. MR 3704255, DOI 10.2140/agt.2017.17.3137
Alexander Kupers and Jeremy Miller, Improved homological stability for configuration spaces after inverting 2, Homology Homotopy Appl. 17 (2015), no. 1, 255–266. MR 3344444, DOI 10.4310/HHA.2015.v17.n1.a12
Alexander Kupers and Jeremy Miller, Sharper periodicity and stabilization maps for configuration spaces of closed manifolds, Proc. Amer. Math. Soc. 144 (2016), no. 12, 5457–5468. MR 3556286, DOI 10.1090/proc/13181
Alexander Kupers, Jeremy Miller, and Trithang Tran, Homological stability for symmetric complements, Trans. Amer. Math. Soc. 368 (2016), no. 11, 7745–7762. MR 3546782, DOI 10.1090/tran/6623
Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91–107. MR 358766, DOI 10.1016/0040-9383(75)90038-5
Jeremy Miller, Nonabelian Poincaré duality after stabilizing, Trans. Amer. Math. Soc. 367 (2015), no. 3, 1969–1991. MR 3286505, DOI 10.1090/S0002-9947-2014-06186-2
Jeremy Miller and Martin Palmer, Scanning for oriented configuration spaces, Homology Homotopy Appl. 17 (2015), no. 1, 35–66. MR 3338540, DOI 10.4310/HHA.2015.v17.n1.a2
Jeremy Miller, Peter Patzt, and Dan Petersen, Representation stability, secondary stability, and polynomial functors, arXiv:1910.05574 [math.AT].
Jeremy Miller and Jennifer C. H. Wilson, Higher-order representation stability and ordered configuration spaces of manifolds, Geom. Topol. 23 (2019), no. 5, 2519–2591. MR 4019898, DOI 10.2140/gt.2019.23.2519
Rohit Nagpal, FI-modules and the cohomology of modular representations of symmetric groups, ProQuest LLC, Ann Arbor, MI, 2015. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR 3358218
Rohit Nagpal and Andrew Snowden, Periodicity in the cohomology of symmetric groups via divided powers, Proc. Lond. Math. Soc. (3) 116 (2018), no. 5, 1244–1268. MR 3805056, DOI 10.1112/plms.12107
Oscar Randal-Williams, “Topological chiral homology” and configuration spaces of spheres, Morfismos 17 (2013), no. 2, 57–69.
Oscar Randal-Williams, Homological stability for unordered configuration spaces, Q. J. Math. 64 (2013), no. 1, 303–326. MR 3032101, DOI 10.1093/qmath/har033
Graeme Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39–72. MR 533892, DOI 10.1007/BF02392088
Michael Weiss, What does the classifying space of a category classify?, Homology Homotopy Appl. 7 (2005), no. 1, 185–195. MR 2175298, DOI 10.4310/HHA.2005.v7.n1.a10

Bibliographic Information

Zachary Himes
Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47906
ORCID: 0000-0001-8633-165X
Email: himesz@purdue.edu
Received by editor(s): March 27, 2021
Received by editor(s) in revised form: May 7, 2022, December 12, 2022, and June 11, 2023
Published electronically: March 20, 2024
Additional Notes: The author was supported in part by a Frederick N. Andrews Fellowship, Department of Mathematics, Purdue University and in part by the ERC under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 756444)
Journal: Trans. Amer. Math. Soc. 377 (2024), 3173-3241
DOI: https://doi.org/10.1090/tran/9015
MathSciNet review: 4744778