Mod 𝑝 homology of unordered configuration spaces of 𝑝 points in parallelizable surfaces

Chen, Matthew; Zhang, Adela

doi:10.1090/proc/16683

Mod $p$ homology of unordered configuration spaces of $p$ points in parallelizable surfaces
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by Matthew Chen and Adela YiYu Zhang;

Proc. Amer. Math. Soc. 152 (2024), 2239-2248

DOI: https://doi.org/10.1090/proc/16683

Published electronically: March 1, 2024

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Abstract:

We provide a short proof that the dimensions of the mod $p$ homology groups of the unordered configuration space $B_k(T)$ of $k$ points in a closed torus are the same as its Betti numbers for $p>2$ and $k\leq p$. Hence the integral homology has no $p$-power torsion in this range. The same argument works for the once-punctured genus $g$ surface with $g\geq 0$, thereby recovering a result of Brantner-Hahn-Knudsen via Lubin-Tate theory.

References

Omar Antolín Camarena, The $\textrm {mod}\,2$ homology of free spectral Lie algebras, Trans. Amer. Math. Soc. 373 (2020), no. 9, 6301–6319. MR 4155178, DOI 10.1090/tran/8131
Greg Arone and Mark Mahowald, The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres, Invent. Math. 135 (1999), no. 3, 743–788. MR 1669268, DOI 10.1007/s002220050300
Mark Behrens, The Goodwillie tower and the EHP sequence, Mem. Amer. Math. Soc. 218 (2012), no. 1026, xii+90. MR 2976788, DOI 10.1090/S0065-9266-2011-00645-3
C.-F. Bödigheimer and F. R. Cohen, Rational cohomology of configuration spaces of surfaces, Algebraic topology and transformation groups (Göttingen, 1987) Lecture Notes in Math., vol. 1361, Springer, Berlin, 1988, pp. 7–13. MR 979504, DOI 10.1007/BFb0083031
C.-F. Bödigheimer, F. R. Cohen, and R. J. Milgram, Truncated symmetric products and configuration spaces, Math. Z. 214 (1993), no. 2, 179–216. MR 1240884, DOI 10.1007/BF02572399
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
L. Brantner, J. Hahn and B. Knudsen, The Lubin-Tate theory of configuration spaces: I, Preprint, arXiv:1908.11321, 2019.
R. R. Bruner, J. P. May, J. E. McClure, and M. Steinberger, $H_\infty$ ring spectra and their applications, Lecture Notes in Mathematics, vol. 1176, Springer-Verlag, Berlin, 1986. MR 836132, DOI 10.1007/BFb0075405
Claude Chevalley and Samuel Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MR 24908, DOI 10.1090/S0002-9947-1948-0024908-8
Michael Ching, Bar constructions for topological operads and the Goodwillie derivatives of the identity, Geom. Topol. 9 (2005), 833–933. MR 2140994, DOI 10.2140/gt.2005.9.833
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
Gabriel C. Drummond-Cole and Ben Knudsen, Betti numbers of configuration spaces of surfaces, J. Lond. Math. Soc. (2) 96 (2017), no. 2, 367–393. MR 3708955, DOI 10.1112/jlms.12066
Yves Félix and Jean-Claude Thomas, Rational Betti numbers of configuration spaces, Topology Appl. 102 (2000), no. 2, 139–149. MR 1741482, DOI 10.1016/S0166-8641(98)00148-5
Brenda Johnson, The derivatives of homotopy theory, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1295–1321. MR 1297532, DOI 10.1090/S0002-9947-1995-1297532-6
Jens Jakob Kjaer, On the odd primary homology of free algebras over the spectral lie operad, J. Homotopy Relat. Struct. 13 (2018), no. 3, 581–597. MR 3856302, DOI 10.1007/s40062-017-0194-y
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
Ben Knudsen, Higher enveloping algebras, Geom. Topol. 22 (2018), no. 7, 4013–4066. MR 3890770, DOI 10.2140/gt.2018.22.4013
N. Konovalov, Algebraic Goodwillie spectral sequence, Preprint, arXiv:2303.06240, 2023.
Igor Kříž, On the rational homotopy type of configuration spaces, Ann. of Math. (2) 139 (1994), no. 2, 227–237. MR 1274092, DOI 10.2307/2946581
J. P. May, The cohomology of restricted Lie algebras and of Hopf algebras, J. Algebra 3 (1966), 123–146. MR 193126, DOI 10.1016/0021-8693(66)90009-3
J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin-New York, 1972. MR 420610, DOI 10.1007/BFb0067491
Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91–107. MR 358766, DOI 10.1016/0040-9383(75)90038-5
R. James Milgram and Peter Löffler, The structure of deleted symmetric products, Braids (Santa Cruz, CA, 1986) Contemp. Math., vol. 78, Amer. Math. Soc., Providence, RI, 1988, pp. 415–424. MR 975092, DOI 10.1090/conm/078/975092
Stewart B. Priddy, Koszul resolutions, Trans. Amer. Math. Soc. 152 (1970), 39–60. MR 265437, DOI 10.1090/S0002-9947-1970-0265437-8
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. MR 331377, DOI 10.1007/BF01390197
Burt Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996), no. 4, 1057–1067. MR 1404924, DOI 10.1016/0040-9383(95)00058-5
A. Y. Zhang, Quillen homology of spectral Lie algebras with application to mod $p$ homology of labeled configuration spaces, Preprint, arXiv:2110.08428, 2021.