Sharper periodicity and stabilization maps for configuration spaces of closed manifolds

Kupers, Alexander; Miller, Jeremy

doi:10.1090/proc/13181

Sharper periodicity and stabilization maps for configuration spaces of closed manifolds
HTML articles powered by AMS MathViewer

by Alexander Kupers and Jeremy Miller PDF

Proc. Amer. Math. Soc. 144 (2016), 5457-5468

Abstract:

In this note we study the homology of configuration spaces of closed manifolds. We sharpen the eventual periodicity results of Nagpal and Cantero-Palmer, prove integral homological stability for configuration spaces of odd-dimensional manifolds and introduce a stabilization map on the homology with $\mathbb {Z}[1/2]$-coefficients of configuration spaces of odd-dimensional manifolds.

References

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
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
Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706, DOI 10.1007/978-1-4612-0647-7
William Browder, Homology operations and loop spaces, Illinois J. Math. 4 (1960), 347–357. MR 120646
Federico Cantero and Martin Palmer, On homological stability for configuration spaces on closed background manifolds, Doc. Math. 20 (2015), 753–805. MR 3398727
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
Thomas Church, Jordan S. Ellenberg, and Benson Farb, FI-modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015), no. 9, 1833–1910. MR 3357185, DOI 10.1215/00127094-3120274
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 0436146
Albrecht Dold, Decomposition theorems for $S(n)$-complexes, Ann. of Math. (2) 75 (1962), 8–16. MR 137113, DOI 10.2307/1970415
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 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
Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
Ben Knudsen, Betti numbers and stability for configuration spaces via factorization homology, preprint (2014), http://arxiv.org/abs/1405.6696.
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, Jeremy Miller, and TriThang Tran, Homological stability for symmetric complements, To appear in Transactions of the American Mathematical Society (preprint 2014).
J. P. May and K. Ponto, More concise algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2012. Localization, completion, and model categories. MR 2884233
Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91–107. MR 358766, DOI 10.1016/0040-9383(75)90038-5
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
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
Burt Totaro, Configuration spaces of algebraic varieties, Topology 35 (1996), no. 4, 1057–1067. MR 1404924, DOI 10.1016/0040-9383(95)00058-5