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Homological stability for symmetric complements


Authors: Alexander Kupers, Jeremy Miller and TriThang Tran
Journal: Trans. Amer. Math. Soc. 368 (2016), 7745-7762
MSC (2010): Primary 55R80; Secondary 55R40
DOI: https://doi.org/10.1090/tran/6623
Published electronically: December 2, 2015
MathSciNet review: 3546782
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Abstract: A conjecture of Vakil and Wood (2015) states that the complements of closures of certain strata of the symmetric power of a smooth irreducible complex variety exhibit rational homological stability. We prove a generalization of this conjecture to the case of connected manifolds of dimension at least 2 and give an explicit homological stability range.


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  • [Arn70] V. I. Arnold, Certain topological invariants of algebrac functions, Trudy Moskov. Mat. Obšč. 21 (1970), 27-46 (Russian). MR 0274462 (43 #225)
  • [Bre97] Glen E. Bredon, Sheaf theory, 2nd ed., Graduate Texts in Mathematics, vol. 170, Springer-Verlag, New York, 1997. MR 1481706 (98g:55005)
  • [CDG13] Gaël Collinet, Aurélien Djament, and James T. Griffin, Stabilité homologique pour les groupes d'automorphismes des produits libres, Int. Math. Res. Not. IMRN (2013), no. 19, 4451-4476. MR 3116169
  • [Chu12] Thomas Church, Homological stability for configuration spaces of manifolds, Invent. Math. 188 (2012), no. 2, 465-504. MR 2909770, https://doi.org/10.1007/s00222-011-0353-4
  • [DK01] James F. Davis and Paul Kirk, Lecture notes in algebraic topology, Graduate Studies in Mathematics, vol. 35, American Mathematical Society, Providence, RI, 2001. MR 1841974 (2002f:55001)
  • [Dol62] Albrecht Dold, Decomposition theorems for $ S(n)$-complexes, Ann. of Math. (2) 75 (1962), 8-16. MR 0137113 (25 #569)
  • [FVB62] Edward Fadell and James Van Buskirk, The braid groups of $ E^{2}$ and $ S^{2}$, Duke Math. J. 29 (1962), 243-257. MR 0141128 (25 #4539)
  • [Ive86] Birger Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986. MR 842190 (87m:14013)
  • [Kal01] Sadok Kallel, Configuration spaces and the topology of curves in projective space, Topology, geometry, and algebra: interactions and new directions (Stanford, CA, 1999) Contemp. Math., vol. 279, Amer. Math. Soc., Providence, RI, 2001, pp. 151-175. MR 1850746 (2003c:55020), https://doi.org/10.1090/conm/279/04559
  • [KM13a] Alexander Kupers and Jeremy Miller, Homological stability for complements of closures, http://arxiv.org/abs/1312.6424.
  • [KM13b] -, Homological stability for topological chiral homology of completions, http://arxiv.org/abs/1311.5203.
  • [KM14] Alexander Kupers and Jeremy Miller, Some stable homology calculations and Occam's razor for Hodge structures, J. Pure Appl. Algebra 218 (2014), no. 7, 1219-1223. MR 3168493, https://doi.org/10.1016/j.jpaa.2013.11.015
  • [Knu14] Ben Knudsen, Betti numbers and stability for configuration spaces via factorization homology, http://arxiv.org/abs/1405.6696v4.
  • [McD75] Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91-107. MR 0358766 (50 #11225)
  • [MS93] I. Moerdijk and J.-A. Svensson, The equivariant Serre spectral sequence, Proc. Amer. Math. Soc. 118 (1993), no. 1, 263-278. MR 1123662 (93f:55023), https://doi.org/10.2307/2160037
  • [RW13] Oscar Randal-Williams, Homological stability for unordered configuration spaces, Q. J. Math. 64 (2013), no. 1, 303-326. MR 3032101, https://doi.org/10.1093/qmath/har033
  • [Seg73] Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213-221. MR 0331377 (48 #9710)
  • [Seg79] Graeme Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39-72. MR 533892 (81c:55013), https://doi.org/10.1007/BF02392088
  • [Tom14] Orsola Tommasi, Stable cohomology of spaces of non-singular hypersurfaces, Adv. Math. 265 (2014), 428-440. MR 3255466, https://doi.org/10.1016/j.aim.2014.08.005
  • [Tra13] Trithang Tran, Homological stability for coloured configuration spaces and symmetric complements, http://arxiv.org/abs/1312.6327.
  • [VW15] Ravi Vakil and Melanie Matchett Wood, Discriminants in the Grothendieck ring, Duke Math. J. 164 (2015), no. 6, 1139-1185. MR 3336842, https://doi.org/10.1215/00127094-2877184
  • [Wei05] Michael Weiss, What does the classifying space of a category classify?, Homology Homotopy Appl. 7 (2005), no. 1, 185-195. MR 2175298 (2007d:57059)

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Additional Information

Alexander Kupers
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305-2125

Jeremy Miller
Affiliation: Mathematics PhD Program, CUNY Graduate Center, New York, New York 10016-4309
Address at time of publication: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907-2067

TriThang Tran
Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

DOI: https://doi.org/10.1090/tran/6623
Received by editor(s): July 14, 2014
Received by editor(s) in revised form: October 20, 2014, and November 12, 2014
Published electronically: December 2, 2015
Additional Notes: The first author was supported by a William R. Hewlett Stanford Graduate Fellowship, Department of Mathematics, Stanford University, and was partially supported by NSF grant DMS-1105058.
Article copyright: © Copyright 2015 American Mathematical Society

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