Nonabelian Poincaré duality after stabilizing

Miller, Jeremy

doi:10.1090/S0002-9947-2014-06186-2

Nonabelian Poincaré duality after stabilizing
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Trans. Amer. Math. Soc. 367 (2015), 1969-1991 Request permission

Abstract:

We generalize the nonabelian Poincaré duality theorems of Salvatore and Lurie to the case of not necessarily grouplike $E_n$-algebras (in the category of spaces). We define a stabilization procedure based on McDuff’s “bringing points in from infinity” maps. For open connected parallelizable $n$-manifolds, we prove that, after stabilizing, the topological chiral homology of $M$ with coefficients in an $E_n$-algebra $A$, $\int _M A$, is homology equivalent to $Map^c(M,B^n A)$, the space of compactly supported maps to the $n$-fold classifying space of $A$.

References

Ricardo Andrade, From manifolds to invariants of En-algebras, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 2898647
C.-F. Bödigheimer, Stable splittings of mapping spaces, Algebraic topology (Seattle, Wash., 1985) Lecture Notes in Math., vol. 1286, Springer, Berlin, 1987, pp. 174–187. MR 922926, DOI 10.1007/BFb0078741
Charles P. Boyer and Benjamin M. Mann, Monopoles, nonlinear $\sigma$ models, and two-fold loop spaces, Comm. Math. Phys. 115 (1988), no. 4, 571–594. MR 933456
Albrecht Dold and René Thom, Quasifaserungen und unendliche symmetrische Produkte, Ann. of Math. (2) 67 (1958), 239–281 (German). MR 97062, DOI 10.2307/1970005
John Francis, The tangent complex and Hochschild cohomology of $\scr E_n$-rings, Compos. Math. 149 (2013), no. 3, 430–480. MR 3040746, DOI 10.1112/S0010437X12000140
Ezra Getzler and Jones John, Operads, homotopy algebra and iterated integrals for double loop spaces, arXiv:hep-th/9403055 (1994).
John L. Harer, Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2) 121 (1985), no. 2, 215–249. MR 786348, DOI 10.2307/1971172
Lars Hesselholt and Ib Madsen, On the $K$-theory of finite algebras over Witt vectors of perfect fields, Topology 36 (1997), no. 1, 29–101. MR 1410465, DOI 10.1016/0040-9383(96)00003-1
Sadok Kallel, Spaces of particles on manifolds and generalized Poincaré dualities, Q. J. Math. 52 (2001), no. 1, 45–70. MR 1820902, DOI 10.1093/qjmath/52.1.45
Jacob Lurie, On the classification of topological field theories, Current developments in mathematics, 2008, Int. Press, Somerville, MA, 2009, pp. 129–280. MR 2555928
J. P. May, The geometry of iterated loop spaces, Lecture Notes in Mathematics, Vol. 271, Springer-Verlag, Berlin-New York, 1972. MR 0420610
Dusa McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975), 91–107. MR 358766, DOI 10.1016/0040-9383(75)90038-5
D. McDuff and G. Segal, Homology fibrations and the “group-completion” theorem, Invent. Math. 31 (1975/76), no. 3, 279–284. MR 402733, DOI 10.1007/BF01403148
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
Paolo Salvatore, Configuration spaces with summable labels, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 375–395. MR 1851264
Graeme Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112. MR 232393
Graeme Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39–72. MR 533892, DOI 10.1007/BF02392088