Parametrized spectra, multiplicative Thom spectra and the twisted Umkehr map

Ando, Matthew¹ ; Blumberg, Andrew² ; Gepner, David³

¹ Department of Mathematics, The University of Illinois at Urbana-Champaign, Urbana, IL, United States
² Department of Mathematics, University of Texas at Austin, Austin, TX, United States
³ Department of Mathematics, Purdue University, West Lafayette, IN, United States

Geometry & topology, Tome 22 (2018) no. 7, pp. 3761-3825.

Résumé

We introduce a general theory of parametrized objects in the setting of $ß$–categories. Although parametrised spaces and spectra are the most familiar examples, we establish our theory in the generality of families of objects of a presentable $\infty$–category parametrized over objects of an $\infty$–topos. We obtain a coherent functor formalism describing the relationship of the various adjoint functors associated to base-change and symmetric monoidal structures.

Our main applications are to the study of generalized Thom spectra. We obtain fiberwise constructions of twisted Umkehr maps for twisted generalized cohomology theories using a geometric fiberwise construction of Atiyah duality. In order to characterize the algebraic structures on generalized Thom spectra and twisted (co)homology, we express the generalized Thom spectrum as a categorification of the well-known adjunction between units and group rings.

Détail

DOI : 10.2140/gt.2018.22.3761

Classification : 55P99, 55R70
Keywords: Thom spectrum, parametrized spectra, twisted Umkehr map

Bibliographie

[1] M Ando, The sigma orientation for analytic circle-equivariant elliptic cohomology, Geom. Topol. 7 (2003) 91 | DOI

[2] M Ando, A J Blumberg, D Gepner, Twists of K–theory and TMF, from: "Superstrings, geometry, topology, and –algebras" (editors R S Doran, G Friedman, J Rosenberg), Proc. Sympos. Pure Math. 81, Amer. Math. Soc. (2010) 27 | DOI

[3] M Ando, A J Blumberg, D J Gepner, M J Hopkins, C Rezk, Units of ring spectra and Thom spectra, preprint (2009)

[4] M Ando, A J Blumberg, D Gepner, M J Hopkins, C Rezk, An ∞–categorical approach to R–line bundles, R–module Thom spectra, and twisted R–homology, J. Topol. 7 (2014) 869 | DOI

[5] M Ando, A J Blumberg, D Gepner, M J Hopkins, C Rezk, Units of ring spectra, orientations and Thom spectra via rigid infinite loop space theory, J. Topol. 7 (2014) 1077 | DOI

[6] B Antieau, D Gepner, Brauer groups and étale cohomology in derived algebraic geometry, Geom. Topol. 18 (2014) 1149 | DOI

[7] B Antieau, D Gepner, J M Gómez, Actions of K(π,n) spaces on K–theory and uniqueness of twisted K–theory, Trans. Amer. Math. Soc. 366 (2014) 3631 | DOI

[8] M F Atiyah, I M Singer, The index of elliptic operators, IV, Ann. of Math. 93 (1971) 119 | DOI

[9] J Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique, II, 315, Soc. Math. France (2007)

[10] C Barwick, S Glasman, D Nardin, Dualizing cartesian and cocartesian fibrations, Theory Appl. Categ. 33 (2018) 67

[11] J C Becker, D H Gottlieb, Transfer maps for fibrations and duality, Compositio Math. 33 (1976) 107

[12] J E Bergner, A model category structure on the category of simplicial categories, Trans. Amer. Math. Soc. 359 (2007) 2043 | DOI

[13] J E Bergner, A survey of (∞,1)–categories, from: "Towards higher categories" (editors J C Baez, J P May), IMA Vol. Math. Appl. 152, Springer (2010) 69 | DOI

[14] G E Bredon, Topology and geometry, 139, Springer (1997) | DOI

[15] A L Carey, B L Wang, Thom isomorphism and push-forward map in twisted K–theory, J. K–Theory 1 (2008) 357 | DOI

[16] D C Cisinski, F Déglise, Triangulated categories of mixed motives, preprint (2009)

[17] B Conrad, Grothendieck duality and base change, 1750, Springer (2000) | DOI

[18] C L Douglas, Twisted parametrized stable homotopy theory, preprint (2005)

[19] D Dugger, Combinatorial model categories have presentations, Adv. Math. 164 (2001) 177 | DOI

[20] W G Dwyer, Transfer maps for fibrations, Math. Proc. Cambridge Philos. Soc. 120 (1996) 221 | DOI

[21] W G Dwyer, D M Kan, Simplicial localizations of categories, J. Pure Appl. Algebra 17 (1980) 267 | DOI

[22] H Fausk, P Hu, J P May, Isomorphisms between left and right adjoints, Theory Appl. Categ. 11 (2003) 107

[23] D S Freed, C Teleman, Relative quantum field theory, Comm. Math. Phys. 326 (2014) 459 | DOI

[24] D S Freed, E Witten, Anomalies in string theory with D–branes, Asian J. Math. 3 (1999) 819 | DOI

[25] N Ganter, The elliptic Weyl character formula, Compos. Math. 150 (2014) 1196 | DOI

[26] D Gepner, M Groth, T Nikolaus, Universality of multiplicative infinite loop space machines, Algebr. Geom. Topol. 15 (2015) 3107 | DOI

[27] D Gepner, J Kock, Univalence in locally cartesian closed ∞–categories, Forum Math. 29 (2017) 617 | DOI

[28] D Gepner, T Lawson, Brauer groups and Galois cohomology of commutative ring spectra, preprint (2016)

[29] R Hartshorne, Residues and duality, 20, Springer (1966) | DOI

[30] A Hirschowitz, C Simpson, Descente pour les n–champs, preprint (1998)

[31] M Hovey, Spectra and symmetric spectra in general model categories, J. Pure Appl. Algebra 165 (2001) 63 | DOI

[32] P Hu, Duality for smooth families in equivariant stable homotopy theory, 285, Soc. Math. France (2003)

[33] T Johnson-Freyd, C Scheimbauer, (Op)lax natural transformations, twisted quantum field theories, and “even higher” Morita categories, Adv. Math. 307 (2017) 147 | DOI

[34] A Joyal, Quasi-categories and Kan complexes, J. Pure Appl. Algebra 175 (2002) 207 | DOI

[35] L G Lewis Jr., J P May, M Steinberger, J E Mcclure, Equivariant stable homotopy theory, 1213, Springer (1986) | DOI

[36] J Lurie, Higher topos theory, 170, Princeton Univ. Press (2009) | DOI

[37] J Lurie, Higher algebra, book project (2012)

[38] M A Mandell, J P May, Equivariant orthogonal spectra and S–modules, 755, Amer. Math. Soc. (2002) | DOI

[39] M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. 82 (2001) 441 | DOI

[40] J P May, Picard groups, Grothendieck rings, and Burnside rings of categories, Adv. Math. 163 (2001) 1 | DOI

[41] J P May, J Sigurdsson, Parametrized homotopy theory, 132, Amer. Math. Soc. (2006) | DOI

[42] J W Milnor, J D Stasheff, Characteristic classes, 76, Princeton Univ. Press (1974)

[43] R Minasian, G Moore, K–theory and Ramond–Ramond charge, J. High Energy Phys. (1997) | DOI

[44] C Rezk, A model for the homotopy theory of homotopy theory, Trans. Amer. Math. Soc. 353 (2001) 973 | DOI

[45] I Rosu, Equivariant elliptic cohomology and rigidity, Amer. J. Math. 123 (2001) 647 | DOI

[46] S Stolz, P Teichner, Supersymmetric field theories and generalized cohomology, from: "Mathematical foundations of quantum field theory and perturbative string theory" (editors H Sati, U Schreiber), Proc. Sympos. Pure Math. 83, Amer. Math. Soc. (2011) 279 | DOI

[47] Z Tamsamani, On non-strict notions of n–category and n–groupoid via multisimplicial sets, preprint (1995)

[48] V Voevodsky, Lectures on cross functors, preprint (2001)

[49] B L Wang, Geometric cycles, index theory and twisted K–homology, J. Noncommut. Geom. 2 (2008) 497 | DOI

[50] C Westerland, A higher chromatic analogue of the image of J, Geom. Topol. 21 (2017) 1033 | DOI

[51] E Witten, Chern–Simons gauge theory as a string theory, from: "The Floer memorial volume" (editors H Hofer, C H Taubes, A Weinstein, E Zehnder), Progr. Math. 133, Birkhäuser (1995) 637 | DOI

[52] E Witten, D–branes and K–theory, J. High Energy Phys. (1998) | DOI