## Algebraic & Geometric Topology

### The factorization theory of Thom spectra and twisted nonabelian Poincaré duality

Inbar Klang

#### Abstract

We give a description of the factorization homology and $E n$ topological Hochschild cohomology of Thom spectra arising from $n$–fold loop maps $f : A → B O$, where $A = Ω n X$ is an $n$–fold loop space. We describe the factorization homology $∫ M Th ( f )$ as the Thom spectrum associated to a certain map $∫ M A → B O$, where $∫ M A$ is the factorization homology of $M$ with coefficients in $A$. When $M$ is framed and $X$ is $( n − 1 )$–connected, this spectrum is equivalent to a Thom spectrum of a virtual bundle over the mapping space $Map c ( M , X )$; in general, this is a Thom spectrum of a virtual bundle over a certain section space. This can be viewed as a twisted form of the nonabelian Poincaré duality theorem of Segal, Salvatore and Lurie, which occurs when $f : A → B O$ is nullhomotopic. This result also generalizes the results of Blumberg, Cohen and Schlichtkrull on the topological Hochschild homology of Thom spectra, and of Schlichtkrull on higher topological Hochschild homology of Thom spectra. We use this description of the factorization homology of Thom spectra to calculate the factorization homology of the classical cobordism spectra, spectra arising from systems of groups and the Eilenberg–Mac Lane spectra $H ℤ ∕ p$, $H ℤ ( p )$ and $H ℤ$. We build upon the description of the factorization homology of Thom spectra to study the ($n = 1$ and higher) topological Hochschild cohomology of Thom spectra, which enables calculations and a description in terms of sections of a parametrized spectrum. If $X$ is a closed manifold, Atiyah duality for parametrized spectra allows us to deduce a duality between $E n$ topological Hochschild homology and $E n$ topological Hochschild cohomology, recovering string topology operations when $f$ is nullhomotopic. In conjunction with the higher Deligne conjecture, this gives $E n + 1$–structures on a certain family of Thom spectra, which were not previously known to be ring spectra.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 5 (2018), 2541-2592.

Dates
Revised: 2 May 2018
Accepted: 20 May 2018
First available in Project Euclid: 30 August 2018

https://projecteuclid.org/euclid.agt/1535594417

Digital Object Identifier
doi:10.2140/agt.2018.18.2541

Mathematical Reviews number (MathSciNet)
MR3848394

Zentralblatt MATH identifier
06935815

#### Citation

Klang, Inbar. The factorization theory of Thom spectra and twisted nonabelian Poincaré duality. Algebr. Geom. Topol. 18 (2018), no. 5, 2541--2592. doi:10.2140/agt.2018.18.2541. https://projecteuclid.org/euclid.agt/1535594417

#### References

• M Ando, A J Blumberg, D Gepner, Parametrized spectra, multiplicative Thom spectra, and the twisted Umkehr map, preprint (2011)
• M Ando, A J Blumberg, D Gepner, M J Hopkins, C Rezk, An $\infty$–categorical approach to $R$–line bundles, $R$–module Thom spectra, and twisted $R$–homology, J. Topol. 7 (2014) 869–893
• 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) 1077–1117
• D Ayala, J Francis, Zero-pointed manifolds, preprint (2014)
• D Ayala, J Francis, Factorization homology of topological manifolds, J. Topol. 8 (2015) 1045–1084
• M Barratt, S Priddy, On the homology of non-connected monoids and their associated groups, Comment. Math. Helv. 47 (1972) 1–14
• J Beardsley, Relative Thom spectra via operadic Kan extensions, Algebr. Geom. Topol. 17 (2017) 1151–1162
• J Beardsley, Topological Hochschild homology of $X(n)$, preprint (2017)
• A J Blumberg, Topological Hochschild homology of Thom spectra which are $E_\infty$ ring spectra, J. Topol. 3 (2010) 535–560
• A J Blumberg, R L Cohen, C Schlichtkrull, Topological Hochschild homology of Thom spectra and the free loop space, Geom. Topol. 14 (2010) 1165–1242
• I Bobkova, A Lindenstrauss, K Poirier, B Richter, I Zakharevich, On the higher topological Hochschild homology of $\mathbb F_p$ and commutative $\mathbb F_p$–group algebras, from “Women in topology: collaborations in homotopy theory” (M Basterra, K Bauer, K Hess, B Johnson, editors), Contemp. Math. 641, Amer. Math. Soc., Providence, RI (2015) 97–122
• F R Cohen, Braid orientations and bundles with flat connections, Invent. Math. 46 (1978) 99–110
• F R Cohen, J P May, L R Taylor, $K({\bf Z},0)$ and $K(Z\sb{2},0)$ as Thom spectra, Illinois J. Math. 25 (1981) 99–106
• R L Cohen, Stable proofs of stable splittings, Math. Proc. Cambridge Philos. Soc. 88 (1980) 149–151
• R L Cohen, J D S Jones, A homotopy theoretic realization of string topology, Math. Ann. 324 (2002) 773–798
• R L Cohen, J D S Jones, Gauge theory and string topology, Bol. Soc. Mat. Mex. 23 (2017) 233–255
• E S Devinatz, M J Hopkins, J H Smith, Nilpotence and stable homotopy theory, I, Ann. of Math. 128 (1988) 207–241
• B I Dundas, A Lindenstrauss, B Richter, Towards an understanding of ramified extensions of structured ring spectra, Math. Proc. Cambridge Philos. Soc. (online publication March 2018)
• A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, Amer. Math. Soc., Providence, RI (1997)
• J Francis, The tangent complex and Hochschild cohomology of $\mathscr E_n$–rings, Compos. Math. 149 (2013) 430–480
• V Franjou, J Lannes, L Schwartz, Autour de la cohomologie de Mac Lane des corps finis, Invent. Math. 115 (1994) 513–538
• V Franjou, T Pirashvili, On the Mac Lane cohomology for the ring of integers, Topology 37 (1998) 109–114
• D Gepner, M Groth, T Nikolaus, Universality of multiplicative infinite loop space machines, Algebr. Geom. Topol. 15 (2015) 3107–3153
• G Ginot, T Tradler, M Zeinalian, Higher Hochschild cohomology, Brane topology and centralizers of $E_n$–algebra maps, preprint (2012)
• G Ginot, T Tradler, M Zeinalian, Higher Hochschild homology, topological chiral homology and factorization algebras, Comm. Math. Phys. 326 (2014) 635–686
• K Gruher, P Salvatore, Generalized string topology operations, Proc. Lond. Math. Soc. 96 (2008) 78–106
• K Hess, P-E Parent, J Scott, CoHochschild homology of chain coalgebras, J. Pure Appl. Algebra 213 (2009) 536–556
• G Horel, Operads, modules and topological field theories, preprint (2014)
• G Horel, Higher Hochschild cohomology of the Lubin–Tate ring spectrum, Algebr. Geom. Topol. 15 (2015) 3215–3252
• G Horel, Factorization homology and calculus à la Kontsevich Soibelman, J. Noncommut. Geom. 11 (2017) 703–740
• P Hu, Higher string topology on general spaces, Proc. London Math. Soc. 93 (2006) 515–544
• P Hu, I Kriz, A A Voronov, On Kontsevich's Hochschild cohomology conjecture, Compos. Math. 142 (2006) 143–168
• B Knudsen, Betti numbers and stability for configuration spaces via factorization homology, Algebr. Geom. Topol. 17 (2017) 3137–3187
• A Kupers, J Miller, $E_n$–cell attachments and a local-to-global principle for homological stability, Math. Ann. 370 (2018) 209–269
• L G Lewis, Jr, J P May, M Steinberger, Equivariant stable homotopy theory, Lecture Notes in Mathematics 1213, Springer (1986)
• J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009)
• J Lurie, Higher algebra, book project (2017) \setbox0\makeatletter\@url http://www.math.harvard.edu/~lurie/papers/HA.pdf {\unhbox0
• M Mahowald, D C Ravenel, P Shick, The Thomified Eilenberg–Moore spectral sequence, from “Cohomological methods in homotopy theory” (J Aguadé, C Broto, C Casacuberta, editors), Progr. Math. 196, Birkhäuser, Basel (2001) 249–262
• E J Malm, String topology and the based loop space, preprint (2011)
• J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer (1972)
• J P May, What precisely are $E_\infty$ ring spaces and $E_\infty$ ring spectra?, from “New topological contexts for Galois theory and algebraic geometry” (A Baker, B Richter, editors), Geom. Topol. Monogr. 16, Geom. Topol. Publ., Coventry (2009) 215–282
• D McDuff, Configuration spaces of positive and negative particles, Topology 14 (1975) 91–107
• J Miller, Nonabelian Poincaré duality after stabilizing, Trans. Amer. Math. Soc. 367 (2015) 1969–1991
• S Priddy, $K({\bf Z}/2)$ as a Thom spectrum, Proc. Amer. Math. Soc. 70 (1978) 207–208
• D C Ravenel, Complex cobordism and stable homotopy groups of spheres, 2nd edition, AMS Chelsea Ser. 347, Amer. Math. Soc., Providence, RI (2004)
• P Salvatore, Configuration spaces with summable labels, from “Cohomological methods in homotopy theory” (J Aguadé, C Broto, C Casacuberta, editors), Progr. Math. 196, Birkhäuser, Basel (2001) 375–395
• P Salvatore, Configuration spaces on the sphere and higher loop spaces, Math. Z. 248 (2004) 527–540
• P Salvatore, N Wahl, Framed discs operads and Batalin–Vilkovisky algebras, Q. J. Math. 54 (2003) 213–231
• C Schlichtkrull, Units of ring spectra and their traces in algebraic $K$–theory, Geom. Topol. 8 (2004) 645–673
• C Schlichtkrull, Higher topological Hochschild homology of Thom spectra, J. Topol. 4 (2011) 161–189
• S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. 80 (2000) 491–511
• G Segal, Locality of holomorphic bundles, and locality in quantum field theory, from “The many facets of geometry” (O García-Prada, J P Bourguignon, S Salamon, editors), Oxford Univ. Press (2010) 164–176
• T Veen, Detecting periodic elements in higher topological Hochschild homology, Geom. Topol. 22 (2018) 693–756