Algebraic & Geometric Topology

Vanishing theorems for representation homology and the derived cotangent complex

Yuri Berest, Ajay C Ramadoss, and Wai-kit Yeung

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Abstract

Let G be a reductive affine algebraic group defined over a field k of characteristic zero. We study the cotangent complex of the derived G –representation scheme DRep G ( X ) of a pointed connected topological space X . We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of DRep G ( X ) to the representation homology HR ( X , G ) : = π O [ DRep G ( X ) ] to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in 3 and generalized lens spaces. In particular, for any finitely generated virtually free group Γ , we show that HR i ( B Γ , G ) = 0 for all i > 0 . For a closed Riemann surface Σ g of genus g 1 , we have HR i ( Σ g , G ) = 0 for all i > dim G . The sharp vanishing bounds for Σ g actually depend on the genus: we conjecture that if g = 1 , then HR i ( Σ g , G ) = 0 for i > r a n k G , and if g 2 , then HR i ( Σ g , G ) = 0 for i > dim Z ( G ) , where Z ( G ) is the center of G . We prove these bounds locally on the smooth locus of the representation scheme Rep G [ π 1 ( Σ g ) ] in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined K –theoretic virtual fundamental class for DRep G ( X ) in the sense of Ciocan-Fontanine and Kapranov (Geom. Topol. 13 (2009) 1779–1804). We give a new “Tor formula” for this class in terms of functor homology.

Article information

Source
Algebr. Geom. Topol., Volume 19, Number 1 (2019), 281-339.

Dates
Received: 15 January 2018
Revised: 20 August 2018
Accepted: 2 September 2018
First available in Project Euclid: 12 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.agt/1549940434

Digital Object Identifier
doi:10.2140/agt.2019.19.281

Mathematical Reviews number (MathSciNet)
MR3910582

Zentralblatt MATH identifier
07053575

Subjects
Primary: 14A20: Generalizations (algebraic spaces, stacks) 14D20: Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14L24: Geometric invariant theory [See also 13A50] 18G55: Homotopical algebra 57M07: Topological methods in group theory
Secondary: 14F17: Vanishing theorems [See also 32L20] 14F35: Homotopy theory; fundamental groups [See also 14H30]

Keywords
representation variety representation homology cotangent complex derived moduli spaces

Citation

Berest, Yuri; Ramadoss, Ajay C; Yeung, Wai-kit. Vanishing theorems for representation homology and the derived cotangent complex. Algebr. Geom. Topol. 19 (2019), no. 1, 281--339. doi:10.2140/agt.2019.19.281. https://projecteuclid.org/euclid.agt/1549940434


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References

  • M André, Homologie des algèbres commutatives, Grundl. Math. Wissen. 206, Springer (1974)
  • Y Berest, G Felder, S Patotski, A C Ramadoss, T Willwacher, Representation homology, Lie algebra cohomology and the derived Harish-Chandra homomorphism, J. Eur. Math. Soc. 19 (2017) 2811–2893
  • Y Berest, G Felder, A Ramadoss, Derived representation schemes and noncommutative geometry, from “Expository lectures on representation theory” (K Igusa, A Martsinkovsky, G Todorov, editors), Contemp. Math. 607, Amer. Math. Soc., Providence, RI (2014) 113–162
  • Y Berest, G Khachatryan, A Ramadoss, Derived representation schemes and cyclic homology, Adv. Math. 245 (2013) 625–689
  • Y Berest, A Ramadoss, Stable representation homology and Koszul duality, J. Reine Angew. Math. 715 (2016) 143–187
  • Y Berest, A C Ramadoss, W-k Yeung, Representation homology of spaces and higher Hochschild homology, preprint (2017)
  • A K Bousfield, D M Kan, Homotopy limits, completions and localizations, Lecture Notes in Math. 304, Springer (1972)
  • A K Bousfield, D M Kan, The homotopy spectral sequence of a space with coefficients in a ring, Topology 11 (1972) 79–106
  • K S Brown, Cohomology of groups, Graduate Texts in Math. 87, Springer (1982)
  • H Cartan, S Eilenberg, Homological algebra, Princeton Univ. Press (1956)
  • I Ciocan-Fontanine, M Kapranov, Virtual fundamental classes via dg–manifolds, Geom. Topol. 13 (2009) 1779–1804
  • F R Cohen, M Stafa, A survey on spaces of homomorphisms to Lie groups, from “Configuration spaces” (F Callegaro, F Cohen, C De Concini, E M Feichtner, G Gaiffi, M Salvetti, editors), Springer INdAM Ser. 14, Springer (2016) 361–379
  • M Culler, P B Shalen, Varieties of group representations and splittings of $3$–manifolds, Ann. of Math. 117 (1983) 109–146
  • J Cuntz, D Quillen, Algebra extensions and nonsingularity, J. Amer. Math. Soc. 8 (1995) 251–289
  • W G Dwyer, J Spaliński, Homotopy theories and model categories, from “Handbook of algebraic topology” (I M James, editor), North-Holland, Amsterdam (1995) 73–126
  • P G Goerss, J F Jardine, Simplicial homotopy theory, Progr. Math. 174, Birkhäuser, Basel (1999)
  • W M Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984) 200–225
  • W M Goldman, Representations of fundamental groups of surfaces, from “Geometry and topology” (J Alexander, J Harer, editors), Lecture Notes in Math. 1167, Springer (1985) 95–117
  • K Habiro, On the category of finitely generated free groups, preprint (2016)
  • D K Harrison, Commutative algebras and cohomology, Trans. Amer. Math. Soc. 104 (1962) 191–204
  • A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
  • S Iyengar, André–Quillen homology of commutative algebras, from “Interactions between homotopy theory and algebra” (L L Avramov, J D Christensen, W G Dwyer, M A Mandell, B E Shipley, editors), Contemp. Math. 436, Amer. Math. Soc., Providence, RI (2007) 203–234
  • D Johnson, J J Millson, Deformation spaces associated to compact hyperbolic manifolds, from “Discrete groups in geometry and analysis” (R Howe, editor), Progr. Math. 67, Birkhäuser, Boston (1987) 48–106
  • D M Kan, A combinatorial definition of homotopy groups, Ann. of Math. 67 (1958) 282–312
  • D M Kan, On homotopy theory and c.s.s. groups, Ann. of Math. 68 (1958) 38–53
  • D M Kan, A relation between $\mathrm{CW}$–complexes and free c.s.s. groups, Amer. J. Math. 81 (1959) 512–528
  • M Kapranov, Injective resolutions of $BG$ and derived moduli spaces of local systems, J. Pure Appl. Algebra 155 (2001) 167–179
  • L Le Bruyn, Qurves and quivers, J. Algebra 290 (2005) 447–472
  • J-L Loday, Cyclic homology, 2nd edition, Grundl. Math. Wissen. 301, Springer (1998)
  • A Lubotzky, A R Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 336, Amer. Math. Soc., Providence, RI (1985)
  • J Lurie, Derived algebraic geometry, PhD thesis, Massachusetts Institute of Technology (2004) Available at \setbox0\makeatletter\@url https://search.proquest.com/docview/305095302 {\unhbox0
  • J Lurie, Higher topos theory, Annals of Mathematics Studies 170, Princeton Univ. Press (2009)
  • J P May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies 11, Van Nostrand, Princeton, NJ (1967)
  • T Pantev, B Toën, M Vaquié, G Vezzosi, Shifted symplectic structures, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 271–328
  • T Pirashvili, Hodge decomposition for higher order Hochschild homology, Ann. Sci. École Norm. Sup. 33 (2000) 151–179
  • J P Pridham, Pro-algebraic homotopy types, Proc. Lond. Math. Soc. 97 (2008) 273–338
  • J P Pridham, Constructing derived moduli stacks, Geom. Topol. 17 (2013) 1417–1495
  • J P Pridham, Presenting higher stacks as simplicial schemes, Adv. Math. 238 (2013) 184–245
  • D G Quillen, Homotopical algebra, Lecture Notes in Math. 43, Springer (1967)
  • D Quillen, Rational homotopy theory, Ann. of Math. 90 (1969) 205–295
  • D Quillen, On the (co)homology of commutative rings, from “Applications of categorical algebra” (A Heller, editor), Amer. Math. Soc., Providence, RI (1970) 65–87
  • R W Richardson, Commuting varieties of semisimple Lie algebras and algebraic groups, Compositio Math. 38 (1979) 311–327
  • S Schwede, Spectra in model categories and applications to the algebraic cotangent complex, J. Pure Appl. Algebra 120 (1997) 77–104
  • P B Shalen, Representations of $3$–manifold groups, from “Handbook of geometric topology” (R J Daverman, R B Sher, editors), North-Holland, Amsterdam (2002) 955–1044
  • A S Sikora, Character varieties, Trans. Amer. Math. Soc. 364 (2012) 5173–5208
  • S Thomas, The functors $\bar {W}$ and $\mathrm{Diag}\circ \mathrm {Nerve}$ are simplicially homotopy equivalent, J. Homotopy Relat. Struct. 3 (2008) 359–378
  • R W Thomason, Une formule de Lefschetz en $K$–théorie équivariante algébrique, Duke Math. J. 68 (1992) 447–462
  • B Toën, Derived algebraic geometry, EMS Surv. Math. Sci. 1 (2014) 153–240
  • B Toën, Derived algebraic geometry and deformation quantization, from “Proceedings of the International Congress of Mathematicians” (S Y Jang, Y R Kim, D-W Lee, I Ye, editors), volume II, Kyung Moon Sa, Seoul (2014) 769–792
  • B Toën, G Vezzosi, Homotopical algebraic geometry, I: Topos theory, Adv. Math. 193 (2005) 257–372
  • B Toën, G Vezzosi, Homotopical algebraic geometry, II: Geometric stacks and applications, Mem. Amer. Math. Soc. 902, Amer. Math. Soc., Providence, RI (2008)
  • C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge Univ. Press (1994)
  • A Weil, Remarks on the cohomology of groups, Ann. of Math. 80 (1964) 149–157
  • W-k Yeung, Representation homology and knot contact homology, PhD thesis, Cornell University (2017) Available at \setbox0\makeatletter\@url https://search.proquest.com/docview/1959333106 {\unhbox0