Geometry & Topology

Infinite loop spaces and positive scalar curvature in the presence of a fundamental group

Johannes Ebert and Oscar Randal-Williams

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/gt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

This is a continuation of our previous work with Botvinnik on the nontriviality of the secondary index invariant on spaces of metrics of positive scalar curvature, in which we take the fundamental group of the manifolds into account. We show that the secondary index invariant associated to the vanishing of the Rosenberg index can be highly nontrivial for positive scalar curvature Spin manifolds with torsionfree fundamental groups which satisfy the Baum–Connes conjecture. This gives the first example of the nontriviality of the group C –algebra-valued secondary index invariant on higher homotopy groups. As an application, we produce a compact Spin 6 –manifold whose space of positive scalar curvature metrics has each rational homotopy group infinite-dimensional.

At a more technical level, we introduce the notion of “stable metrics” and prove a basic existence theorem for them, which generalises the Gromov–Lawson surgery technique, and we also give a method for rounding corners of manifolds with positive scalar curvature metrics.

Article information

Source
Geom. Topol., Volume 23, Number 3 (2019), 1549-1610.

Dates
Received: 21 February 2018
Revised: 30 August 2018
Accepted: 7 October 2018
First available in Project Euclid: 5 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.gt/1559700278

Digital Object Identifier
doi:10.2140/gt.2019.23.1549

Mathematical Reviews number (MathSciNet)
MR3956897

Zentralblatt MATH identifier
07079063

Subjects
Primary: 19K35: Kasparov theory ($KK$-theory) [See also 58J22] 19K56: Index theory [See also 58J20, 58J22] 53C27: Spin and Spin$^c$ geometry 58J22: Exotic index theories [See also 19K56, 46L05, 46L10, 46L80, 46M20]

Keywords
positive scalar curvature Gromov–Lawson surgery cobordism categories diffeomorphism groups Madsen–Weiss-type theorems secondary index invariant Rosenberg index Baum–Connes conjecture

Citation

Ebert, Johannes; Randal-Williams, Oscar. Infinite loop spaces and positive scalar curvature in the presence of a fundamental group. Geom. Topol. 23 (2019), no. 3, 1549--1610. doi:10.2140/gt.2019.23.1549. https://projecteuclid.org/euclid.gt/1559700278


Export citation

References

  • D W Anderson, E H Brown, Jr, F P Peterson, The structure of the Spin cobordism ring, Ann. of Math. 86 (1967) 271–298
  • N Bárcenas, R Zeidler, Positive scalar curvature and low-degree group homology, Ann. K–Theory 3 (2018) 565–579
  • P Baum, M Karoubi, On the Baum–Connes conjecture in the real case, Q. J. Math. 55 (2004) 231–235
  • R Bieri, Homological dimension of discrete groups, 2nd edition, Department of Pure Mathematics, Queen Mary College, London (1981)
  • B Blackadar, $K$–theory for operator algebras, 2nd edition, Mathematical Sciences Research Institute Publications 5, Cambridge Univ. Press (1998)
  • B Botvinnik, J Ebert, O Randal-Williams, Infinite loop spaces and positive scalar curvature, Invent. Math. 209 (2017) 749–835
  • B Botvinnik, P B Gilkey, The eta invariant and metrics of positive scalar curvature, Math. Ann. 302 (1995) 507–517
  • L Buggisch, The spectral flow theorem for families of twisted Dirac operators, PhD thesis, Westfälische Wilhelms-Universität Münster (2019)
  • U Bunke, A $K$–theoretic relative index theorem and Callias-type Dirac operators, Math. Ann. 303 (1995) 241–279
  • V Chernysh, On the homotopy type of the space $\mathcal{R}^+(M)$, preprint (2004)
  • V Chernysh, A quasifibration of spaces of positive scalar curvature metrics, Proc. Amer. Math. Soc. 134 (2006) 2771–2777
  • A Connes, G Skandalis, The longitudinal index theorem for foliations, Publ. Res. Inst. Math. Sci. 20 (1984) 1139–1183
  • J Dixmier, A Douady, Champs continus d'espaces hilbertiens et de $C\sp{\ast} $–algèbres, Bull. Soc. Math. France 91 (1963) 227–284
  • J Ebert, Elliptic regularity for Dirac operators on families of noncompact manifolds, preprint (2016)
  • J Ebert, The two definitions of the index difference, Trans. Amer. Math. Soc. 369 (2017) 7469–7507
  • J Ebert, Index theory in spaces of manifolds, Math. Ann. (online publication January 2019)
  • J Ebert, O Randal-Williams, The positive scalar curvature cobordism category, preprint (2019)
  • S Echterhoff, Bivariant $\mathit{KK}$–theory and the Baum–Connes conjecture, from “$K$–theory for group $C^*$–algebras and semigroup $C^*$–algebras” (J Cuntz, S Echterhoff, X Li, G Yu, editors), Oberwolfach Seminars 47, Springer (2017) 81–147
  • S Führing, A smooth variation of Baas–Sullivan theory and positive scalar curvature, Math. Z. 274 (2013) 1029–1046
  • P Gajer, Riemannian metrics of positive scalar curvature on compact manifolds with boundary, Ann. Global Anal. Geom. 5 (1987) 179–191
  • S Galatius, O Randal-Williams, Stable moduli spaces of high-dimensional manifolds, Acta Math. 212 (2014) 257–377
  • S Galatius, O Randal-Williams, Homological stability for moduli spaces of high dimensional manifolds, II, Ann. of Math. 186 (2017) 127–204
  • S Gallot, D Hulin, J Lafontaine, Riemannian geometry, 3rd edition, Springer (2004)
  • M Gromov, H B Lawson, Jr, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983) 83–196
  • M Gromov, H B Lawson, Jr, Spin and scalar curvature in the presence of a fundamental group, I, Ann. of Math. 111 (1980) 209–230
  • M Gromov, H B Lawson, Jr, The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. 111 (1980) 423–434
  • J-C Hausmann, D Husemoller, Acyclic maps, Enseign. Math. 25 (1979) 53–75
  • D W Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Bull. Amer. Math. Soc. 75 (1969) 759–762
  • N Higson, J Roe, Analytic $K$–homology, Oxford Univ. Press (2000)
  • N Higson, J Roe, $K$–homology, assembly and rigidity theorems for relative eta invariants, Pure Appl. Math. Q. 6 (2010) 555–601
  • M W Hirsch, Differential topology, Graduate Texts in Mathematics 33, Springer (1976)
  • M Joachim, S Stolz, An enrichment of $\mathit{KK}$–theory over the category of symmetric spectra, Münster J. Math. 2 (2009) 143–182
  • D D Joyce, Compact $8$–manifolds with holonomy ${\rm Spin}(7)$, Invent. Math. 123 (1996) 507–552
  • M Karoubi, A descent theorem in topological $K$–theory, $K$–Theory 24 (2001) 109–114
  • G G Kasparov, The operator $K$–functor and extensions of $C\sp{\ast} $–algebras, Izv. Akad. Nauk SSSR Ser. Mat. 44 (1980) 571–636 In Russian; translated in Math. USSR-Izv. 16 (1981) 513–572
  • M A Kervaire, Le théorème de Barden–Mazur–Stallings, Comment. Math. Helv. 40 (1965) 31–42
  • M Land, The analytical assembly map and index theory, J. Noncommut. Geom. 9 (2015) 603–619
  • H B Lawson, Jr, M-L Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton Univ. Press (1989)
  • W Lück, H Reich, The Baum–Connes and the Farrell–Jones conjectures in $K$– and $L$–theory, from “Handbook of $K$–theory” (E M Friedlander, D R Grayson, editors), volume 2, Springer (2005) 703–842
  • M Matthey, The Baum–Connes assembly map, delocalization and the Chern character, Adv. Math. 183 (2004) 316–379
  • N Perlmutter, Parametrized Morse theory and positive scalar curvature, preprint (2017)
  • A Phillips, Submersions of open manifolds, Topology 6 (1967) 171–206
  • P Piazza, T Schick, Groups with torsion, bordism and rho invariants, Pacific J. Math. 232 (2007) 355–378
  • J Rosenberg, $C\sp{\ast} $–algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. 58 (1983) 197–212
  • T Schick, Real versus complex $K$–theory using Kasparov's bivariant $\mathit{KK}$–theory, Algebr. Geom. Topol. 4 (2004) 333–346
  • T Schick, $L^2$–index theorems, $\mathit{KK}$–theory, and connections, New York J. Math. 11 (2005) 387–443
  • S Stolz, Concordance classes of positive scalar curvature metrics, unpublished manuscript (1998) Available at \setbox0\makeatletter\@url http://www3.nd.edu/~stolz/preprint.html {\unhbox0
  • C T C Wall, Finiteness conditions for ${\rm CW}$–complexes, Ann. of Math. 81 (1965) 56–69
  • C T C Wall, Geometrical connectivity, I, J. London Math. Soc. 3 (1971) 597–604
  • M Walsh, Metrics of positive scalar curvature and generalised Morse functions, I, Mem. Amer. Math. Soc. 983, Amer. Math. Soc., Providence, RI (2011)
  • M Walsh, Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics, Proc. Amer. Math. Soc. 141 (2013) 2475–2484
  • M Walsh, The space of positive scalar curvature metrics on a manifold with boundary, preprint (2014)
  • S Weinberger, G Yu, Finite part of operator $K$–theory for groups finitely embeddable into Hilbert space and the degree of nonrigidity of manifolds, Geom. Topol. 19 (2015) 2767–2799
  • Z Xie, G Yu, R Zeidler, On the range of the relative higher index and the higher rho-invariant for positive scalar curvature, preprint (2017)