Hostname: page-component-8678677fb7-wnmm5 Total loading time: 0 Render date: 2026-10-09T15:13:46.503Z Has data issue: false hasContentIssue false
  • English
  • Français

Twisted K-theory and obstructions against positive scalar curvature metrics

Published online by Cambridge University Press:  17 April 2014

Ulrich Pennig
Ulrich Pennig*
Affiliation:
Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstraße 62, 48149 Münster, Germany, u.pennig@uni-muenster.de
Get access

Abstract

We decompose θ(M), the twisted index obstruction to a positive scalar curvature metric for closed oriented manifolds with spin universal cover, into a pairing of a twisted K-homology with a twisted K-theory class and prove that θ(M) does not vanish if M is a closed orientable enlargeable manifold with spin universal cover.

Keywords

enlargeable manifoldindex theoryDirac operatorpositive scalar curvaturetwisted index obstructiontwisted K-theory19L5019K56

Information

Type
Research Article
Information
Journal of K-Theory , Volume 14 , Issue 1 , August 2014 , pp. 47 - 71
Copyright
Copyright © ISOPP 2014 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Atiyah, M. & Hopkins, M.A variant of K-theory: K±.In: Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser. 308, pp. 5–17, Cambridge Univ. Press, Cambridge 2004.Google Scholar
2.Atiyah, M. & Segal, G.Twisted K-theory. Ukr. Mat. Visn. 1(3) (2004), 287–330.Google Scholar
3.Blackadar, B.K-theory for operator algebras, Mathematical Sciences Research Institute Publications 5, Cambridge University Press, Cambridge 1998, second edition.Google Scholar
4.Bouwknegt, P., Carey, A. L., Mathai, V., Murray, M. K. & Stevenson, D.TwistedK-theory and K-theory of bundle gerbes. Comm. Math. Phys. 228(1) (2002), 17–45.Google Scholar
5.Busby, R. C. & Smith, H. A.Representations of twisted group algebras. Trans. Amer. Math. Soc. 149 (1970), 503–537.Google Scholar
6.Carey, A. L. & Wang, B.-L.Thom isomorphism and push-forward map in twisted K-theory. J. K-Theory 1(2) (2008), 357–393.Google Scholar
7.Donovan, P. & Karoubi, M.Graded Brauer groups and K-theory with local coefficients. Inst. Hautes Études Sci. Publ. Math. 38 (1970), 5–25.CrossRefGoogle Scholar
8.Fuchs, J., Nikolaus, T., Schweigert, C. & Waldorf, K.Bundle gerbes and surface holonomy. In: European Congress of Mathematics, pp. 167–195, Eur. Math. Soc., Zürich 2010.Google Scholar
9.Gromov, M. & Lawson, H. B. Jr.Spin and scalar curvature in the presence of a fundamental group. I. Ann. of Math. (2) 111(2) (1980), 209–230.Google Scholar
10.Gromov, M.. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83–196 (1984).Google Scholar
11.Hanke, B. & Schick, T.Enlargeability and index theory. J. Differential Geom. 74(2) (2006), 293–320.Google Scholar
12.Hanke, B.. Enlargeability and index theory: infinite covers. K-Theory 38(1) (2007), 23–33.Google Scholar
13.Karoubi, M.Algèbres de Clifford et K-théorie. Ann. Sci. École Norm. Sup. (4)1 (1968), 161–270.Google Scholar
14.Karoubi, M.. Twisted K-theory—old and new. In: K-theory and noncommutative geometry, pp. 117–149. EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich 2008.Google Scholar
15.Lance, E. C.Hilbert C*-modules, London Mathematical Society Lecture Note Series 210, Cambridge University Press, Cambridge 1995.CrossRefGoogle Scholar
16.Lawson, H. B. Jr. & Michelsohn, M.-L.Spin geometry, Princeton Mathematical Series 38, Princeton University Press, Princeton NJ 1989.Google Scholar
17.Mathai, V., Murray, M. K. & Stevenson, D.Type-I D-branes in an H-flux and twisted KO-theory. J. High Energy Phys. (11) (2003), 053, 23 pp. (electronic).Google Scholar
18.Mishchenko, A. S. & Fomenko, A. T.The index of elliptic operators over C*-algebras. Izv. Akad. Nauk SSSR Ser. Mat. 43(4) (1979), 831–859, 967.Google Scholar
19.Murray, M. K.Bundle gerbes. J. London Math. Soc. (2) 54(2) (1996), 403–416.Google Scholar
20.Murray, M. K.. An introduction to bundle gerbes. In: The many facets of geometry, pp. 237–260, Oxford Univ. Press, Oxford 2010.Google Scholar
21.Murray, M. K. & Singer, M. A.Gerbes, Clifford modules and the index theorem. Ann. Global Anal. Geom. 26(4) (2004), 355–367.Google Scholar
22.Rosenberg, J.C*-algebras, positive scalar curvature, and the Novikov conjecture. III. Topology 25(3) (1986), 319–336.CrossRefGoogle Scholar
23.Rosenberg, J.. Manifolds of positive scalar curvature: a progress report. In: Surveys in differential geometry. Vol. XI, Surv. Differ. Geom. 11, pp. 259–294, Int. Press, Somerville, MA 2007.Google Scholar
24.Schick, T.A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture. Topology 37(6) (1998), 1165–1168.Google Scholar
25.Schick, T.. L 2-index theorems, KK-theory, and connections. New York J. Math. 11 (2005), 387–443 (electronic).Google Scholar
26.Stolz, S. Concordance classes of positive scalar curvature metrics. preprintGoogle Scholar
27.Stolz, S.. Simply connected manifolds of positive scalar curvature. Ann. of Math. (2) 136(3) (1992), 511–540.Google Scholar
28.Waldorf, K.More morphisms between bundle gerbes. Theory Appl. Categ. 18(9) (2007), 240–273.Google Scholar