Moduli spaces of invariant metrics of positive scalar curvature on quasitoric manifolds

Wiemeler, Michael

doi:10.1090/proc/14123

Moduli spaces of invariant metrics of positive scalar curvature on quasitoric manifolds
HTML articles powered by AMS MathViewer

by Michael Wiemeler PDF

Proc. Amer. Math. Soc. 146 (2018), 4521-4534 Request permission

Abstract:

We show that the higher homotopy groups of the moduli space of torus-invariant positive scalar curvature metrics on certain quasitoric manifolds are nontrivial.

References

L. Bérard Bergery, Scalar curvature and isometry group, Spectra of Riemannian Manifolds, Kaigai Publications, Tokyo, 1983, pp. 9–28.
Boris Botvinnik, Johannes Ebert, and Oscar Randal-Williams, Infinite loop spaces and positive scalar curvature, Invent. Math. 209 (2017), no. 3, 749–835. MR 3681394, DOI 10.1007/s00222-017-0719-3
Boris Botvinnik, Bernhard Hanke, Thomas Schick, and Mark Walsh, Homotopy groups of the moduli space of metrics of positive scalar curvature, Geom. Topol. 14 (2010), no. 4, 2047–2076. MR 2680210, DOI 10.2140/gt.2010.14.2047
Jean-Pierre Bourguignon, Une stratification de l’espace des structures riemanniennes, Compositio Math. 30 (1975), 1–41 (French). MR 418147
Victor M. Buchstaber and Taras E. Panov, Torus actions and their applications in topology and combinatorics, University Lecture Series, vol. 24, American Mathematical Society, Providence, RI, 2002. MR 1897064, DOI 10.1090/ulect/024
Victor M. Buchstaber and Taras E. Panov, Toric topology, Mathematical Surveys and Monographs, vol. 204, American Mathematical Society, Providence, RI, 2015. MR 3363157, DOI 10.1090/surv/204
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Suyoung Choi, Mikiya Masuda, and Dong Youp Suh, Quasitoric manifolds over a product of simplices, Osaka J. Math. 47 (2010), no. 1, 109–129. MR 2666127
Michael W. Davis and Tadeusz Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), no. 2, 417–451. MR 1104531, DOI 10.1215/S0012-7094-91-06217-4
David G. Ebin, The manifold of Riemannian metrics, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 11–40. MR 0267604
F. T. Farrell and W. C. Hsiang, On the rational homotopy groups of the diffeomorphism groups of discs, spheres and aspherical manifolds, Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976) Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978, pp. 325–337. MR 520509
S. Goette, Morse theory and higher torsion invariants I, Preprint, arXiv:math/0111222, 2001.
Saibal Ganguli and Mainak Poddar, Blowdowns and McKay correspondence on four dimensional quasitoric orbifolds, Osaka J. Math. 50 (2013), no. 2, 397–415. MR 3080806
Bernhard Hanke, Positive scalar curvature with symmetry, J. Reine Angew. Math. 614 (2008), 73–115. MR 2376283, DOI 10.1515/CRELLE.2008.003
Bernhard Hanke, Thomas Schick, and Wolfgang Steimle, The space of metrics of positive scalar curvature, Publ. Math. Inst. Hautes Études Sci. 120 (2014), 335–367. MR 3270591, DOI 10.1007/s10240-014-0062-9
Wilderich Tuschmann and David J. Wraith, Moduli spaces of Riemannian metrics, Oberwolfach Seminars, vol. 46, Birkhäuser Verlag, Basel, 2015. Second corrected printing. MR 3445334, DOI 10.1007/978-3-0348-0948-1
Mark Walsh, Metrics of positive scalar curvature and generalised Morse functions, Part I, Mem. Amer. Math. Soc. 209 (2011), no. 983, xviii+80. MR 2789750, DOI 10.1090/S0065-9266-10-00622-8
Mark Walsh, Cobordism invariance of the homotopy type of the space of positive scalar curvature metrics, Proc. Amer. Math. Soc. 141 (2013), no. 7, 2475–2484. MR 3043028, DOI 10.1090/S0002-9939-2013-11647-3
Mark Walsh, $H$-spaces, loop spaces and the space of positive scalar curvature metrics on the sphere, Geom. Topol. 18 (2014), no. 4, 2189–2243. MR 3268776, DOI 10.2140/gt.2014.18.2189
Michael Wiemeler, Exotic torus manifolds and equivariant smooth structures on quasitoric manifolds, Math. Z. 273 (2013), no. 3-4, 1063–1084. MR 3030690, DOI 10.1007/s00209-012-1044-6
Michael Wiemeler, Circle actions and scalar curvature, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2939–2966. MR 3449263, DOI 10.1090/tran/6666
David Wraith, Non-negative versus positive scalar curvature, Preprint, arXiv:1607.00657, 2016.