Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-25T06:55:15.548Z Has data issue: false hasContentIssue false
  • English
  • Français

THE GROMOV-LAWSON-ROSENBERG CONJECTURE FOR THE SEMI-DIHEDRAL GROUP OF ORDER 16

Part of: Differential topology Homology and cohomology theories Applied homological algebra and category theory Topological $K$-theory Fiber spaces and bundles

Published online by Cambridge University Press:  18 December 2014

ARJUN MALHOTRA  and
KIJTI RODTES
ARJUN MALHOTRA
Affiliation:
7 Rusholme Grove, London SE19 1HYUnited Kingdom E-mail: arjun.malhotra2807@gmail.com
KIJTI RODTES
Affiliation:
Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand E-mail: kijtir@nu.ac.th
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that the Gromov-Lawson-Rosenberg conjecture for the Semi-Dihedral group of order 16 is true.

MSC classification

Primary: 19L41: Connective $K$-theory, cobordism
Secondary: 19L47: Equivariant $K$-theory 19L64: Computations, geometric applications 55N15: $K$-theory 55N91: Equivariant homology and cohomology 55R35: Classifying spaces of groups and $H$-spaces 55U20: Universal coefficient theorems, Bockstein operator 57R90: Other types of cobordism
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 57 , Issue 2 , May 2015 , pp. 365 - 386
Copyright
Copyright © Glasgow Mathematical Journal Trust 2014 

References

REFERENCES

1.Bayen, D., The connective real K-homology of finite groups, Phd. Thesis (Wayne State University, 1994).Google Scholar
2.Botvinnik, B. and Gilkey, P., The eta invariant and metrics of positive scalar curvature Math. Ann. 302 (1995), 507–517.Google Scholar
3.Botvinnik, B., Gilkey, P. and Stolz, S., The Gromov Lawson Rosenberg conjecture for groups with periodic cohomology, J. Differ. Geom. 46 (3) (1997), 374–405.Google Scholar
4.Bruner, R. R. and Greenlees, J. P. C., The connective K-theory of finite groups, Mem. Am. Math. Soc. 165 (785) (2003), 127.Google Scholar
5.Bruner, R. R. and Greenlees, J. P. C., The connective real K-theory of finite groups Math. Surv. Monogr. 169 (2010).Google Scholar
6.Dwyer, W., Schick, T. and Stolz, S., Remarks on a conjecture of Gromov and Lawson, High dimensional manifold topology (Word Scientific publishing, River edge, New Jersey, 2003) 159–176.Google Scholar
7.Donnelly, H., Eta invariants for G-spaces Indiana Univ. Math. J. 27 (1978), 889–918.Google Scholar
8.Evens, L. and Priddy, S., The cohomology of the semi-dihedral group Contemp. Math. 37 (1985).Google Scholar
9.Gromov, M. and Lawson, H. B., The classification of simply connected manifolds of positive scalar curvature Ann. Math. 111 (1980), 423–434.CrossRefGoogle Scholar
10.Hitchin, N., Harmonic spinors Adv. Math. 14 (1974), 1–55.CrossRefGoogle Scholar
11.Joachim, M. and Malhotra, A., The Gromov-Lawson-Rosenberg conjecture for dihedral groups, In preparation.Google Scholar
12.Kreck, M. and Stolz, S., ℍℙ2 bundles and elliptic homology Acta Math. 171 (1993), 231–261.CrossRefGoogle Scholar
13.Kwasik, S. and Scultz, R., Positive scalar curvature and periodic fundamental groups Math. Helv. 65 (1990), 271–286.Google Scholar
14.Lawson, H. B. and Michelsohn, M. L., Spin geometry, Princeton Mathematical Series, vol. 38 (Princeton University Press, 1989)Google Scholar
15.Lichnerowicz, A., Spineurs harmonique C. R. Acad. Sci. Paris 257 (1963), 7–9.Google Scholar
16.Malhotra, A., The Gromov-Lawson-Rosenberg conjecture for some finite groups, Thesis (The university of Sheffield, UK, 2011), available online at http://arxiv.org/abs/1305.0455.Google Scholar
17.Rodtes, K., The connnective K theory of semidihedral groups, Thesis (The university of Sheffield, UK, 2010), available online at http://etheses.whiterose.ac.uk/1103/2/Rodtes,_Kijtie.pdf.Google Scholar
18.Rosenberg, J., C*-Algebras, positive scalar curvature and the Novikov conjecture I, Inst. Étaudes Sci. Publ. Math. 58 (1983), 197–212.Google Scholar
19.Rosenberg, J., C*-Algebras, positive scalar curvature and the Novikov conjecture II, in Geometric methods in operator algebras (Araki, H. and Effros, E., Editors) Pitman Research Notes in Mathematics Series, vol. 123 (Longman, Harlow, 1986) 341–374.Google Scholar
20.Rosenberg, J., C*-Algebras, positive scalar curvature and the Novikov conjecture III Topology 25 (1986), 319–336.Google Scholar
21.Rosenberg, J. and Stolz, S., Manifolds of positive scalar curvature, Algebraic topology and its applications (Carlson, G., Cohen, R., Hsiang, W. C. and Jones, J. D. S, Editors) Mathematical Sciences Research Institute Publications, vol 27, (Springer, New York, 1994) 241–267.CrossRefGoogle Scholar
22.Rosenberg, J. and Stolz, S., A stable version of the Gromov-Lawson conjecture Contemp. Math. 181 (1995), 405–418.Google Scholar
23.Schick, T., A counterexample to the (unstable) Gromov-Lawson-Rosenberg conjecture, Topology 37 (6) (1998), 1165–1168.CrossRefGoogle Scholar
24.Stolz, S., Simply connected manifolds of positive scalar curvature Ann. Math. 136 (1992), 511–540.Google Scholar
25.Stolz, S., Splitting certain MSpin module spectra Topology 33 (1994), 159–180.CrossRefGoogle Scholar