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CLASSIFICATION OF SPIN STRUCTURES ON FOUR-DIMENSIONAL ALMOST-FLAT MANIFOLDS

Part of: Other groups of matrices Global differential geometry

Published online by Cambridge University Press:  14 February 2018

R. Lutowski ,
N. Petrosyan  and
A. Szczepański
R. Lutowski
Affiliation:
Institute of Mathematics, University of Gdańsk, Gdańsk, Poland email [email protected]
N. Petrosyan
Affiliation:
Department of Mathematics, University of Southampton, Southampton, U.K. email [email protected]
A. Szczepański
Affiliation:
Institute of Mathematics, University of Gdańsk, Gdańsk, Poland email [email protected]
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Abstract

Almost-flat manifolds were defined by Gromov as a natural generalization of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental group. Utilizing this, we classify the spin structures on all four-dimensional almost-flat manifolds that are not flat. Out of 127 orientable families, we show that there are exactly 15 that are non-spin, the rest are, in fact, parallelizable.

MSC classification

Primary: 53C27: Spin and Spin$^c$ geometry 20H25: Other matrix groups over rings
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 253 - 266
Copyright
Copyright © University College London 2018 

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References

Atiyah, M. F., Bott, R. and Shapiro, A., Clifford modules. Topology (1) 3 1964, 3–38.CrossRefGoogle Scholar
Auslander, L., Bieberbach’s theorems on space groups and discrete uniform subgroups of Lie groups. Ann. of Math. (2) 71 1960, 579–590.CrossRefGoogle Scholar
Borel, A. and Hirzebruch, F., Characteristic classes and homogeneous spaces. II. Amer. J. Math. 81 1959, 315–382.CrossRefGoogle Scholar
Buser, P. and Karcher, H., Gromov’s Almost Flat Manifolds (Astérisque 81 ), Société Mathématique de France (Paris, 1981).Google Scholar
Cheeger, J., Fukaya, K. and Gromov, M., Nilpotent structures and invariant metrics on collapsed manifolds. J. Amer. Math. Soc. 5(2) 1992, 327–372.CrossRefGoogle Scholar
Chevalley, C., The Algebraic Theory of Spinors, Columiba University Press (New York, NY, 1954).CrossRefGoogle Scholar
Dekimpe, K., Almost-Bieberbach Groups: Affine and Polynomial Structures (Lecture Notes in Mathematics 1639 ), Springer (Berlin, 1996).CrossRefGoogle Scholar
Dekimpe, K. and Eick, B., Computational aspects of group extensions and their applications in topology. Exp. Math. 11(2) 2002, 183–200.CrossRefGoogle Scholar
Dekimpe, K. and Eick, B., Aclib – a GAP package, Version 1.2 (2012), http://www.icm.tu-bs.de/ beick/so.html.Google Scholar
Friedrich, T., Dirac Operators in Riemannian Geometry (Graduate Studies in Mathematics 25 ), American Mathematical Society (Providence, RI, 2000). Translated from the 1997 German original by Andreas Nestke.CrossRefGoogle Scholar
GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.8.3(2016), http://www.gap-system.org/.Google Scholar
Gąsior, A., Petrosyan, N. and Szczepański, A., Spin structures on almost-flat manifolds. Algebr. Geom. Topol. 16(2) 2016, 783–796.CrossRefGoogle Scholar
Gromov, M., Almost flat manifolds. J. Differ. Geom. 13(2) 1978, 231–241.Google Scholar
Hirzebruch, F. and Hopf, H., Felder von Flächenelementen in 4-dimensionalen Mannigfaltigkeiten. Math. Ann. 136 1958, 156–172.CrossRefGoogle Scholar
Kirby, R. C., The Topology of 4-manifolds (Lecture Notes in Mathematics 1374 ), Springer (Berlin, 1989).CrossRefGoogle Scholar
Lutowski, R. and Putrycz, B., Spin structures on flat manifolds. J. Algebra 436 2015, 277–291.CrossRefGoogle Scholar
Milnor, J. W. and Stasheff, J. D., Characteristic Classes (Annals of Mathematics Studies 76 ), Princeton University Press, University of Tokyo Press (Princeton, NJ and Tokyo, 1974).CrossRefGoogle Scholar
Pfäffle, F., The Dirac spectrum of Bieberbach manifolds. J. Geom. Phys. 35(4) 2000, 367–385.CrossRefGoogle Scholar
Putrycz, B. and Szczepański, A., Existence of spin structures on flat four-manifolds. Adv. Geom. 10(2) 2010, 323–332.CrossRefGoogle Scholar
Ruh, E. A., Almost flat manifolds. J. Differ. Geom. 17(1) 1982, 1–14.Google Scholar
Szczepański, A., Geometry of Crystallographic Groups (Algebra and Discrete Mathematics 4 ), World Scientific (Hackensack, NJ, 2012).CrossRefGoogle Scholar