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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 rafal.lutowski@mat.ug.edu.pl
N. Petrosyan
Affiliation:
Department of Mathematics, University of Southampton, Southampton, U.K. email N.Petrosyan@soton.ac.uk
A. Szczepański
Affiliation:
Institute of Mathematics, University of Gdańsk, Gdańsk, Poland email aszczepa@mat.ug.edu.pl
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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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