Algebraic & Geometric Topology

Spin structures on almost-flat manifolds

Anna Gąsior, Nansen Petrosyan, and Andrzej Szczepański

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at msp.org/agt.

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We give a necessary and sufficient condition for almost-flat manifolds with cyclic holonomy to admit a Spin structure. Using this condition we find all 4–dimensional orientable almost-flat manifolds with cyclic holonomy that do not admit a Spin structure.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 2 (2016), 783-796.

Dates
Received: 5 November 2014
Revised: 6 July 2015
Accepted: 7 September 2015
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841152

Digital Object Identifier
doi:10.2140/agt.2016.16.783

Mathematical Reviews number (MathSciNet)
MR3493407

Zentralblatt MATH identifier
1361.53040

Subjects
Primary: 53C27: Spin and Spin$^c$ geometry
Secondary: 20H25: Other matrix groups over rings

Keywords
almost-flat manifolds infra-nilmanifolds Spin structures

Citation

Gąsior, Anna; Petrosyan, Nansen; Szczepański, Andrzej. Spin structures on almost-flat manifolds. Algebr. Geom. Topol. 16 (2016), no. 2, 783--796. doi:10.2140/agt.2016.16.783. https://projecteuclid.org/euclid.agt/1510841152


Export citation

References

  • L Auslander, Bieberbach's theorems on space groups and discrete uniform subgroups of Lie groups, Ann. of Math. 71 (1960) 579–590
  • P Buser, H Karcher, Gromov's almost flat manifolds, Astérisque 81, Soc. Math. France, Paris (1981)
  • J Cheeger, K Fukaya, M Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992) 327–372
  • C,W Curtis, I Reiner, Representation theory of finite groups and associative algebras, Wiley, New York (1988)
  • K Dekimpe, Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics 1639, Springer, Berlin (1996)
  • K Dekimpe, M Sadowski, A Szczepański, Spin structures on flat manifolds, Monatsh. Math. 148 (2006) 283–296
  • S,M Gagola, Jr, S,C Garrison, III, Real characters, double covers, and the multiplier, J. Algebra 74 (1982) 20–51
  • A G\kasior, A Szczepański, Tangent bundles of Hantzsche–Wendt manifolds, J. Geom. Phys. 70 (2013) 123–129
  • J Griess, R,L, A sufficient condition for a finite group to have a nontrivial Schur multiplier, Not. Amer. Math. Soc. 17 (1970) 644
  • M Gromov, Almost flat manifolds, J. Differential Geom. 13 (1978) 231–241
  • G Hiss, A Szczepański, Spin structures on flat manifolds with cyclic holonomy, Comm. Algebra 36 (2008) 11–22
  • R,C Kirby, The topology of $4$–manifolds, Lecture Notes in Mathematics 1374, Springer, Berlin (1989)
  • R,J Miatello, R,A Podestá, Spin structures and spectra of $Z\sp k\sb 2$–manifolds, Math. Z. 247 (2004) 319–335
  • R,J Miatello, R,A Podestá, The spectrum of twisted Dirac operators on compact flat manifolds, Trans. Amer. Math. Soc. 358 (2006) 4569–4603
  • J,W Milnor, J,D Stasheff, Characteristic classes, Ann. Math. Studies 76, Princeton Univ. Press (1974)
  • B Putrycz, A Szczepański, Existence of spin structures on flat four-manifolds, Adv. Geom. 10 (2010) 323–332
  • E,A Ruh, Almost flat manifolds, J. Differential Geom. 17 (1982) 1–14
  • C-H Sah, Homology of classical Lie groups made discrete, I: Stability theorems and Schur multipliers, Comment. Math. Helv. 61 (1986) 308–347
  • A Szczepański, Geometry of crystallographic groups, Algebra and Discrete Mathematics 4, World Scientific, Hackensack, NJ (2012)