Orientation reversal of manifolds

Müllner, Daniel

doi:10.2140/agt.2009.9.2361

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Orientation reversal of manifolds

Daniel Müllner

Algebraic & Geometric Topology 9 (2009) 2361–2390

DOI: 10.2140/agt.2009.9.2361

Abstract

We call a closed, connected, orientable manifold in one of the categories TOP, PL or DIFF chiral if it does not admit an orientation-reversing automorphism and amphicheiral otherwise. Moreover, we call a manifold strongly chiral if it does not admit a self-map of degree $- 1$ . We prove that there are strongly chiral, smooth manifolds in every oriented bordism class in every dimension $\geq 3$ . We also produce simply-connected, strongly chiral manifolds in every dimension $\geq 7$ . For every $k \geq 1$ , we exhibit lens spaces with an orientation-reversing self-diffeomorphism of order $2^{k}$ but no self-map of degree $- 1$ of smaller order.

Keywords

orientation, reversal, oriented, manifold, chiral, chirality, amphicheiral, amphicheirality, achiral, degree

Mathematical Subject Classification 2000

Primary: 55M25

Secondary: 57S17, 57N65, 57R19

References

Publication

Received: 30 July 2009

Revised: 24 September 2009

Accepted: 1 October 2009

Published: 3 November 2009

Authors

Daniel Müllner
Hausdorff Research Institute for Mathematics Poppelsdorfer Allee 82 53115 Bonn Germany
http://www.math.uni-bonn.de/people/muellner

Volume 9, issue 4 (2009)