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Orientation reversal of manifolds

Daniel Müllner

Algebraic & Geometric Topology 9 (2009) 2361–2390

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 3. We also produce simply-connected, strongly chiral manifolds in every dimension 7. For every k 1, we exhibit lens spaces with an orientation-reversing self-diffeomorphism of order 2k but no self-map of degree 1 of smaller order.

orientation, reversal, oriented, manifold, chiral, chirality, amphicheiral, amphicheirality, achiral, degree
Mathematical Subject Classification 2000
Primary: 55M25
Secondary: 57S17, 57N65, 57R19
Received: 30 July 2009
Revised: 24 September 2009
Accepted: 1 October 2009
Published: 3 November 2009
Daniel Müllner
Hausdorff Research Institute for Mathematics
Poppelsdorfer Allee 82
53115 Bonn