Orientation reversal of manifolds - Universität Bonn

1
Introduction
In this work, the phenomenon of orientation reversal of manifolds is studied.
We call an orientable manifold amphicheiral if it admits an orientationreversing
self-map and chiral if it does not. Below, this definition is extended
by attributes, e. g. “topologically chiral” or “smoothly amphicheiral” that express
various degrees of restriction on the orientation-reversing map. Many familiar
manifolds like spheres or orientable surfaces are amphicheiral: in these cases
mirror-symmetric embeddings into R n exist, and reflection at the “equatorial”
hyperplane reverses orientation. On the other hand, examples of chiral manifolds
have been known for many decades, e. g. the complex projective spaces
CP 2k or some lens spaces in dimensions congruent 3 mod 4. However, this
phenomenon has not been studied systematically.
In the next chapter, we start with a survey of known results and examples of
chiral manifolds. This cannot encompass every result which is related to chirality
and amphicheirality of manifolds. Still we try to give a broad overview, state
the most important results in this context and give reasons why the problems
that are dealt with in the following chapters are relevant.
A fundamental question is in which dimensions there are chiral manifolds.
The solution to this problem is the first main result of this work and the content
of Chapter 3:
Theorem A
A single point, considered as an orientable 0-dimensional manifold, is chiral.
In dimensions 1 and 2, every closed, orientable, smooth manifold admits
an orientation-reversing diffeomorphism. In every dimension ≥ 3, there is
a closed, connected, orientable, smooth manifold which does not admit a
continuous map to itself with degree −1, i. e. it is chiral.
The construction of these chiral manifolds is divided into even and odd
dimensions. First we construct odd-dimensional chiral manifolds in every
dimension n ≥ 3 as mapping tori of (n − 1)-dimensional tori T n−1 . The fundamental
group of the total space is a semidirect product of abelian groups.
If we restrict the monodromy maps H 1 (T n−1 ) → H 1 (T n−1 ) to certain maps,
the effect of endomorphisms of the fundamental group on the orientation of
1