Orientation reversal of manifolds.

*(English)*Zbl 1284.57027
Bonner Mathematische Schriften 392. Bonn: Univ. Bonn, Mathematisches Institut; Bonn: Univ. Bonn, Mathematisch-Naturwissenschaftliche Fakultät (Diss. 2008). v, 126 p. (2009).

Summary: Chiral manifolds can be studied in various categories by restricting the orientation-reversing map to homotopy equivalences, homeomorphisms or diffeomorphisms. The various notions of chirality do not coincide, and we extend the definition of chiral and amphicheiral manifolds by attributes, e.g. “topologically chiral” or “smoothly amphicheiral” that express the various restrictions on the orientation-reversing map.

We start with a survey of known results and examples of chiral manifolds, observing the basic facts that the point in dimension 0 is chiral and every closed, orientable 1- and 2-dimensional manifold is amphicheiral. A fundamental question is whether there are chiral manifolds in every dimension \(\geq 3\), and we prove this as the first main result. Our general aim is to produce manifolds which are chiral in the strongest sense, so we construct manifolds in every dimension \(\geq 3\) which do not admit a self-map of degree \(-1\).

The obstruction to orientation reversal in the constructed manifolds lies in the fundamental group since, e.g., the odd-dimensional examples are Eilenberg-MacLane spaces, and the proof of chirality uses as a substantial ingredient that the effect of a self-map on homology is completely determined by the induced map on the fundamental group. Therefore, we next ask for obstructions other than the fundamental group and restrict the analysis to simply-connected manifolds.

In dimensions 3, 5 and 6, every simply-connected (closed, orientable, smooth) manifold is amphicheiral by a diffeomorphism, and a topological 4-manifolds is amphicheiral if and only if its signature is zero. In all dimensions \(\geq 7\), we prove the existence of a simply-connected manifold which does not allow a self-map of degree \(-1\).

Next, in order to further characterise the properties of manifolds which allow or prevent orientation reversal, we consider the question whether every manifold is bordant to a chiral one. This allows also an approximation to the (not mathematically precise) question “how many” manifolds are chiral or if “the majority” of manifolds is chiral or amphicheiral. We prove that in every dimension \(\geq 3\), every closed, smooth, oriented manifold is oriented bordant to a manifold of this type which is connected and chiral.

The majority of the theorems so far aimed at proving that certain manifolds or families of manifolds are chiral. The opposite problem, however, namely proving amphicheirality in nontrivial circumstances, is also an interesting question. In general, this is even more challenging since not only one obstruction to orientation reversal must be identified and realised but for the opposite direction every possible obstruction must vanish. By using surgery theory, we prove the following theorem: Every product of 3-dimensional lens spaces whose orders of the fundamental groups are odd and coprime admits an orientation-reversing self-diffeomorphism.

In the last chapter, we add a new facet to the results by showing that the order of an orientation-reversing map can be relevant: For every positive integer \(k\), there are infinitely many lens spaces which admit an orientation-reversing diffeomorphism of order \(2^k\) but no orientation-reversing self-map of smaller order.

We start with a survey of known results and examples of chiral manifolds, observing the basic facts that the point in dimension 0 is chiral and every closed, orientable 1- and 2-dimensional manifold is amphicheiral. A fundamental question is whether there are chiral manifolds in every dimension \(\geq 3\), and we prove this as the first main result. Our general aim is to produce manifolds which are chiral in the strongest sense, so we construct manifolds in every dimension \(\geq 3\) which do not admit a self-map of degree \(-1\).

The obstruction to orientation reversal in the constructed manifolds lies in the fundamental group since, e.g., the odd-dimensional examples are Eilenberg-MacLane spaces, and the proof of chirality uses as a substantial ingredient that the effect of a self-map on homology is completely determined by the induced map on the fundamental group. Therefore, we next ask for obstructions other than the fundamental group and restrict the analysis to simply-connected manifolds.

In dimensions 3, 5 and 6, every simply-connected (closed, orientable, smooth) manifold is amphicheiral by a diffeomorphism, and a topological 4-manifolds is amphicheiral if and only if its signature is zero. In all dimensions \(\geq 7\), we prove the existence of a simply-connected manifold which does not allow a self-map of degree \(-1\).

Next, in order to further characterise the properties of manifolds which allow or prevent orientation reversal, we consider the question whether every manifold is bordant to a chiral one. This allows also an approximation to the (not mathematically precise) question “how many” manifolds are chiral or if “the majority” of manifolds is chiral or amphicheiral. We prove that in every dimension \(\geq 3\), every closed, smooth, oriented manifold is oriented bordant to a manifold of this type which is connected and chiral.

The majority of the theorems so far aimed at proving that certain manifolds or families of manifolds are chiral. The opposite problem, however, namely proving amphicheirality in nontrivial circumstances, is also an interesting question. In general, this is even more challenging since not only one obstruction to orientation reversal must be identified and realised but for the opposite direction every possible obstruction must vanish. By using surgery theory, we prove the following theorem: Every product of 3-dimensional lens spaces whose orders of the fundamental groups are odd and coprime admits an orientation-reversing self-diffeomorphism.

In the last chapter, we add a new facet to the results by showing that the order of an orientation-reversing map can be relevant: For every positive integer \(k\), there are infinitely many lens spaces which admit an orientation-reversing diffeomorphism of order \(2^k\) but no orientation-reversing self-map of smaller order.