For a (smoothly) triangulated $n$ manifold $M$, I'll say that the triangulation is amphichiral if it admits an orientation-reversing automorphism. I'll say that the triangulation is locally amphichiral if every link admits an orientation-reversion automorphism (as a simplicial complex). I'll say that a manifold is locally amphichiral if it admits a locally amphichiral triangulation.
My question is what manifolds are locally amphichiral. I am particulatly interested in 3-manifolds, but any information will be useful. These are a few facts I know:
- The Lens spaces $L(p,q)$ are locally amphichiral: triangulate the 3-ball using $2p$ tetrahedra, $p$ for the northern and $p$ for the southern hemisphere. Gluing the top and bottom hemispheres with a $2\pi q/p$ rotation yields a locally amphichiral triangulation.
- In dimensions 0 mod 4 I managed to convince myself that manifolds with nonzero Pontyagin numbers are not locally amphichiral. The argument is to endow the manifold with a metric that is flat everywhere but in a neighborhood of the $n-1$ dimensional skeleton, and choose the top-dimensional curvature forms vanish everywhere but a neighborhood of the vertices. Defining the top Pontyagin classes via Chern-Weil theory, contributions to the integral $\int p_1^{a_1}\cdots p_k^{a_k}$ will come only from a neighborhood of the vertex, and vanish by the orientation-reversing symmetry around the vertex.
- By brute-force search over triangulations, I see that some Seifert fibrations over the sphere are locally amphirchiral (I assume that all of them are), but no additional examples for manifolds that are locally amphichiral but not amphichiral.
