On Fixed-Point Sets of $Z_2$-Tori in Positive CurvatureMPIM

by Catherine Searle (Wichita State University/MPIM)

Thursday Jul 17, 2025, 5:00 PM → 5:30 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Lecture Hall (Max Planck Institute for Mathematics)

Summer Geometry Day

In recent work of Kennard, Khalili Samani, and the last author, they generalize the Half-Maximal Symmetry Rank result of Wilking for torus actions on positively curved manifolds to $\mathbb{Z}_2$-tori with a fixed point. They show that if the rank is approximately one-fourth of the dimension of the manifold, then fixed point set components of small co-rank subgroups of the $Z_2$-torus  are homotopy equivalent to spheres, real projective spaces, complex projective spaces, or lens spaces. In this paper, we lower the bound on the rank of the $\mathbb{Z}_2$-torus to approximately $1/6$ and $1/8$ of the dimension of the manifold and are able to classify either the integral cohomology ring or the $\mathbb{Z}_2$-cohomology ring, respectively, of the fixed point set of the $\mathbb{Z}_2$-torus. This is joint work with Austin Bosgraaf and Christine Escher.