Rotation index, Milnor--Munkres--Novikov pairing, and group actions on manifolds

TL;DR

The paper introduces the rotation index invariant for commuting matrices and applies it with the Milnor--Munkres--Novikov pairing to analyze group actions on manifolds, addressing problems like Nielsen realization and higher-rank Anosov actions.

Contribution

It develops a new invariant called the rotation index and combines it with existing pairings to study complex group actions on manifolds.

Findings

The rotation index provides new insights into group actions.

Applications include solutions to Nielsen realization and extension problems.

The approach advances understanding of higher-rank Anosov actions.

Abstract

We introduce an invariant of a pair of commuting invertible matrices that we call the rotation index. We apply this invariant, together with the Milnor--Munkres--Novikov pairing, to the study of some questions about group actions of $\mathbb{Z}^2$, specifically the Nielsen realization problem, higher-rank Anosov actions, and extending actions from the sphere $S^{d-1}$ to the disk $D^d$.