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

Mauricio Bustamante, Bena Tshishiku

TL;DR

The paper defines the rotation index ρ for commuting pairs A,B ∈ SL_d(R) and connects it to the second Stiefel--Whitney class w_2 of representations Z^2 → SL_d(R), yielding path-component distinctions in Hom(Z^2, SL_d(R)) for d ≥ 3. It then uses ρ together with the Milnor--Munkres--Novikov pairing to study actions of Z^2 on exotic tori, proving obstructions to Nielsen realization and showing nonrealizability of certain higher-rank Anosov actions as smooth actions on exotic tori. The authors construct explicit commuting hyperbolic matrices with ρ=1 to produce Anosov Z^2-actions that cannot be realized as linearizations on exotic tori, and they produce Z^2-actions on S^{d-1} isotopic to the identity that do not extend to D^d, with the MMN pairing providing the detectable obstruction. The analysis combines perturbation theory of linear operators, spectral projections, and central extensions to relate dynamical, geometric, and topological obstructions, advancing Borel-type questions about how π_1(M) governs symmetry realizations on aspherical and exotic manifolds.

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 $D^d$.

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

TL;DR

The paper defines the rotation index ρ for commuting pairs A,B ∈ SL_d(R) and connects it to the second Stiefel--Whitney class w_2 of representations Z^2 → SL_d(R), yielding path-component distinctions in Hom(Z^2, SL_d(R)) for d ≥ 3. It then uses ρ together with the Milnor--Munkres--Novikov pairing to study actions of Z^2 on exotic tori, proving obstructions to Nielsen realization and showing nonrealizability of certain higher-rank Anosov actions as smooth actions on exotic tori. The authors construct explicit commuting hyperbolic matrices with ρ=1 to produce Anosov Z^2-actions that cannot be realized as linearizations on exotic tori, and they produce Z^2-actions on S^{d-1} isotopic to the identity that do not extend to D^d, with the MMN pairing providing the detectable obstruction. The analysis combines perturbation theory of linear operators, spectral projections, and central extensions to relate dynamical, geometric, and topological obstructions, advancing Borel-type questions about how π_1(M) governs symmetry realizations on aspherical and exotic manifolds.

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 , specifically the Nielsen realization problem, higher-rank Anosov actions, and extending actions from the sphere to the disk .
Paper Structure (12 sections, 12 theorems, 52 equations, 1 figure)

This paper contains 12 sections, 12 theorems, 52 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Rotation index
  3. Homotopy Anosov actions on exotic tori
  4. Nielsen realization and Borel's theme
  5. Bordism of group actions
  6. The rotation index
  7. Perturbation theory and eigenspaces
  8. Nielsen realization (Proof of Theorem \ref{['thm:nielsen']})
  9. Obstruction to lifting when $\rho\neq0$
  10. Constructing lifts when $\rho=0$
  11. Homotopy Anosov actions (Proof of Theorem \ref{['thm:anosov']})
  12. Bordism of group actions (Theorem \ref{['thm:bordism']})

Key Result

Theorem A

For all $d\geq 2$, the function $\rho:\mathrm{SL}_d(\mathbb{R})\times\mathrm{SL}_d(\mathbb{R})\to\mathbb{Z}/2\mathbb{Z}$ is continuous when restricted to the subspace of commuting pairs.

Figures (1)

  • Figure 1: Loop homotopic to $[\eta_1,\eta_2]$ in $\mathop{\mathrm{SO}}\nolimits(3)\cong\mathbb{R} P^3$, viewed as the quotient of the unit 3-ball by the antipodal map on its boundary. A point $v$ in the ball corresponds to the rotation with axis $v$ and angle $|v|\pi$ (counterclockwise according to the right-hand rule). The pictured loop is homotopically nontrivial.

Theorems & Definitions (18)

  • Theorem A
  • Theorem B
  • Corollary 1.1
  • Theorem C
  • Theorem D
  • Remark 1.2
  • Theorem E
  • Lemma 2.1
  • proof
  • Corollary 2.2
  • ...and 8 more