On powers of the Euler class for flat circle bundles

TL;DR

This paper proves that powers of the Euler class are nonzero in integer cohomology for flat circle bundles with analytic diffeomorphisms, contrasting with previous results in rational cohomology, and provides a simplified proof of Morita's theorem.

Contribution

It establishes the non-vanishing of Euler class powers in integer cohomology for flat circle bundles with analytic structure, and offers a concise proof of Morita's theorem.

Findings

Powers of the Euler class are nonzero in integer cohomology.

Contrasts with rational cohomology where the cube vanishes.

Provides a simplified proof of Morita's theorem.

Abstract

Apparently a lost theorem of Thurston states that the cube of the Euler class $e^3\in H^6(BDiff^{\delta}_{\omega}(S^1);\mathbb{Q})$ is zero where $Diff^{\delta}_{\omega}(S^1)$ is the analytic orientation preserving diffeomorphisms of the circle with the discrete topology. This is in contrast with Morita's theorem that the powers of the Euler class are nonzero in $H^*(BDiff^{\delta}(S^1);\mathbb{Q})$ where $Diff^{\delta}(S^1)$ is the orientation preserving $C^{\infty}$- diffeomorphisms of the circle with the discrete topology. The purpose of this short note is to prove that the powers of the Euler class $e^k \in H^*(BDiff^{\delta}_{\omega}(S^1);\mathbb{Z})$ in fact are nonzero in cohomology with integer coefficients. We also give a short proof of Morita's theorem.