The topology of compact Lie group actions through the lens of finite models

TL;DR

This paper explores the algebraic models of manifolds with Lie group actions, revealing their cohomological and formal properties, with applications to Sasakian manifolds and surface singularities.

Contribution

It introduces a method to construct algebraic models for manifolds with Lie group actions and analyzes their formality and cohomology properties under certain conditions.

Findings

Compact Sasakian manifolds are (n-1)-formal.

Fundamental groups of these manifolds are filtered-formal.

Applications to weighted-homogeneous surface singularities.

Abstract

Given a compact, connected Lie group $K$, we use principal $K$-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let $M$ be a compact, connected, smooth manifold which supports an almost free $K$-action. Under a partial formality assumption on the orbit space and a regularity assumption on the characteristic classes of the action, we describe an algebraic model for $M$ with commensurate finiteness and partial formality properties. The existence of such a model has various implications on the structure of the cohomology jump loci of $M$ and of the representation varieties of $\pi_1(M)$. As an application, we show that compact Sasakian manifolds of dimension $2n+1$ are $(n-1)$-formal, and that their fundamental groups are filtered-formal. Further applications to the study of weighted-homogeneous isolated surface singularities are also given.