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

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.

39 pages; v3: new results, examples and references added

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Cited by in corpus (6)

• Formality properties of finitely generated groups and Lie algebras
• Cohomology jump loci of 3-manifolds
• Infinitesimal finiteness obstructions
• Rank two topological and infinitesimal embedded jump loci of quasi-projective manifolds
• Cohomology, Bocksteins, and resonance varieties in characteristic 2
• Poincaré duality and resonance varieties