On the rational homotopy Lie algebra of a fixed point set of a torus action

Abstract

Let X X be a simply connected topological space, and let L ∗ ( X ) {\mathcal {L}_{\ast }}(X) be its rational homotopy Lie algebra. Suppose that a torus acts on X X with fixed points, and suppose that F F is a simply connected component of the fixed point set. If L ∗ ( X ) {\mathcal {L}_{\ast }}(X) is finitely presented and if F F is full, then it is shown that L ∗ ( F ) {\mathcal {L}_{\ast }}(F) is finitely presented, and that the numbers of generators and relations in a minimal presentation of L ∗ ( F ) {\mathcal {L}_{\ast }}(F) do not exceed the numbers of generators and relations (respectively) in a minimal presentation of L ∗ ( X ) {\mathcal {L}_{\ast }}(X) . Various other related results are given.