An action of a group $G$ on a space $X$ is said to be equivariantly formal if the induced map from Borel equivariant cohomology of the action to singular cohomology of $X$ is surjective. This situation is much to desired in various geometric settings, where it can allow the integral of an invariant function to be reduced to one over a lower-dimensional or even finite fixed-point set.

It turns out (work of Fok) one can approach the question in terms of equivariant $K$-theory: the action of a compact Lie group $G$ on a finite $G-CW$ complex $X$ is equivariantly formal with rational coefficients if and only if some each power of every vector bundle over $X$ admits a stable equivariant structure. This correspondence allows us an alternate proof, with a slightly stronger conclusion, of a result of Adem–Gómez on the equivariant $K$-theory of actions with maximal-rank isotropy.

In the case of the left translation ("isotropy") action of a connected group $H$ on a homogeneous space $G/H$, the correspondence also allows us to more simply reobtain results of Goertsches–Noshari on generalized symmetric spaces. In the realm of rational homotopy theory, we are able to show that equivariant formality of the isotropy action implies $G/H$ is a formal space, allowing us to improve a sufficient condition of Shiga–Takahashi to an equivalence.

This work is joint with Chi-Kwong Fok.

Rearrangement Lemma is a very deep result that first was stated and proved by Connes. He proved it to integrate a vector valued function while studying the geometry of noncommutative 2-tori with a conformally perturbed metric. In our recent work joint with Masoud Khalkhali and Ali Moatadelro, on the noncommutative 3-torus, we have proved a slightly different version of that. In this talk I will give a proof of that version of Rearrangement Lemma.