## Upcoming Seminars

Dec
12
Jeffrey Carlson
Dec 12, 15:30 - 16:30
MC 107
Equivariant formality in rational cohomology and $K$-theory
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 Ademâ€“GÃ³mez 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 Goertschesâ€“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 Shigaâ€“Takahashi to an equivalence. This work is joint with Chi-Kwong Fok.

Jan
16
Christin Bibby
Jan 16, 15:30 - 16:30
MC 107
TBA

Jan
30
Hiraku Abe
Jan 30, 15:30 - 16:30
MC 107
TBA