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