# Equivariant K-theory for products of groups?

Let $$X$$ be a $$(G \times H)$$-space. What is known about the connection between the groups $$K_G(X)$$, $$K_H(X)$$ and $$K_{G \times H}(X)$$? The $$G$$ and $$H$$ action on $$X$$ come from the canonical inclusions $$G \to G \times H$$ and $$H \to G \times H$$.

There is the Künneth theorem relating $$K_G(X)$$, $$K_H(Y)$$ and $$K_{G \times H}(X \times Y)$$, where $$X$$ is a $$G$$-space and $$Y$$ is a $$H$$-space. This only reduces to the above case when either $$G$$ or $$H$$ acts trivially (and the corresponding $$X$$ or $$Y$$ is a point).

I can't find anything in the literature, for K-theory or equivariant cohomology. Is there a specific reason why it is so hard to get information about $$K_{G \times H}(X)$$ from $$K_G(X)$$ and $$K_H(X)$$?

• Let me mention that a $G\times H$-action carries potentially more structures than a $G$-action and an $H$-action. At least, there is a commutativity between two actions (and in higher algebra, such as the K-theory spectrum that you want to study, commutativity is an extra structure).
– Z. M
Oct 18 at 13:33

A Künneth formula for equivariant $$K$$-theory. Haruo Minami. Osaka J. Math. 6(1): 143-146 (1969).
As a final comment note that if $$G$$ acts on spaces $$X$$ and $$Y$$ it is too much hard to obtain $$K_G(X\times Y)$$ in terms of $$K_G(X)$$ and $$K_G(Y)$$
• OP is asking for the case when both $G$ and $H$ are acting on the same space $X$, and interested in the $G\times H$ equivariant $K$-theory of $X$ (and not that of $X \times X$). Nov 13, 2022 at 19:49