How to extend an equivariant map from a compact Lie group

Question

Let $G$ be a compact Lie group and let $H$ be a closed subgroup of it. Let $g$ be a torsion element of $G$ and $C_G(g)$ the centralizer of it. Let $Y$ be a $C_G(g)-$space. I'm working on the space $$Map_{C_G(g)}(G, Y)^H,$$ the fixed point space of $Map_{C_G(g)}(G, Y)$ by $H$.

I studied the image of each double coset $C_G(g)\alpha H$. All the equivariant properties of the maps in $Map_{C_G(g)}(G, Y)$ are satisfied if and only if the image of each $\alpha$ is in $Y^{\alpha H\alpha^{-1}\cap C_G(g)}$. I don't have a good idea how to put all these pieces together regarding the topology on $G$ to make it a continuous map.

One question is: when is this space $Map_{C_G(g)}(G, Y)^H$ nonempty even if each $Y^{\alpha H\alpha^{-1}\cap C_G(g)}$ is nonempty? What additional properties should it satisfy?

Another questions: Let $X$ be a $G-$subspace of $G$, which is the union of several double cosets $C_G(g)\alpha H$. Given a map in $Map_{C_G(g)}(X, Y)^H$, when can it be extended to a map in $Map_{C_G(g)}(G, Y)^H$? I have no good idea so far.

Thanks.

Answer 1

First, if I understand your conventions correctly, the thing that you are interested in can be described as $\text{Map}_P(X,Y)$, where $P=C_G(g)$ and $X=G/H$ (considered as a $P$-space by left multiplication) and $Y$ is some other $P$-space.

In this level of generality, I do not think that you gain anything from the fact that $P$ is the centraliser of a torsion element and that $X$ arises as $G/H$. If you have a particular choice of $G$, $H$, $g$ and $Y$ in mind, then it might be possible to say something specific. If not, then you are just left with methods that apply to $\text{Map}_P(X,Y)$ for any compact Lie group $P$, and any $P$-spaces $X$ and $Y$. Two questions which might be relevant:

• Is $Y$ a manifold?
• Are you primarily interested in the set $[X,Y]^P=\pi_0(\text{Map}_P(X,Y))$ of equivariant homotopy classes, or the the actual geometry of the space $\text{Map}_P(X,Y)$?

Some approaches you could use:

• Study the spaces $\text{Map}_Q(X,Y)\supseteq\text{Map}_P(X,Y)$ and the maps $\text{Map}_Q(X,Y)\to\text{Map}(X^Q,Y^Q)$ for various subgroups $Q\leq P$.
• Find a $P$-equivariant CW structure on $X$, with skeleta $X_k$ say, and use equivariant obstruction theory to understand the difference between $\pi_0\text{Map}_P(X_k,Y)$ and $\pi_0\text{Map}_P(X_{k+1},Y)$ in terms of the homotopy groups of various spaces $Y^Q$.
• Study the equivariant $K$-theory rings $K_P(X)$ and $K_P(Y)$, possibly including their Adams operations. Any equivariant map $f\colon X\to Y$ will give a ring map $f^*\colon K_P(Y)\to K_P(X)$, which will be compatible with Adams operations. Depending on the details of your situation, this may give useful information about the possibilities for $f$.
• Given any $P$-equivariant fibre bundle $F\to E\to B$, there are useful relationships between $[X,F]^P$, $[X,E]^P$ and $[X,B]^P$ (especially if $E^P\neq\emptyset$). If there are naturally occurring fibre bundles where $Y$ appears as $F$, $E$ or $B$, then this can be a good way to approach $[X,Y]^P$.