# How to extend an equivariant map from a compact Lie group

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.

• From your formulas, it seems that the centralizer acts from the left on G, and H acts only on G, and from the right. Is this correct? Next, to be more precise, you say that a map f from your space is determined by its value on one representative of each double coset, and you want this value to lie in the fixpoint set you specified. – Sebastian Goette Oct 6 '15 at 17:56
• ... The points in C\G/H come with stabilizers aHa^{-1}\cap C. These groups are ordered partially by inclusion. I would suggest to fix f for those points with greatest stabilizers first. Then you get boundary values for the subspaces with smaller stabilisers, and you can work your way up to the generic points with the smallest possible stabilisers. This should produce a continuous map f for you. – Sebastian Goette Oct 6 '15 at 18:03
• Thanks. But I'm not familiar with what you said. Why the points with the greatest stabilizers give the boundary values for the subspaces with smaller ones? So if $C_G(g)\backslash G/H$ does not have generic point, we just end till reach the points with the smallest stabilizer group? BTW, does $C_G(g)\backslash G/H$ usually have generic points.... I appreciate a lot if you can recommend some reference. – Megan Oct 7 '15 at 1:16
• If you follow the approach I outlined, you will need some technology. So you could follow Strickland's suggestion and consider P- or PxH-CW structures on X and G. Those also start with small cells with large stabilisers. I don't know which references are good, but maybe you start with Matumoto, Takao, On G-CW complexes and a theorem of J. H. C. Whitehead, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971), 363–374, or Illman, Sören Equivariant singular homology and cohomology for actions of compact Lie groups, Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, 403–415. – Sebastian Goette Oct 7 '15 at 9:46

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 $[X,Y]^P$.