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Currently I know nothing about homotopy fixed points except for its definition: given a $G$-space $X$, the set of homotopy fixed points is defined as the space of equivariant maps from $EG$ to $X$.

I was wondering under which (maybe simple) conditions, homotopy fixed points could be “identified” with (ordinary) fixed point sets, in a suitable sense.

I ask this question since I know that if the action of $G$ on $X$ is free, then the Borel construction $(EG\times X)/G$ is homotopy equivalent to $X/G$. This is how I understand equivariant (co)homology as a good replacement of the (co)homology of $X/G$ in case the action is not free.

It seems the condition dual to “free” (no kernel) should be “transitive” (no cokernel). Is there a good correspondence between homotopy fixed points and ordinary fixed points when $G$ acts transitively on $X$? Or any other conditions and results?

If the above naive attempt is not in the correct way, how should I understand homotopy fixed points as a natural generalization/replacement of ordinary fixed points?

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If $G$ is discrete and $T$ is a transitive $G$-orbit with $\left\lvert T \right\rvert > 1$, then $T^{hG} \cong T^G$, but in this case both fixed points are empty. It's easy to see that the latter is empty, and for the former note that the only maps from the connected space $EG$ to the discrete space $T$ are the constant maps, and they aren't $G$-equivariant if $\left\lvert T \right\rvert > 1$. So this case is not very interesting.

The correct dual to "free" should be "cofree": a $G$-space $C$ is cofree if there is some ordinary space $\bar{C}$ such that $$\operatorname{Map}_G(X,C) \simeq \operatorname{Map}(X, \bar{C})$$ for all $G$-spaces $X$ in a natural way. For example, $$C = \operatorname{Map}(G, \bar{C})$$ is cofree for any $\bar{C}$.

From this definition, we can show that if $C$ is cofree, then $$C^{hG} = \operatorname{Map}_G(EG, C) \simeq \operatorname{Map}(EG, \bar{C}) \simeq \bar{C}$$ since $EG \simeq *$ non-equivariantly.

Here's one way to think about homotopy fixed points intuitively. Homotopy fixed points can be regarded as a relaxed version of ordinary fixed points. Instead of requiring a fixed point $x$ to satisfy equalities $x = gx$ for all $g \in G$, we simply require $x$ and $gx$ to be connected by paths in a coherent manner (these paths are part of the data) for it to be a homotopy fixed point

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  • $\begingroup$ Great, thank you! In your definition of cofree action, taking $X=pt$ we see $\bar{C}\simeq C^G$. $\endgroup$
    – user188722
    Mar 4, 2021 at 20:50

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