I am no expert in equivariant homotopy theory. Let's say, I am planing to give a talk on homotopy fixed points and orbits. My audience will be graduate students who are doing algebraic topology or others who have fair knowledge of basic algebraic topology.
I want to give lot of computational examples of fairly basic but illuminating examples homotopy orbits and homotopy fixed points and illustrate that
- Homotopy orbits or fixed points differ from ordinary fixed points and ordinary orbits.
- Examples two homotopy equivalent spaces, where the $G$-fixed points of respective spaces are not homotopic or the $G$-orbits of respective spaces are not homotopic.
However, I want to avoid examples involving spectra.
Please help me in creating a substantial list of examples/counterexamples/results (of any kind) related to homotopy fixed points and orbits that is suitable for an introductory/motivational talk. References will also be appreciated.
Best Answer
You might mention the case where the group is infinite cyclic with generator $t$. In this case a point in the homotopy fixed space $X^{hG}$can be taken to be a pair $(x,\alpha)$ where $x\in X$ and $\alpha:I\to X$ is a path from $x$ to $\alpha(x)$.
You might make the point that a homotopy fixed point space is in some sense more computable than a fixed point space, partly because if a $G$-map $X\to Y$ is (nonequivariantly) a homotopy equivalence then the induced map $X^{hG}\to Y^{hG}$ is also an equivalence. In this connection you might note that when $G$ is infinite cyclic then there is a long exact sequence relating homotopy groups of $X$ and $X^{hG}$. This generalizes to other groups, but you might need spectral sequences.
Will you mention the (solved) Sullivan Conjecture?