Need examples of homotopy orbit and fixed points

Question

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

1. Homotopy orbits or fixed points differ from ordinary fixed points and ordinary orbits.
2. 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.

Answer 1

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?

Answer 2

Homotopy fixed points and orbits make sense in a much wider (but not necessarily less elementary!) context than for spaces and spectra. In particular, they make sense for categories. Some examples:

• Suppose $G$ is a group acting on another group $N$. The homotopy quotient of this action (thinking of $N$ as a one-object category) is the semidirect product $N \rtimes G$. (This can be stated as a fact about spaces, where it says that the homotopy quotient of $BN$ by an action of $G$ induced by an action on $N$ is $B(N \rtimes G)$.)
• Suppose $G$ is a group, $A$ is an abelian group on which $G$ acts, and $\alpha \in H^2(BG, A)$ is a $2$-cocycle. The $2$-cocycle can be used to modify the action of $G$ on $A$ in such a way that the homotopy quotient of this action is the extension of $G$ by $A$ determined by $\alpha$. (This can also be stated as a fact about spaces.)
• Again suppose $G$ is a group acting on another group $N$. This induces an action of $G$ on $\text{Rep}(N)$. The homotopy fixed points of this action is $\text{Rep}(N \rtimes G)$.
• Let $K \to L$ be a finite Galois extension with Galois group $G$. Then $G$ naturally acts on the category $\text{Lie}(L)$ of Lie algebras over $L$. The homotopy fixed points of this action is $\text{Lie}(K)$.

If you insist on examples involving spaces, to my mind the nicest examples come from homotopy quotients by actions of $\mathbb{Z}$. Such actions correspond to homeomorphisms $f : X \to X$ from a space to itself, and the homotopy quotient can be modeled by the mapping torus of $f$. One of the many great things about the mapping torus construction is that if it takes as input a diffeomorphism of a smooth manifold, it returns as output a smooth manifold of one dimension higher which only depends on the class of the diffeomorphism in the mapping class group. In particular, the mapping torus turns mapping class group elements of surfaces into $3$-manifolds, and knowledge about how mapping class group elements behave turns into knowledge about the corresponding $3$-manifolds.

You can draw pretty pictures here, too. For example, the mapping class group of an $n$-punctured disk is the braid group $B_n$, and you can draw very vividly what the mapping torus of an element of the braid group looks like: it's the complement of the link determined by the braid group element in a solid torus.

By contrast, the ordinary quotient of a space by the action of a homeomorphism can be quite ugly, and this is enough to give examples of both of the kinds of phenomena you ask for. Consider, for example, an irrational rotation of $S^1$: the quotient space is not even Hausdorff, but the homotopy quotient space is the torus $S^1 \times S^1$. The irrational rotation is homotopy equivalent as an action to the identity, and the corresponding quotient space is now $S^1$.

One last example involving spaces which is also relatively easy to visualize. Consider the action of $\mathbb{Z}_2$ on $S^1$ by reflection. The quotient by this action is the interval, and in particular is contractible. The orbifold / stacky quotient should look like the interval, but where the two endpoints are orbifold / stacky points with stabilizer $\mathbb{Z}_2$, and so one might guess that the homotopy quotient should be the wedge sum of two copies of $B \mathbb{Z}_2$, or equivalently $B (\mathbb{Z}_2 \ast \mathbb{Z}_2)$. Thinking of $S^1$ as $B \mathbb{Z}$ and using the first example I gave above, the homotopy quotient is in fact $B(\mathbb{Z} \rtimes \mathbb{Z}_2)$. But no problem: these groups are isomorphic!

Answer 3

I mention this paper "Groupoids and Faa di Bruno formulae for Green functions in bialgebras of trees" which uses groupoids and homotopy orbits.

@Qiaochu : Another aspect of the last example given in your answer is that the fundamental groupoid on two points of the orbit space comes out nicely from the notion of orbit groupoid $\Gamma//G$ given by the action of a group $G$ on a groupoid $\Gamma$, as discussed in Chapter 11 of Topology and Groupoids. An older version of this chapter is available as arXiv:math/0212271. One example computed there is the fundamental group of the symmetric square of a space $X$: under reasonable conditions, it is the fundamental group of $X$ made abelian.