Reference for iterated homotopy fixed points? - MathOverflow

Reference for iterated homotopy fixed points?

2009-11-11T19:01:32Z

What are (good) references for results about iterated homotopy fixed points? That is, suppose G is a topological group acting on a space (or spectrum) X, and H is a normal subgroup of G. Then one would like to first compute the homotopy fixed points of X with respect to H, and use that as a stepping stone to compute the homotopy fixed points of X with respect to G.

(I am independently interested in both the space and spectrum versions, so am happy with pointers, comments regarding either.)

Answer by Tyler Lawson for Reference for iterated homotopy fixed points?

2009-11-11T20:30:26Z

I'm going to assume the groups are discrete because I don't want to worry about G-CW-structures restricting to H-CW-structures.

Say X is an "object" with a G-action and H a normal subgroup of G. Let EG be a free contractible CW-G-space, E(G/H) the same for G/H and EG x E(G/H) have the diagonal G-action.

Then homotopy fixed points of X are the G-equivariant functions F^G(EG,X) (where if X is a spectrum I want to add a disjoint basepoint to EG).

Then the projection map from EG x E(G/H) to EG is a G-equivariant equivalence, and so we get a diagram as follows. $$ F^G(EG,X) \simeq F^G(EG \times E(G/H),X) $$$$ \simeq F^{G/H}(E(G/H), F^H(EG,X)) $$ (where G/H acts on the latter function space by ${}^gf = g f g^{-1}$).

As EG is also a version of EH, this says that the G/H-homotopy fixed points of the H-homotopy fixed points is the same as the G-homotopy fixed points.

Answer by Reid Barton for Reference for iterated homotopy fixed points?

2009-11-12T04:39:03Z

The statement X^hG = (X^hH)^hG/H is true for any G-object X of any complete (∞,1)-category C. An object of C with a G-action is the same as a functor BG → C where BG represents the category (or (∞,1)-category if G is not discrete) with a single object with automorphism group G. The G-fixed points are the homotopy limit of this functor, or equivalently its right Kan extension along the functor BG → •. We can factor this latter functor as p: BG → B(G/H) followed by q: B(G/H) → •. So

$X^{hG} = (qp)_* X = q_* p_* X = (p_* X)^{hG/H}$

It remains to compute the right Kan extension of X along p. On the object • of B(G/H), it is given as the limit of the diagram X over the category • ↓ G, which is the translation groupoid of G acting on G/H, or equivalently BH. So indeed $p_* X = X^{hH}$. Identifying the action of G/H is left as an exercise for the reader. :)