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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.)

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  • $\begingroup$ I realized I don't know what you mean. Do you mean that you want a proof that the iterated fixed-point object is fixed points for G, or are you interested in computational methodologies? $\endgroup$
    – Tyler Lawson
    Commented Nov 11, 2009 at 21:54
  • $\begingroup$ Primarily the former, that (X^hH)^hG/H = X^hG, but I'd also be happy for any references to computational approaches or examples. $\endgroup$
    – cdouglas
    Commented Nov 11, 2009 at 23:21

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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 FG(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.

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The statement XhG = (XhH)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. :)

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