Reference for iterated homotopy fixed points? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T09:36:23Z http://mathoverflow.net/feeds/question/5087 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5087/reference-for-iterated-homotopy-fixed-points Reference for iterated homotopy fixed points? cdouglas 2009-11-11T19:01:32Z 2009-11-12T04:42:24Z <p>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.</p> <p>(I am independently interested in both the space and spectrum versions, so am happy with pointers, comments regarding either.)</p> http://mathoverflow.net/questions/5087/reference-for-iterated-homotopy-fixed-points/5100#5100 Answer by Tyler Lawson for Reference for iterated homotopy fixed points? Tyler Lawson 2009-11-11T20:30:26Z 2009-11-11T23:43:01Z <p>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.</p> <p>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.</p> <p>Then homotopy fixed points of X are the G-equivariant functions F<sup>G</sup>(EG,X) (where if X is a spectrum I want to add a disjoint basepoint to EG).</p> <p>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 ${}^gf = g f g^{-1}$).</p> <p>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.</p> http://mathoverflow.net/questions/5087/reference-for-iterated-homotopy-fixed-points/5152#5152 Answer by Reid Barton for Reference for iterated homotopy fixed points? Reid Barton 2009-11-12T04:39:03Z 2009-11-12T04:39:03Z <p>The statement X<sup>hG</sup> = (X<sup>hH</sup>)<sup>hG/H</sup> is true for any G-object X of any complete (&infin;,1)-category C. An object of C with a G-action is the same as a functor BG &rarr; C where BG represents the category (or (&infin;,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 &rarr; &bull;. We can factor this latter functor as p: BG &rarr; B(G/H) followed by q: B(G/H) &rarr; &bull;. So</p> <p>$X^{hG} = (qp)_* X = q_* p_* X = (p_* X)^{hG/H}$</p> <p>It remains to compute the right Kan extension of X along p. On the object &bull; of B(G/H), it is given as the limit of the diagram X over the category &bull; &darr; 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. :)</p>