Subject: Re: two postings Date: Fri, 27 Feb 2004 10:38:39 -0500 From: Greg Friedman Organization: Yale University Department of Mathematics To: Don Davis A reply to Tom Goodwillie's question: I've always been a fan of "Elements of Algebraic Topology" by Munkres. It essentially starts from scratch with simplicial complexes (complices?) and does cover Cech cohomology by the end. There might be a lot more in there than needed, but it's all covered quite well. Greg Friedman Don Davis wrote: > Two postings: A text question and a response to yesterday's > question...DMD > _______________________________________________ > > Date: Thu, 26 Feb 2004 07:42:22 -0500 > From: Tom Goodwillie > > For the list: > > David Mumford, who does applied math (pattern recognition) here at > Brown, asks the following: > > "I have a student who has never done topology but needs to master > some because of Gunnar Carlsson's latest stuff with clustering in > high dimensional spaces. Can you recommend a good intro book, > starting with basic finite simplicial complexes, lots of examples and > -- if possible -- doing Cech coho?" > > I'll bet people on the list have some good answers. > > Tom Goodwillie > ____________________________________________________________ > > Date: Thu, 26 Feb 2004 17:18:58 +0100 (CET) > From: John Rognes > > On Thu, 26 Feb 2004, Don Davis wrote: > > >>Subject: Question for the mailing list >>Date: Wed, 25 Feb 2004 16:44:33 -0500 >>From: Greg Arone >> >>[..] Suppose that the homotopy orbit spectrum E \smash_{G} EG_+ is >>contractible. Does it follow that the homotopy orbits of the dual >>spectrum D(E) \smash_G EG_+ is contractible? [..] If it is not true in > > >>general, can one give sufficient conditions on E for this to be true? > > > Dear Greg, > > If EG_+ smash_G E = E_{hG} is contractible, then so is its functional > dual > D(E_{hG}) = F(EG_+ smash_G E, S) = F(EG_+, F(E, S))^G = D(E)^{hG}, i.e., > > the homotopy fixed point spectrum of D(E). So from the norm sequence > > D(E)_{hG} -N-> D(E)^{hG} --> D(E)^{tG} > > the contractibility of D(E)_{hG} is equivalent to the contractibility of > > the Tate spectrum D(E)^{tG}. (Sometimes it is denoted t_G D(E)^G or > hat H(G, D(E).) > > If D(E) is a G-equivariant ring spectrum, e.g. if E is the suspension > spectrum of a finite G-space, then the Tate spectrum is an algebra over > the homotopy fixed point spectrum, so if the latter is contractible, > surely so is the Tate spectrum. Thus in this case D(E)_{hG} will be > contractible. > > As you seem to be aware, if E is equivalent to a finite free G-spectrum, > > or equivalently, if D(E) is equivalent to a finite free G-spectrum, then > > the Tate spectrum is also contractible, hence so is the homotopy orbit > spectrum. > > - John > -- Greg Friedman "Where life had no value, death, sometimes, had its price." -For a Few Dollars More