Subject: Re: 1 or 2 postings
Date: Thu, 26 Feb 2004 17:18:58 +0100 (CET)
From: John Rognes <rognes@math.uio.no>
To: Don Davis <dmd1@lehigh.edu>
CC: John Rognes <rognes@math.uio.no>

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 <zga2m@virginia.edu>
>
> [..] 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