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Let $G$ be a finite group and let $G$ act on a vector space $V$. Let $S(V)$ be the representation sphere. In the monoidal category of $RO(G)$-graded spectra $S(V)$ is invertible, so it's dualizable too.

Is $S(V)$ dualizable in the monoidal category of naive $G$-spectra?

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Let $G$ be of order $2$, and let $V$ be the nontrivial one-dimensional representation. I'll write $S(V)$ for the unit sphere in $V$ (which is just $G$); the one-point compactification of $V$ is then the unreduced suspension of $G$, which I'll call $S^V$. We can only form spectra from $G$-spaces with fixed basepoint, so we need to consider $S(V)_+=G_+$ rather than $S(V)$ itself. There is a cofibration $G_+\to S^0\to S^V$, and $S^0$ is its own dual, so the question for $S^V$ is equivalent to the question for $G_+$.

In the category of genuine $G$-spectra, $G_+$ is its own dual, which means that there are unit and counit maps $S^0\xrightarrow{\eta}G_+\wedge G_+\xrightarrow{\epsilon}S^0$ making certain diagrams commute. In the naive category, $\{S^0,G_+\wedge G_+\}^G$ is the colimit of the unstable mapping sets $[S^n,S^n\wedge G_+\wedge G_+]^G$, which are trivial because $S^n\wedge G_+\wedge G_+$ is free away from the basepoint. Thus, we cannot construct the required map $\eta$, and $G_+$ is not self-dual. The same applies if we replace one of the $G_+$'s by something else, so $G_+$ is not dualisable.

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  • $\begingroup$ Neil, in the naive category don't we have to take a cofibrant replacement of $S^n$ to calculate homotopy classes of maps? $\endgroup$
    – Tyler Lawson
    Commented Jul 13, 2012 at 13:19
  • $\begingroup$ @Tyler: what foundations do you have in mind? For finite groups I think we can do something very similar to Boardman's original setup, where a naive $G$-spectrum consists of $G$-CW complexes with equivariant structure maps $\Sigma X_n\to X_{n+1}$. A genuine $G$-spectrum has $X_n$ defined only when $|G|$ divides $n$, and structure maps $S^{\mathbb{R}[G]}\wedge X_n\to X_{n+|G|}$. My argument is valid in that context. With EKMM foundations you would need cofibrant replacement wrt a model structure based on the trivial universe, which would not make a difference at the end of the day. $\endgroup$
    – Neil Strickland
    Commented Jul 13, 2012 at 16:53
  • $\begingroup$ Naive Spanier-Whitehead works fine equivariantly and gives the full subcategory of finite G-CW spectra (up to equivalence) in any other good construction. Neil's answer is fine, with R the regular representation, but Gaunce's result I cited is valid for orbits of compact Lie groups, not just finite ones. The pre-EKMM Lewis-May foundations work for compact Lie groups and any universe, the suspension G-spectra of spheres with trivial $G$-action are cofibrant there, and space level maps can be used via adjunction. Passing to EKMM doesn't clarify things here. $\endgroup$
    – Peter May
    Commented Jul 13, 2012 at 17:10
  • $\begingroup$ @Neil: My apologies. My comment was based on misinterpreting the question: I was considering a different weak equivalence structure on naive G-spectra (using underlying weak equivalences, rather than equivariant ones). $\endgroup$
    – Tyler Lawson
    Commented Jul 13, 2012 at 19:07
  • $\begingroup$ @TL: Is that the same as spectra over BG? In this category the analog of $G_+$ has a dual? $\endgroup$
    – user25092
    Commented Jul 14, 2012 at 16:15
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There is a strongly related result of Gaunce Lewis that I would like to advertise. We have different kinds of $G$-spectra for different ``universes'' $U$, which are countable sums of representations that include $\mathbf{R}$ and countably many copies of each representation they contain. Naive $G$-spectra are indexed on the trivial universe, genuine ones are indexed on the complete universe, which contains all irreducible representations. Gaunce proved that the $U$-suspension $G$-spectrum of an orbit $G/H_+$ (I've added a disjoint basepoint) is dualizable if and only if $G/H$ embeds as a sub $G$-space of a representation in the universe $U$. Neil's example is the simplest illustration of this general result. (Tyler, no problem with cofibrancy here: Neil is working in the Spanier-Whitehead category; Gaunce is working with Lewis-May $G$-spectra.)

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