$S$-dual of filtered spectra

Question

I hope this is research level. Suppose $E$ is the direct limit of finite spectra, say $E=\mathrm{colim }\ E_i$, which itself is not finite. I wonder how much and under which conditions the inverse limit $\mathrm{lim}\ D(E_i)$ is a good candidate for playing role of $D(E)$? Here, I write $D$ for the $S$-duality functor. I feel there is problem with the possibility of existence of phantom maps here. I will be very grateful for any advise on this, or possibly some references on the topic.

As to what one might expect. For instance, if $f:X\to Y$ is a map between finite spectra then $f_*=0$ if and only if $D(f)_*=0$. Or if we know about dimension of bottom cell of $E$ then that would tell about the dimension of the top cell of the dual spectrum. Here, $f_*$ is the map induced in ordinary homology and I consider CW spectra. And if $D(E)$ can always be identified with the function spectrum $F(E,S^0)$?!

I also would like to know what features of duality for finite spectra fail to hold in general. For the moment, homological behavior is very interesting for me. I also wonder if my worry about phantom maps has any point?!?

Answer 1

One key property of duality is that the canonical map from $E$ to its double dual $D(D(E))$ is a homotopy equivalence when $E$ is a finite spectrum. This fails (in general) for infinite spectra. Moreover, the map $E\to D^2(E)$ is not always a monomorphism in the homotopy category, or even injective on homotopy groups. For example, it is known that the SW-dual of the Eilenberg-Mac Lane spectrum $H{\mathbb Z/p}$ is contractible (T.Y. Lin, Duality and Eilenberg-Mac Lane spectra). So it is not true that if $D(f)$ is null then so is $f$.

Regarding phantom maps: suppose $E$ is the homotopy direct limit of finite spectra $E_i$. Then the homotopy groups of $D(E)$ fit into a lim$^1$ short exact sequence

$$0\to {\lim}^1 \pi_{*+1}(DE_i)\to \pi_*(DE) \to \lim \pi_*(DE_i) \to 0$$

The image of ${\lim}^1 \pi_1(D(E_i))$ in $\pi_0(DE)$ represents precisely the phantom maps from $E$ to $S^0$.

Another key property of duality: in good cases the map $D(E) \wedge X \to F(E, X)$ is a homotopy equivalence. Here $F(-, -)$ stands for function spectrum. This holds if either $E$ or $X$ is a finite spectrum, but not in general.

Answer 2

Here are a couple more interesting examples.

For a finite group $G$ the dual of the suspension spectrum of $BG$ is described by the (solved) Segal Conjecture. One of the most striking features of the answer is that the dual spectrum is connective. For example, even though the dual of $\mathbb RP^n$ has cohomology down to dimension $-n$, the dual of $\mathbb RP^\infty$ is a lot like (the suspension spectrum of) $\mathbb RP^\infty$.

The dual of $E=H\mathbb Q$ can be determined by viewing $E$ as a direct limit of copies of the sphere spectrum; it is a corresponding inverse limit of copies of the sphere spectrum, and its only nontrivial homotopy group is $\pi_{-1}$.

Answer 3

Yet more fun examples ...

If $A$ is an abelian torsion group and $n \ge 2$, the suspension spectrum of $K(A,n)$ has trivial dual: this is a result of Chun-Nip Lee [Amer.J.Math 114 (1992)]. (This implies the result of T.Y. Lin that Greg mentioned.)

An example that I observed in [Progress in Math 215 (2003) (arXiv)] goes as follows: if $K$ and $X$ are finite CW complexes, and the connectivity of $X$ is greater than the dimension of $K$, then the suspension spectrum of the space of continuous maps from $K$ to $X$ is an $S$-dual. (For example, $\Sigma^\infty \Omega^{2n-1}\mathbb CP^{m}/\mathbb CP^n$ is an $S$-dual, for all $m>n$.) In fact, it is an $S$-dual of a filtered spectrum as a functor of $X$, and the dual of this filtration yields the Goodwillie tower.

Regarding intuition behind the Segal conjecture, it can be rephrased as saying that a map from a fixed point spectrum to an associated homotopy fixed point spectrum is an appropriate sort of completion. So, for example, there is an easily defined sensible map $\mathbb RP^{\infty} \vee S^0 \rightarrow D(\mathbb RP^{\infty})$ that turns out to be an equivalence after completing the sphere at 2.