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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?!?

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    $\begingroup$ For any spectrum $E$ you can define a spectrum $D(E)$ such that maps $X\to D(E)$ correspond to maps $X\wedge E\to S$, or to maps $E\to D(X)$, for all spectra $X$. If $E$ is the direct limit of $E_i$ then $D(E)$ will be the inverse limit of $D(E_i)$. But some things that are true for finite $E$ will be false in general. Can you make your question any more precise? $\endgroup$ – Tom Goodwillie Aug 8 '18 at 14:54
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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.

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  • $\begingroup$ By analogy with "plain" algebra I would expect a converse to the last part: I believe it is the case that, for given $E$, its finiteness is implied by $D(E)\wedge X\to F(E,X)$ being an equivalence for all $X$? $\endgroup$ – მამუკა ჯიბლაძე Aug 8 '18 at 20:34
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    $\begingroup$ Both functors $D(E)\wedge -$ and $F(E, -)$ commute with cofibration sequences. The map is clearly an equivalence when $X=S^0$. Use induction on cells. $\endgroup$ – Gregory Arone Aug 8 '18 at 20:40
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    $\begingroup$ Yes, it preserves fibration sequences, and those are the same as cofibration sequences in spectra. $\endgroup$ – Gregory Arone Aug 8 '18 at 20:48
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    $\begingroup$ There is a converse as in the first comment. If $D(E)\wedge E\to F(E,E)$ is an equivalence then there's an element of $\pi_0(D(E)\wedge E)$ mapping to the element of $\pi_0F(E,E)$ corresponding to the identity map. Any element of $\pi_0(D(E)\wedge E)$ is in the image of some element of $\pi_0(A\wedge B)$ for some finite spectra $A$ and $B$ and maps $f:A\to D(E)$ and $g:B\to E$. $f$ corresponds to a map $f^\ast:E\to D(A)$. An element of $\pi_0(A\wedge B)$ corresponds to a map $h:D(A)\to B$. The map $g\circ h\circ f^\ast$ is the identity. Thus $E$ is a retract of a finite spectrum, hence finite. $\endgroup$ – Tom Goodwillie Aug 9 '18 at 3:10
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    $\begingroup$ @Tom Goodwillie thanks for this latter argument. $\endgroup$ – user51223 Aug 9 '18 at 9:05
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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 $\pi_{-1}$.

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    $\begingroup$ I've seen the example of the dual of $\mathbf{R}P^\infty$ before, but I don't have any intuition for why this should be true. Do you have an intuitive explanation for this fact? Thanks! $\endgroup$ – skd Aug 9 '18 at 2:03
  • $\begingroup$ Sorry, I don't. $\endgroup$ – Tom Goodwillie Aug 9 '18 at 2:53
  • $\begingroup$ I guess for Thom spectra, say $E=X^\xi$ the bundle map $\xi\times(-\xi)\to 0$ upon Thomification leads to a duality map which may explain this. $\endgroup$ – user51223 Aug 9 '18 at 9:03
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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.

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