Cfrgtkky

Suppose $X$ is a finite dimensional CW-complex with top cell at dimmension $n$ and consider its S-dual denoted by $DX$. I wonder if there are any obstructions to find a space $Y$ and an interger $kgeqslant n$ so that
$$Sigma^kD(X)simeqSigma^infty Y_+ ?$$
For example, in the case of $X=S^m$ the answer is positive.

EDIT In the case of finite dimensional projective spaces, one may start from spaces $mathbb{R}P_m^n$ and choosing $m$ and $n$ in accordance with James periodicity, we can actually compute such a $k$.

The question is that is there any such $k$ for a given CW-complex of finite type (which of course after Neil's answer below I realise the answer is positive if one is to choose $k$ enough large, and obstructions are for specific $k$'s)

I would be very grateful for any references.

Firstly, you say that the answer is negative for finite-dimensional projective spaces. However, in this case, and for any finite complex $X$, the answer will be positive if we take $k$ sufficiently large. Perhaps you are just thinking of the case $k=n$? Anyway, I will assume that we have fixed some particular $kgeq n$ and we want to know whether $Sigma^kDX$ is a suspension spectrum.

The most obvious point is that $H^*(Sigma^kD(X);mathbb{F}_p)$ needs to be an unstable module over the Steenrod algebra. This is easy to check if you have a good understanding of $H^*(X;mathbb{F}_p)$ as a Steenrod module. Similarly, the module $M=K^0(Sigma^kDX)$ has naturally defined Adams operations $psi^qcolon Mto M[q^{-1}]$, and if $Sigma^kDX$ is a suspension spectrum then these will lift in a compatible way to give operations $psi^qcolon M to M$. If this does not settle the question then one can consider additive unstable operations in $BP$ theory or $MU$ theory. These are harder to handle explicitly but the basic slogan is as follows: $MU^*(Sigma^kDX)$ is functorial for isomorphisms of formal groups, and if $Sigma^kDX$ is a suspension spectrum then this extends to give functoriality for all homomorphisms of formal groups.