Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles? The question is inspired by an answer to The concept of Duality
It is explained in that answer that the Spanier-Whitehead dual of a compact manifold is given by the Thom spectrum of normal bundles of the manifold.
Spanier-Whitehead duals exist more generally for e.g. all finite CW-complexes.
Is there known a generalization of the notion of normal bundle to e. g. all finite CW-complexes which would generalize the above description of the Spanier-Whitehead dual?
 A: Let $X$ be a finite complex. Then the functor
$$\lim_X:\operatorname{Fun}(X,\operatorname{Sp})\to \operatorname{Sp}$$
sending a local system of spectra $E$ to its limit preserves all colimits. Indeed it preserves all finite colimits by stability, and it preserves all filtered colimits by the finiteness of $X$. Therefore it is of the form
$$\lim_X E \cong \operatorname*{colim}_X(E\otimes \omega_X)$$
for a certain local system of spectra $\omega_X$ (this is by the universal property of the presheaf category and the fact that $\operatorname{Fun}(X,\operatorname{Sp})$ is the stabilization of the presheaf category). The local system $\omega_X$ is called the dualizing sheaf of $X$. Now if we let $\mathbb{S}_X$ be the constant local system at the sphere spectrum $\mathbb{S}$ we have
$$\mathbb{D}X_+=\operatorname{map}(\Sigma^\infty_+X,\mathbb{S})\cong\lim_X \mathbb{S}_X\cong \operatorname*{colim}_X \omega_X$$
where the right hand side is some sort of generalized Thom spectrum.
When $\omega_X$ is invertible (i.e. all its stalks are spheres), it is called the Spivak normal fibration of $X$, and $X$ is said to be a Poincaré complex. Note that $\omega_X$ is rather explicit: it follows formally from the definition that
$$\omega_X(x)=\lim_{z\in X^{op}} \Sigma^\infty_+ \operatorname{Map}_X(z,x)$$
where $\operatorname{Map}_X(z,x)$ is the space of paths from $z$ to $x$. Moreover one can show that for a closed topological manifold $X$ we have a natural equivalence
$$\omega_X(x)\cong \mathbb{D}\left(X/X\smallsetminus\{x\}\right)$$
where $X/X\smallsetminus\{x\}$ is the cofiber of the inclusion $X\smallsetminus\{x\}\subseteq X$.

This is equivalent to the construction explained in Gregory Arone's comment. A reference for this material is

J. R. Klein, The dualizing spectrum of a topological group, Mathematische Annalen 319 (2001), no. 3, 421–456, DOI 10.1007/PL00004441.

Another useful reference are the lecture notes for Jacob Lurie's class on Algebraic L-theory and manifold topology. In particular Lecture 26 is relevant.
