The question is inspired by [an answer](https://mathoverflow.net/a/262700/41291) to [The concept of Duality](https://mathoverflow.net/q/73711/41291)


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?



Are Spanier-Whitehead duals of general spaces expressible through some generalization of normal bundles?
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 $X\smallsetminus\{x\}\subseteq X$.




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This is equivalent to the construction explained in Gregory Arone's [comment](https://mathoverflow.net/questions/381598/are-spanier-whitehead-duals-of-general-spaces-expressible-through-some-generaliz#comment969953_381598). A reference for this material is



> 
> J. R. Klein, [*The dualizing spectrum of a topological group*](https://link.springer.com/article/10.1007/PL00004441), 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*](https://www.math.ias.edu/%7Elurie/287x.html). In particular [Lecture 26](https://www.math.ias.edu/%7Elurie/287xnotes/Lecture26.pdf) is relevant.


