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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?

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    $\begingroup$ I have occasionally wondered if the Spivak normal fibration could be used for this purpose: en.wikipedia.org/wiki/Normal_invariant#Homotopy_theory $\endgroup$ – Qiaochu Yuan Jan 19 at 6:57
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    $\begingroup$ You can identify the spherical normal bundle of a manifold $M$ as the inclusion $\partial T\hookrightarrow T$, where $T$ is a tubular neighborhood of $M$. This point of view can be extended to finite CW complexes. Define a thickening of $K$ to be an equivalence $K\simeq T$, where $(T, \partial T)$ is a Poincare pair. Turn the inclusion $\partial T\hookrightarrow T$ into a fibration. In a suitable sense, the stable homotopy type of this fibration depends only on $K$. This fibration is a generalization of the spherical normal bundle. You can recover the SW dual of $K$ from it. $\endgroup$ – Gregory Arone Jan 19 at 8:12
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    $\begingroup$ One source is the paper "Normal fibrations for complexes." by N. Levitt $\endgroup$ – Gregory Arone Jan 19 at 8:14
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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.

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  • $\begingroup$ Both your answer and the Klein paper are fascinating, thanks! Could you just briefly explain in which formalism is $\operatorname{Fun}(X,\mathrm{Sp})$ understood? $\infty$-categories? Klein does not seem to have this. Also, what is $\mathbb S$? And, when $\omega_X$ is not invertible, does not this create any problems? $\endgroup$ – მამუკა ჯიბლაძე Jan 19 at 11:26
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    $\begingroup$ @მამუკაჯიბლაძე I just mean the limit of the constant functor at $\mathbb{S}$ (it's a general fact in spectra that if $E$ is a spectrum, the limit over a space $Y$ of the constant functor at $Y$ is $\operatorname{map}(\Sigma^\infty_+Y,E)$ and the colimit is $\Sigma^\infty_+Y\otimes E$. I've added another useful reference, unfortunately some of this is folklore (but it has to be said that the classical theory of Verdier duality for complexes goes through quite verbatim, without needing to do much work to adapt it). $\endgroup$ – Denis Nardin Jan 19 at 11:37
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    $\begingroup$ @მამუკაჯიბლაძე Perhaps more interestingly, one can also recover Atiyah duality for compact manifolds with boundary (i.e. spectral Lefschetz duality) in this way, using instead of $\lim_X$ the functor sending $E$ to the fiber of $\lim_XE\to \lim_Y f^*E$, where $f:Y\to X$ is a map of finite complexes (generalizing $\partial M\to M$) $\endgroup$ – Denis Nardin Jan 21 at 7:35
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    $\begingroup$ @მამუკაჯიბლაძე I suspect the theory of constructible sheaves will be more relevant in that case, as they behave much better in presence of local singularities (that the theory of local systems is unable to see). But this is getting long for a comment thread... $\endgroup$ – Denis Nardin Jan 21 at 7:54
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    $\begingroup$ @მამუკაჯიბლაძე If I understand correctly what you mean by "resolves", this is true for every spectrum. For a general $X$, the local system $\omega_X$ can be quite arbitrary. $\endgroup$ – Denis Nardin Jan 21 at 8:43

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