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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?

John Pardon
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მამუკა ჯიბლაძე
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    I have occasionally wondered if the Spivak normal fibration could be used for this purpose: https://en.wikipedia.org/wiki/Normal_invariant#Homotopy_theory – Qiaochu Yuan Jan 19 '21 at 06:57
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    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. – Gregory Arone Jan 19 '21 at 08:12
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    One source is the paper "Normal fibrations for complexes." by N. Levitt – Gregory Arone Jan 19 '21 at 08:14

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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.

Denis Nardin
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  • 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? – მამუკა ჯიბლაძე Jan 19 '21 at 11:26
  • @მამუკაჯიბლაძე Yes, I'm using ∞-categories (Klein's paper is a bit too old for this, but it's easy enough to translate his paper in the new language). As usual in stable homotopy theory $\mathbb{S}$ is the sphere spectrum (so that SW duality sends a spectrum $X$ to $\operatorname{map}(X,\mathbb{S})$). When $\omega_X$ is not invertible, you get problems in the sense that you cannot use it to put a duality on $\operatorname{Fun}(X,\operatorname{Sp})$. I probably should also note that this story can be generalized to manifolds with boundary, and even more to a full six functor formalism. – Denis Nardin Jan 19 '21 at 11:30
  • Although I should probably mention that to get a full six functor formalism you need to use constructible sheaves instead of local systems, of course, to get a spectral version of Verdier duality. – Denis Nardin Jan 19 '21 at 11:30
  • So $\mathbb S$ is just a single spectrum? What is $\lim_X\mathbb S$ then? Sort of "product of $X$ copies of $\mathbb S$"? Also, is the formalism of constructible sheaves of spectra developed somewhere? – მამუკა ჯიბლაძე Jan 19 '21 at 11:33
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    @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). – Denis Nardin Jan 19 '21 at 11:37
  • So what happens with a finite CW-complex which is manifestly non-Poincaré? Does this construction still recover the SW-dual for it? I am aware that any finite CW can be modeled on a manifold with boundary, so equivalently I might ask whether one does get the dual of a manifold with boundary this way. – მამუკა ჯიბლაძე Jan 21 '21 at 06:28
  • @მამუკაჯიბლაძე Yes you recover the SW dual, as it is explained in this answer. However I'm not sure it is going to be too interesting, since the formula is useful exactly when you have a good control on $\omega_X$ – Denis Nardin Jan 21 '21 at 07:30
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    @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 $\partial M\to M$) – Denis Nardin Jan 21 '21 at 07:35
  • Well it might be interesting for example if one could use it for a better description of higher analogs of the $e$-invariant given by Laures in his manifolds-with-corners paper – მამუკა ჯიბლაძე Jan 21 '21 at 07:51
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    @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... – Denis Nardin Jan 21 '21 at 07:54
  • I see, thanks. Just one more, sorry :D I wonder whether there might exist some analog of the $K$-theoretic splitting principle for other cohomology theories, up to stable cohomotopy - it might well be that even when $\omega_X$ is not invertible, it is locally a bouquet of invertibles? – მამუკა ჯიბლაძე Jan 21 '21 at 07:56
  • The splitting principle holds for all complex-orientable cohomology theories, but I don't see the connection with $\omega_X$. – Denis Nardin Jan 21 '21 at 08:00
  • So $\omega_X$ stably resolves by sums of spherical fibrations? – მამუკა ჯიბლაძე Jan 21 '21 at 08:03
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    @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. – Denis Nardin Jan 21 '21 at 08:43