I would like to know what motivated or led Thom to think that the (un)oriented cobordism groups would correspond with the homotopy groups of some structure (Thom spectum), or with the coefficient groups of a cohomology theory.
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4$\begingroup$ Surely Pontryagin's work on (co)homotopy groups of spheres was relevant, as was Whitney's work on the topology of manifolds $\endgroup$– Thomas RotCommented Dec 24, 2021 at 13:03
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6$\begingroup$ Thom never spoke of spectra. I think it is Atiyah who coined the concept of spectrum, exactly to reinterpret Thom's construction and use it to prove a generalized version of Poincaré duality, namely what we call nowadays Atiyah duality. $\endgroup$– D.-C. CisinskiCommented Dec 24, 2021 at 14:27
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4$\begingroup$ @ThomasRot You'd think so, but it seems Thom was unaware of Pontryagin's work until after he finished his famous paper. The only concrete thing I can add is that one of Thom's goals was solving the Steenrod problem. $\endgroup$– Denis NardinCommented Dec 24, 2021 at 15:38
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2$\begingroup$ @D.-C.Cisinski: My understanding is spectra were defined by Mike Boardman. The first time I met Boardman (and Mike Hopkins), I was a grad student and had no idea who he was or what he had done in math. I introduced myself in a way that made clear I did not like current definitions of spectra, and I would like to see a better one. He handled it in good humour. $\endgroup$– Ryan BudneyCommented Dec 24, 2021 at 22:55
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3$\begingroup$ @RyanBudney Boardman was the first to contruct the stable homotopy category. Spectra, however, predate him by some years. Whitehead attributes the notion to Lima in 1958 or 1959. I'm not sure what Lima's exact definition was. By 1961, Whitehead's definition (in his paper Generalized Homology Theories) was basically the modern one. Boardman's thesis was not until 1964. $\endgroup$– Dan RamrasCommented Dec 26, 2021 at 2:46
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1 Answer
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I did not know Thom so I can't speak to all his personal motivations. But I can speak as someone that has read much of his work carefully and I think I have some insights into this.
The primary motivation for his theorem boils down to thinking carefully about the implicit function theorem, and asking when a subset of Euclidean space is the pre-image of a regular value of a smooth function.
These are the problems most students grapple with when learning about smooth manifolds for the first time, and Thom's considerations flowed out of these kinds of naive considerations. Thom just went a little further than most.
The first step is realizing a submanifold of $S^n$ is the pre-image of a regular value of a smooth function
$$f : S^n \to S^m$$
if and only if it has no boundary, is compact and has a trivial tubular neighbourhood. And this is the key insight into the Pontriagin construction.
The next step is asking about general manifolds, and this is where you take the step up to maps to Thom spaces. Specifically the structure of the normal bundle is key in the Pontriagin construction. So if you have a non-trivial normal bundle you need some feature of your map to take that into account. Grassman manifolds are the key object related to maps out of vector bundles. But once you see that Grassman manifolds are the key analogue to the Pontriagin construction that gives you your map defined in a tubular neighbourhood of your submanifold. Coning off as in the Pontriagin construction gives you the Thom space.
From this point of view you can see his theorem as a direct extrapolation from the trivial tubular neighbourhood case. I think for people that like to think categorically it could maybe feel unintuitive since the Thom space is a departure from manifolds.
As has been mentioned, spectra came after the fact, as the idea was germinating around that time. In Thom's paper he largely phrased things in the language of stable homotopy groups.
answered Dec 24, 2021 at 22:43