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I'm trying to find the details of the Pontriagin-Thom construction proof about the isomorphism between framed cobordism groups and homotopy groups of spheres and I can't find any good reference.

I was reading Milnor's Topology From the Differential Viewpoint but the construction there is just for homotopy classes and not for homotopy classes of pointed maps.

Also I'm reading Bredon's Topology and Geometry but I think there are many gaps in the proof.

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  • $\begingroup$ I seem to recall that Davis-Kirk's Lecture Notes in Algebraic Topology covered some of this. I can't remember how much detail they provided, and how much the sketched, however. You can always read the original papers. some of which appear in English in Novikov's Topological library: Cobordisms and their Applications. $\endgroup$
    – Tyrone
    Jun 7, 2018 at 10:20

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The fundamental group of a space acts on all homotopy groups $\pi_n(M)$ with $n>0$. How to do this? You take an element $\gamma\in \pi_1(M)$ and $\eta\in \pi_n(M)$. Form a new element $\mu=\gamma\eta\overline\gamma$ by first tracing the path $\gamma$, then $\eta$ and then $\gamma$ in the other way.

We have a bijection (For sure this is in Hatcher)

$$ [S^n,M]\cong\pi_n(M)/\pi_1(M) $$ where on the left hand the based maps are meant. Since $S^k$ is simply connected for $k>1$ it doesn't matter if we compute based or non-based maps. For $k=1$ one needs a small argument that the action is trivial (basically because the fundamental group is abelian).

The Pontryagin-Thom construction as in Milnor is not a based construction, but the end result will compute the based homotopy classes of maps.

You can upgrade the actual PT construction to based maps, but I don't know where this is written down.

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