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I have a question about an argument that occurred in the discussion about consequences of Bott periodicity in A Concise Course in Algebraic Topology by P. May on page 221. Here is the excerpt:

Firstly: we denote by $TO(q)$ the Thom space of the universal $q$-bundle $gamma_q: E_q to G_q(mathbbR)=BO(q)$.

In the excerpt May constructs a map $Tf_M circ t: S^n+q to TO(q)$ where $f_M: M to BO(q)$ is the classifying map.

My question is why is this map cobordant? Namely why does it respect the cobordism relations?

Especially that would mean that if we have two $n$-manifolds $M,N$ and a $n+1$-manifold $W$ with boundary $partial W =M coprod N $ then the map $Tf_W circ t$ should be nullhomotopic.

Intuitively I think the author means that the intuition behind it is that the homotopy is reached by "going along to inner" of $W$ from $M$ to $N$ but honestly it's to sloppy for me. I'm not sure why it should work globally.

Could anybody explain the argument and why it works ?

This is the Pontryagin-Thom construction, which is a general mechanism for showing that a cobordism theory is given by homotopy groups of a Thom spectrum. I'll talk about some of the aspects but leave a lot of details out.

It can be easier to treat $mathcalN_n$ as the embedded cobordism groups, where each closed manifold $M^n$ is assumed to be embedded in $mathbbR^infty$, and a cobordism between $M$ and $N$ is a $W^n+1$ embedded in $mathbbR^infty times [0,1]$ such that $W|_0 cong M$ and $W|_1cong N$. You can show using the Whitney embedding theorem this is the same as considering abstract manifolds without embeddings.

Given an $Msubset mathbbR^infty$, by compactness $M$ is actually contained in $mathbbR^n+k$ for some finite $k$, with a $k$-dimensional normal bundle $nu$. If we pick $k$ high enough then the embedding is unique up to isotopy and so $nu$ is unique up to isomorphism. The bundle $nu$ admits a bundle map $varphicolonnu to gamma_k$ covering some $fcolon M to BO(k)$, and so there is an induced map on Thom spaces $Tfcolon Tnu to Tgamma_k$. If we choose an open tubular neighbourhood $U$ of $M$ in $mathbbR^n+k$, or equivalently in $S^n+k$, we get a collapse map $$ccolon S^n+k to S^n+k/U^c cong D(nu)/S(nu)=Tnu$$
In other words, for a chosen embedding $Msubset mathbbR^n+k$ we get a continuous function $S^n+kto Tgamma_k$ given by $Tfcirc c$, and if we pick $k$ high enough its homotopy class doesn't depend on the embedding. If we picked another $k'$ we would get a continuous function between different spaces $S^n+k'to Tgamma_k'$ but the idea is we end up getting an element of the limit
$$[Tfcirc c]inpi_n MO = lim_ktoinfty pi_n+kTgamma_k$$

Given a cobordism $W$ from $M$ to $N$, again we can pick some $k$ so that $W$ actually lives in $mathbbR^ktimes[0,1]$, and collapsing a tubular neighbourhood gives a map $Tf_Wcirc c_Wcolon S^n+ktimes [0,1] to Tgamma_k$ which is a homotopy between $Tf_Mcirc c_M$ and $Tf_Ncirc c_N$. This is roughly the sense in a cobordism of manifolds produces a homotopy of functions.

On the other hand, if $Hcolon S^n+ktimes [0,1] to Tgamma_k$ is a homotopy, we can assuming wlog that it is transverse to the $0$-section of $gamma_k$, i.e. the smooth manifold $BO(k)$, and then take the transverse intersection to produce a manifold $W$ of dimension $n+1$. The boundary is divided into two closed $n$-manifolds, one is $M=H_0^-1(BO(k))$ and the other is $N=H_1^-1(BO(k))$. This is roughly the way in which a homotopy of maps $S^n+kto Tgamma_k$ produces a cobordism.