I am studying cobordism theory and I basically follow Milnor's book, Characteristic classes.
I want to prove that cobordism is a transitive relation so I need the collar neighborhood theorem. In Milnor's book (theorem 17.1) the proof is omitted because is similar to the proof of the transversality theorem (theorem 11.1). Although I proved transversality theorem, I don't understand why the collar neighborhood theorem follows from it. I checked the same proof in Lee's book "introduction to smooth manifolds", but he uses flows on manifolds and I don't know anything about that. Any hints?
In general, any recommendations about an introductory book on Cobordism theory?
Thank you in advance.
Any manifold book worth its salt should prove this.
Put a Riemannian metric on $X$; then for every point in the boundary, there is a unique tangent vector in $T_p X$ that is orthogonal to the boundary, of norm 1, and points inwards. This provides a trivialization of the normal bundle of $\partial X$ in $X$. Now, how would one prove the tubular neighborhood theorem here? (Note that we can't invoke the standard one without modifications, since we have a neighborhood of "half" the normal bundle.) Flow along this vector field... (I think Lee's proof is the "standard" one.) A proof that completely avoids flows should work if you live as subsets of $\Bbb R^n$. See Bredon for a proof along these lines, IIRC. I don't think Bredon does much with flows in his book.