8
$\begingroup$

Let $M$ be a smooth manifold and $V \to [0,1] \times M$ be a smooth vector bundle. The homotopy invariance states that the restrictions $V_0$ and $V_1$ to the bottom and top of the cylinder are isomorphic.

One can prove that using parallel transport: pick a connection on $V$. For each $x \in M$, take the curve $c_x:t \mapsto (t,x)$. Parallel transport defined by the connection along the curves $c_x$ gives an isomorphism $V_0 \cong V_1$.

This is of course an easy exercise, and my question is to find a self-contained reference that is accessible for undergraduates who know the notions of manifolds, vector bundles and integral curves of vector fields on manifolds (I wish to assign this as a seminar talk).

$\endgroup$
6
$\begingroup$

Greub, Halperin, Vanstone, Connections, curvature and cohomology, Volume II, ch VII.7.18, Theorem I.

$\endgroup$
2
$\begingroup$

You can check pages 33-35 of my course notes. Also you can check section 3.4 of Husemoller's book Fiber Bundles, Graduate Texts in Math, vol. 20, Springer Verlag, 1994. In my notes I prove the result for arbitrary compact spaces, while Husemoller does this for paracompact spaces. In both cases no smoothness is assumed.

$\endgroup$
1
  • $\begingroup$ Thanks, but that does not answer my question. I am well aware of the most general statement and its proof. $\endgroup$ – Johannes Ebert Jul 5 '12 at 10:01

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.