# Homotopy invariance of vector bundles by parallel transport: reference needed for my students.

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).

## 2 Answers

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

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.

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