I have a question while reading the proof of Theorem 3.4 in Milnor's book Lectures on the h-cobordism theorem. (https://www.maths.ed.ac.uk/~v1ranick/surgery/hcobord.pdf)

Theorem 3.4. If the Morse number $\mu$ of the triad $(W;V_0,V_1)$ is zero, then $(W;V_0,V_1)$ is a product cobordism.

Proof) $(\cdots)$ We obtain an integral which satisfies $f(\psi(s))=s$. Each integral curve can be extended uniquely over a maximal interval, which, since $W$ is compact, must be $[0,1]$. $(\cdots)$

My question is: How did the compactness of $W$ used to show that the maximal domain of an integral curve must be $[0,1]$?


1 Answer 1


The function $f$ is assumed to have value $0$ on $V_0$ and $1$ on $V_1$. An integral curve of an ODE/vector field on a compact manifold with boundary can be extended until it hits the boundary. Since $f(\psi(s))=s$ this will happen precisely at $s=0$ and at $s=1$.


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