On the minimal number of critical points of functions on $h$-cobordisms

Let $(W,M_0,M_1)$ be a non-trivial $h$-cobordism (i.e., the Whitehead torsion of $(W,M_0)$ is non-zero) with $W$ compact, connected and $\dim W \ge 6$. We prove that every smooth function $f: W \to [0,1]$, $f(M_0)=0, f(M_1)=1$ has at least 2 critical points. This estimate is sharp: $W$ possesses a function as above with precisely two critical points.

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Published 1 January 2002