I've taken a course on Morse theory a couple of years ago, but I have no idea about how to solve the following problem. Could you give some hints?

Problem: Let $M$ be a smooth manifold homeomorphic but not diffeomorphic to the $4$-dimensional sphere $S^4$. Show that every Morse function on $M$ has at least $4$ critical points.

Many thanks in advance.


1 Answer 1


Recall $\chi(S^4) = 2$. Let $\beta_i$ be the number of critical points of index $i$ (for $i = 0, 1, 2, 3, 4$). Then $\beta_0 - \beta_1 + \beta_2 - \beta_3 + \beta_4 = 2$. Note that $\beta_0$ and $\beta_4$ are $\geq 1$ (since $f$ must have a minimum and a maximum). Suppose for now that $\beta_0 = \beta_4 = 1$. Then $\beta_1 + \beta_3 = \beta_2$. It is then easily seen that the only way to have $< 4$ critical points is if $\beta_1 = \beta_2 = \beta_3 = 0$, but that would make $M$ diffeomorphic to $S^4$ since $f$ would have only two critical points and $4$ is small enough.

Now let's deal with the case $\beta_0 > 1$ (if this is not true, but $\beta_4 > 1$ instead, switch from $f$ to $-f$). But then clearly there must be a critical point of index between 1 and 3 or the Euler characteristic would be too big. So we have at least $4$ critical points (at least two of index 0, at least one of index 4, and at least something in between).

  • $\begingroup$ What do you mean when you say: "...but that would make $M$ diffeomorphic to $S^4$ since ff would have only two critical points and $4$ is small enough"? Reeb's theorem says that if $M$ (of dimension $n$) has two critical points then it is (only) homeomorphic to a $n$-sphere. $\endgroup$
    – manifold
    Dec 25, 2015 at 20:02
  • $\begingroup$ Moreover we don't know if exotic $4$-spheres exist. $\endgroup$
    – manifold
    Dec 25, 2015 at 20:25
  • 1
    $\begingroup$ The theorem can be extended to obtain a diffeomorphism to the sphere in dimensions $\leq 6$. See, for instance, Matsumoto's An Introduction to Morse Theory, p. 83, theorem 3.6 $\endgroup$
    – Pedro
    Dec 26, 2015 at 9:07
  • $\begingroup$ I think this problem is related to the famous Cerf Theorem : any orientation preserving diffeomorphism of the 3-sphere is isotopic to the identity. How can one prove that a 4-manifold with a Morse function with two critical point is diffeomorphic to the sphere without this ? $\endgroup$
    – Thomas
    Dec 27, 2015 at 7:46

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