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Let $f: M \to R$ be a morse function from a closed n-manifold $M$ with a critical value $c$. Is it possible that there are two critical points $P_1$, $P_2$ with different index, but with the same critical value?

A possible counter example is below with $M$ diffeomorphic to the torus and $f$ the height function.

enter image description here

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  • $\begingroup$ Your example looks fine to me! $\endgroup$
    – Kenny Wong
    Commented Apr 13, 2017 at 18:01
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    $\begingroup$ Looks to me as if both of the critical points you marked have index 1. But you can get a counterexample by pushing up the local minimum on the right (index 0) so that it has the same height as $p_1$ (index 1). $\endgroup$
    – Jack Lee
    Commented Apr 13, 2017 at 20:20
  • $\begingroup$ Dear Professor Lee, I am delighted to hear from the author of my favorite geometry textbook; I am a bit confused by your comment. I agree $P_1$ has index $1$. I think $P_2$ has index $2$: Use the local coordinates that one gets by projecting onto the 'ground'. Then $f$ is an upside down paraboloid on this chart. Thus on this chart $f \circ \text{inverse image of projection onto ground}(x_1,x_2)=f(P_2)-x_1^2-x_2^2$. I do agree that the critical point on the extreme right has index zero as it looks like a paraboloid. Kind Regards, $\endgroup$
    – De Yang
    Commented Apr 14, 2017 at 1:06
  • $\begingroup$ oh I see: I am visualizing a different manifold than the one that I drew over here, which has a saddle at $P_2$. I was visualizing the manifold at $P_2$ as the one you would get by taking an ordinary upright torus and pushing up with a pencil on the bottom right. $\endgroup$
    – De Yang
    Commented Apr 14, 2017 at 2:06

1 Answer 1

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The simplest example is on $\mathbb{R}$. Try to investigate the function $f:\mathbb{R}\rightarrow \mathbb{R}$ defined by $f(x)=x(x^2-1)^2.$

Nice exercise: Why is degree 5 the lowest degree of a polynomial which is Morse and has critical points with same value but different index?

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