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Let $f:M\to \Bbb{R}$ be a Morse function, where $M$ is a $k$-manifold. The index $i_{f,p}$ is defined to be the number of negative eigenvalues of the Hessian $H_f$ at the critical point $p$. For instance, let $f:\Bbb{D}^2\to\Bbb{R}$ be defined as $(x,y)\to x+y$. Then there are no critical points of this function, and the Hessian $\begin{pmatrix} 0&0\\0&0\end{pmatrix}$ has no negative eigenvalues.

The Euler characteristic of $M$ is defined to be $\sum\limits_{i=0}^n {(-1)^i T_{f,i}}$, where $T_{f,i}$ is the number of critical points of index $i$.

I wonder why this is a valid definition. For example, we know that the Euler characteristic of $\Bbb{D^2}$ is $1$. However, the map $f:\Bbb{D^2}\to\Bbb{R}$ has no critical points. Hence, according to the formula above, its Euler characteristic should come out to be $0$. Isn't this a contradiction?

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    $\begingroup$ I think that is true only on compact manifolds. $\endgroup$
    – user99914
    Commented Nov 26, 2014 at 7:27
  • $\begingroup$ @John- How does the situation change if we take $\overline{\Bbb{D^2}}$? $\endgroup$
    – user67803
    Commented Nov 26, 2014 at 7:46
  • $\begingroup$ Oh sorry, I should say compact without boundary. $\endgroup$
    – user99914
    Commented Nov 26, 2014 at 7:51
  • $\begingroup$ Euler characteristic is usually defined for compact manifolds. For manifolds with boundary, formulas for computing it usually take into account the boundary in some form. If you look at the statement of Poincare-Hopf theorem given at Wikipedia, what you say should be true for the closed disk for $f$ constant on the boundary, which $x+y$ isn't. If we take $f(x,y)=x^2+y^2$, then $f$ is constant on the boundary and indeed the formula works. $\endgroup$
    – tomasz
    Commented Nov 26, 2014 at 8:19
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    $\begingroup$ Compactness aside, Morse functions do not have degenerate critical points. Thus, if it did have a critical point, then it wouldn't be a Morse function anyway. $\endgroup$
    – Robin Goodfellow
    Commented Nov 26, 2014 at 20:07

1 Answer 1

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Here is a correct statement. Let $M$ be a compact manifold with boundary. Let $f,g$ be a pair of a Morse function and metric such that $-\nabla f\pitchfork \partial M$. Assume now that the negative gradient flow points inwards at the boundary. Then the alternating sum of the number of critical points of index $k$ equals the Euler characteristic of the manifold. If the gradient points outwards, the same holds, but you should compute the Euler characteristic of the manifold relative to the boundary.

You can see now what should be true for your example I think.

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