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A Morse function $f: M \rightarrow \mathbb R$ on a connected closed manifold $M$ is called $\mathit{perfect}$ with respect to the field $\mathbb F$ if all of the Morse inequalities are equalities, i.e. the number of critical points of $f$ with index $k$ coincides with the $k$-th Betti number of $M$ with respect to $\mathbb F$-coefficients for all $k$.

Now assume that $f: M \rightarrow \mathbb R$ is a perfect Morse function on a closed connected Riemannian manifold and that $N \subset M$ is a closed submanifold, such that $N$ contains all the critical points of $f$ and the restriction $f|_N$ is a Morse function on $N$. If in addition the gradient $\nabla f$ is tangent to $N$ along $N$, i.e. $\nabla f|_N \in \Gamma(TN)$, then $N$ is a union of flow lines and the critical points of $f|_N$ are exactly the critical points of $f$.

Now my question is: Is it possible to deduce perfectness of $f|_N$ from perfectness of $f$ in this setting, or are there additional conditions under which this is possible?

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up vote 8 down vote accepted

Isn't there a counterexample with $(M,N)=(\mathbb CP^2,\mathbb RP^2)$?

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Ok, I see. Thank you. The problem is that the attaching map restricts to a map of degree 2. Is that a problem of orientability? Can one avoid problems like that by using $\mathbb Z/2\mathbb Z$-coefficients, for instance? –  Stephan Wiesendorf Jan 10 '11 at 20:54
    
The fact that $\mathbb RP^2$ is not orientable is irrelevant, since I could just as well take $(\mathbb CP^3,\mathbb RP^3)$. –  Tom Goodwillie Jan 11 '11 at 4:17
    
For that matter, I am pretty sure there is an example of a metric and perfect Morse function on $M=S^1\times S^3$ such some submanifold $N$ diffeomorphic to $S^1$ consists of the four critical points and four trajectories of the gradient flow. –  Tom Goodwillie Jan 11 '11 at 4:21
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