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Could an inverse Morse inequality hold in some sense? More precisely I wish the following result to be true:

Problem $M$ is a smooth simply connected compact manifold, $dim(M)=n$, $f$ is a morse function on $M$, the number of critical points with $k$-th morse index of $f$ is $m_k$ $0\leq k\leq n$,the standard morse inequality(weak form) said, $$rank(H^k(M)\leq m_k)$$ I wish, for some function $g$ is there some inequality hold for all $M$,$$rank(H^k(M))\geq g(m_1,...,m_n)?$$

Motivation

The Motivation why I need this type of result is due to I need to find if there is a manifold $M$ which is simply connected, compact, the cohomology ring of $M$ is isomorphism to $\mathbb {CP}^3$ but $M$ is not (topology) homomorphism to $\mathbb{CP^3}$. If I have this type of result, then thanks to the CW-structure of topology manifold I reduce this question to complicated calculate but can be finished in finite times.

Attempt

I follow the analytic proof of Witten on the Morse inequality and wish to find a analytic explain of the difference of Morse index and rank of cohomology group.

In the analytic proof, Witten consider the chain homotopy of the following Witten defamation from,

$$(\Omega^*(M),d):0 \stackrel{d}{\to} \Omega^0(M)\stackrel{d}{\to} \Omega^1(M)\stackrel{d}{\to} ...\stackrel{d}{\to} \Omega^{dim(M)}(M)\stackrel{d}{\to} 0$$

to,

$$(\Omega^*(M),d_{Tf}):0\stackrel{d_{Tf}}{\to} \Omega^0(M)\stackrel{d_{Tf}}{\to}\Omega^1(M)\stackrel{d_{Tf}}{\to} ...\stackrel{d_{Tf}}{\to} \Omega^{dim(M)}(M)\stackrel{d_{Tf}}{\to} 0$$

Where $d_{Tf}=e^{-Tf}de^{Tf}$.

Then the first observation is Dirac operator $D=d+d^*$ have a induced defamation operator $D_{Tf}=d_{Tf}+d^*_{Tf}$. And for all $T\in [0,\infty]$ we can prove a hodge split result for $D_{Tf}$.

The second observation is on the local of critical point, when $T\to \infty$, $D_{Tf}$ coverage to the solution of harmonic oscillator.

The third observation is

Witten stable lemma For any $c > 0$, there exists $T_0 > 0$ such that when $T\geq T_0$, the number of eigenvalues in $[0,c]$ of $D_{Tf}|_{\Omega^i(M)}$, $0\leq i \leq n$, equals to $m_i$.

For any integer $i$ such that $0 \leq i\leq n$,let $$F^{[0,c]}_{Tf,i}\subset \Omega^*(M)$$ denote the $m_i$ dimensional vector space generated by the eigenspaces of $D_{Tf}|_{\Omega^i(M)}$, associated with eigenvalues in $[0,c ]$ .

On the other hand, there is a natural restriction map from $(\Omega(M),d_{Tf})$ to $(F_{Tf}^{[0,c]},d_{Tf})$, and the hodge decomposition theorem still hold then we get the weak Morse inequality, and moreover due to the proof the analytic meaning of $m_i-\beta_i$ seems to be the term vanish at infinity, but how to compute this term rigorously and find it?

I will appreciate for any advice and comments!

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    $\begingroup$ You might need to reformulate your question a bit; the function $g=0$ seems to work. What kinds of lower bounds are you trying to find? $\endgroup$ Dec 16, 2017 at 19:37

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There are some weak things that one can say using elementary linear algebra (rank + nullity theorem). Eg $dim(H^k(M)) \geq m_k -(m_{k-1} + m_{k+1})$.

With regard to your motivating problem: There are certainly (smooth) manifolds homotopy equivalent to $CP^3$ that are not homeomorphic to $CP^3$. You can find a comprehensive discussion in Chapter 14C of Wall's book, Surgery on Compact Manifolds. Wall does the PL case, but in this dimension, the PL and smooth classifications are equivalent; it's also not hard to compare the PL and topological classifications.

Wall's proof is not constructive, but leans on transversality via Sullivan's analysis of the homotopy type of the space $G/PL$. If you need an actual construction, I would bet you can extract one out of the literature of the time. There is a nice exposition that gives explicit constructions for fake $CP^5$ and $CP^6$. Even better might be Montgomery-Yang, Differentiable actions on homotopy seven spheres which I think gives constructions for fake $CP^3$s.

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