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!



Could an inverse of (weak) Morse inequality exists in some special case?
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](http://www.map.mpim-bonn.mpg.de/images/2/2f/Regensburg2012Talk17.pdf) that gives explicit constructions for fake $CP^5$ and $CP^6$. Even better might be Montgomery-Yang, [Differentiable actions on homotopy seven spheres](http://www.ams.org/journals/tran/1966-122-02/S0002-9947-1966-0200934-1/S0002-9947-1966-0200934-1.pdf) which I think gives constructions for fake $CP^3$s.


