In Peter Petersen words, **Gromov Betti number estimate** is considered one of the deepest and most beautiful results in Riemannian geometry; which asserts that



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> **Theorem (Gromov 1981)**. There is a constant $C(n)$ such that
> any complete manifold $(M, g)$ with $\sec\geq 0$ and for any field $\Bbb F$ of coefficients satisfies
> $$\sum\_{i=0}^n b\_i(M,\Bbb F)\leq C(n).$$
> and conjectured that the best upper bound is $C(n)=2^n$.
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I want to know



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> What would be the consequences if Gromov upper bound Conjecture be true? Will the new results be different from the non exact upper bound $C(n)$?
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Consequences of Gromov's Conjecture
As Igor mentioned knowing the optimal bound is always better than knowing a non-optimal one such as the bound provided by Gromov's proof. It rules out a lot more examples. A proof of the sharp bound would also likely imply a rigidity result that if the sum of the Betti numbers is exactly $2^n$ then $M$ is a torus. This is quite out of reach with the currently known bound.


Conceptually more interesting is the relation of Gromov's conjecture to several other conjectures.
The strongest of these is a conjecture of Bott that a simply connected nonnegatively curved manifold is rationally elliptic. This means that that the total sum of ranks of rational homotopy groups is finite. This holds for homogeneous spaces for example. Rational ellipticity is a very strong condition.
It's been extensively studied in rational homotopy theory.
It in particular implies that the sum of the rational Betti numbers is at most $2^n$. So Bott's conjecture would imply Gromov's conjecture over $\mathbb Q$.
Ellipticity of $M$ also implies that $\chi(M)\ge 0$. So Bott's conjecture also implies Chern's conjecture that a nonnegatively curved manifold must have nonnegative Euler characteristic. Moreover it's known that a rationally elliptic space $M$ has $\chi(M)> 0$ iff all odd Betti numbers of $M$ vanish. This is also conjectured but not known for manifolds of nonnegative curvature.
Lastly, let me mention that Bott's conjecture is much stronger than Gromov's conjecture. For example it's easy to see that connected sum of at least 3 copies of $\mathbb{CP}^n$ is rationally hyperbolic (i.e. not elliptic) so it should not admit nonnegative sectional curvature according to Bott's conjecture. On the other hand Gromov's conjecture does not rule out the connected sum of $k$ copies of $\mathbb{CP}^n$ unless $k> \frac{2^{2n}-2}{n-1}$.


