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

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 $C(n)=2^n$.

I want to know

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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    $\begingroup$ The $2^n$ bound would show that many known simply-connected manifolds admit no metric of nonnegative curvature. Gromov's constant $C(n)$ is explicit and huge (if memory serves, it is double exponential in $n$). There are many manifolds that don't violate Gromov's bound but would violate the $2^n$ bound. Mayer-Vietoris relates Betti numbers of connected sum to Betti numbers of the summand, and roughly they add up (one has to subtract/add $\le 2$ in some cases). Start taking e.g. iterated commected sum of $CP^n$ and see how quickly it fails the $2^n$ bound. $\endgroup$
    – Igor Belegradek
    Commented Jul 2, 2020 at 11:23
  • $\begingroup$ I guess $n$ is the dimension of $M$, right? $\endgroup$
    – Eduardo Longa
    Commented Jul 2, 2020 at 16:08
  • $\begingroup$ @EduardoLonga: yes, $n=dim(M)$ so I should be talking about $CP^{n/2}$. Gromov's theorem is about closed manifolds. Any open complete nonnegatively curved manifold deformation retracts onto a compact boundaryless nonnegatively curved submanifold, so $n$ is the dimension of that submanifold. $\endgroup$
    – Igor Belegradek
    Commented Jul 2, 2020 at 17:37

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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 $k> \frac{2^{2n}-2}{n-1}$.

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