Lower bounds for Betti numbers of a manifold given its boundary?

Question

Let $B$ be some compact, path connected $n$-manifold without boundary such that its cobordism class is trivial, so that there exists some other $n+1$ manifold $M$ with $\partial M= B$. While there is not a unique choice for $M$ (for example $\mathbb{S}^1$ can be seen as the boundary of the connected sum of $n\geq 0$ torus $\mathbb{T}^2$) I was wondering if given $B$, some lower bounds could be obtained for the Betti numbers of $M$.

My intuition tells me that complicated manifolds cannot be the boundary of very simple manifolds, but I don't know if there are any results regarding my question. I have tried using the Mayer–Vietoris sequence but I have not get to anything menaningful.

On principle $B$ is not endowed with any particular structure $G$, but I could be very interested if such bounds were to depend also on $G$ apart from the topology of $B$.

Answer 1

Let me assume that both $M$ and $B$ are orientable. From the long exact sequence of the pair $(M,B)$, Poincaré–Lefschetz duality, and the universal coefficient theorem, for every $k$ we get an exact sequence (I'll use rational or real coefficients throughout): $$ H_{n-k}(M)^{\vee} \cong H^{n-k}(M)\cong H_{k+1}(M,B) \to H_k(B) \to H_k(M), $$ so $b_{n-k}(M) + b_k(M) \ge b_k(B)$.

You will get a similar statement for $\mathbb{F}_2$-Betti numbers if you drop the assumption of orientability.

If you look at 3-manifolds, you can get something better by looking at the torsion in $H_1$ and at the linking form. You can show that a rational homology 3-sphere (i.e. $b_1 = b_2 = 0$) cannot bound a rational homology 4-ball (i.e. $b_1 = b_2 = b_3 = 0$) unless the order of its first homology is a square. (In fact, the linking form on it has to be metabolic, which goes in the direction of the half lives/half dies principle mentioned in Fernando Muro's comment.) There are even finer obstructions coming from gauge theory telling you that certain integer homology 3-spheres (i.e. $H_1 = 0$) cannot bound smooth rational homology 4-balls.

I don't know enough surgery theory to be able to say that in higher dimensions you "frequently" get equality in the inequality $b_{n-k}(M) + b_k(M) \ge b_k(B)$. Kervaire proved that you can always get equality if $B$ is an integer homology sphere and $n \ge 4$. Anyone wants to pitch in with more information?