Calculating Homology of the Boundary of a Handlebody

Question

Given a manifold $M$ with boundary $W = \partial M$, I know that having a handle decomposition of $M$ allows one to compute its homology, at least in nice cases, by - for example - using the Morse Homology of its critical points. Is it similarly easy to compute the homology of $W$, since the handle decomposition of $M$ provides a surgery description of $W$?

If it helps, I'm interested in a particularly simple case: $M$ is a smooth $2n$-manifold which is described by simultaneously adding some number of $n$-handles to $D^{2n}$. However, I am primarily concerned with the integer homology, rather than over $\mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Q}$.

Answer 1

For $n>1$, manifold itself is determined by the linking numbers between the attaching spheres of your handles, and the framings of those handles. (For $n=1$, you have to take into account knotting and linking of the attaching circles.) The framings are in 1-1 correspondence with elements of $\pi_n(SO(n+1))$. The homology depends on the image of the framings under the map $\pi_n(SO(n+1))\to \pi_n(S^n)$. Geometrically, this means to take the linking number of your attaching sphere with a pushoff given by the first vector of the given framing. Then the homology is presented by the $(-1)^{n-1}$ symmetric matrix of linking numbers and images of these framings. The proof is an exercise using Poincar\'e duality and the long exact sequence of the pair $(W,M)$.

This is a standard fact in the case $n=2$, in which case the map $\pi_1(SO(2))\to \pi_1(S^1)$ is a bijection, and I think you can find it in such sources as Gompf-Stipsicz. For the general high-dimensional case, look at Classification of (n-1)-Connected 2n-manifolds by C.T.C. Wall, Annals of Math Vol. 75, No. 1 (1962), pp. 163-189.