Homology of connected sum

Homology in low and high dimensions

In all dimensions other than $n$ and $n-1$, we have the following formula:

${\tilde {H}}_{i}(M_{1}\sharp M_{2})={\tilde {H}}_{i}(M_{1})\oplus {\tilde {H}}_{i}(M_{2})$

This does not require any conditions on the manifolds, and only uses the fact that the point-deletion inclusion (inclusion of manifold minus a point into the manifold) induces isomorphism on all homologies other than $n,n-1$.

In the second highest dimension

In dimension $n-1$, we need to know about the nature of the map from $S^{n-1}$ into $M_{i}\setminus p$ as far as $(n-1)^{th}$ homology is concerned. Clearly, the inclusion of $S^{n-1}$ inside $M$ is nullhomotopic, because it factors through a contractible open set.

If $M_{i}$ is a compact connected orientable manifold then the inclusion of $M_{1}\setminus p$ induces isomorphism on the $(n-1)^{th}$ homology, hence the induced map $H_{n-1}(S^{n-1})\to H_{n-1}(M_{1}')$ is zero. Thus if both manifolds are compact connected orientable, then Mayer-Vietoris yields that:

${\tilde {H}}_{n-1}(M_{1}\sharp M_{2})={\tilde {H}}_{n-1}(M_{1})\oplus {\tilde {H}}_{n-1}(M_{2})$

If both $M_{1}$ and $M_{2}$ are compact connected manifolds and $M_{1}$ is non-orientable but $M_{2}$ is orientable, then the sequence:

$0\to {\tilde {H}}_{n-1}(S^{n-1})\to {\tilde {H}}_{n-1}(M_{1}\setminus p)\to {\tilde {H}}_{n-1}(M_{1})\to 0$

is exact, and this yields, along with Mayer-Vietoris, that:

${\tilde {H}}_{n-1}(M_{1}\sharp M_{2})={\tilde {H}}_{n-1}(M_{1})\oplus {\tilde {H}}_{n-1}(M_{2})$

If both are non-orientable, however, then an exceptional situation occurs.

In the highest dimension

The observations given above yield that when both $M_{1}$ and $M_{2}$ are compact connected orientable, then the top homology of their connected sum is again $\mathbb {Z}$, viz the connected sum is again orientable. This can also be seen directly by the definition of orientability.

Euler characteristic

The Euler characteristics are related by the following formula when both $M_{1}$ and $M_{2}$ are compact connected manifolds:

$\chi (M_{1}\sharp M_{2})=\chi (M_{1})+\chi (M_{2})-\chi (S^{n})$