Connected sum of manifolds

Definition

Let $M_1$ and $M_2$ be connected manifolds. A connected sum of $M_1$ and $M_2$, denoted $M_1\mathrm{♯}{M}_2$, is constructed as follows. Let $f_i:{\mathbb{R}}^n\to U_i$ be homeomorphisms where $U_i$ are open subsets of $M_i$. Let ${M_{i^{\prime }}}^{\prime }$ denote the complement in $M_i$ of the image of the open unit ball in ${\mathbb{R}}^n$, under $f_i$. Then the connected sum is the quotient of $M_1\bigsqcup M_2$ under the identification of the boundary $S^{n-1}$s with each other, via the composite $f_2\circ f_1^{-1}$.

In general, the homotopy type of the connected sum of two manifolds depends on the choice of open neighbourhoods and on the way of gluing together.

Homology

The homology of the connected sum can be computed using the Mayer-Vietoris homology sequence for open sets obtained by enlarging the ${M_{i^{\prime }}}^{\prime }$s slightly, and using the fact that ${M_{i^{\prime }}}^{\prime }$ is a strong deformation retract of $M_i$ minus a point.

The interesting phenomena occur at $n$ and $n-1$, because this is where the gluing is occurring.

Homology in low and high dimensions

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

${\tilde{H}}_i(M_1\mathrm{♯}{M}_2)={\tilde{H}}_i(M_1)\oplus {\tilde{H}}_i(M_2)$

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 $H_{n-1}(S^{n-1})\to H_{n-1}({M_{1^{\prime }}}^{\prime })$ is zero. Thus if both manifolds are compact connected orientable, then Mayer-Vietoris yields that:

${\tilde{H}}_{n-1}(M_1\mathrm{♯}{M}_2)={\tilde{H}}_{n-1}(M_1)\oplus {\tilde{H}}_{n-1}(M_2)$

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:

${\tilde{H}}_{n-1}(M_1\mathrm{♯}{M}_2)={\tilde{H}}_{n-1}(M_1)\oplus {\tilde{H}}_{n-1}(M_2)$

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:

$\chi (M_1\mathrm{♯}{M}_2)=\chi (M_1)+\chi (M_2)-\chi (S^n)$