Connected sum of manifolds

Definition

Let $M_1$ and $M_2$ be connected manifolds. A connected sum of $M_1$ and $M_2$, denoted Failed to parse (syntax error): {\displaystyle M_1#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 $f_2\circ f_1^{-1}$.

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:

Failed to parse (syntax error): {\displaystyle \tilde{H}_i(M_1 # 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 deleted-point inclusion (inclusion of manifold minus a point into the manifold) induces isomorphism on all homologies uptil $n-2$.

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:

Failed to parse (syntax error): {\displaystyle \tilde{H}_{n-1}(M_1 # M_2) = \tilde{H}_{n-1}(M_1) \oplus \tilde{H}_{n-1}(M_2)}

It turns out that the result holds for compact connected manifolds even if one of them is non-orientable; this requires a little more argument.

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.