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

Further information: Homology of connected sum

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.

Fundamental group

Further information: Fundamental group of connected sum

The fundamental group of the connected sum of two manifolds is the free product of their fundamental groups. In particular, the fundamental group of a connected sum of simply connected manifolds is simply connected.

Related notions

• Fiber sum
• Symplectic sum
• Knot sum