Connected sum of manifolds

Definition

Let $M_1$ and $M_2$ be connected manifolds. A connected sum of $M_1$ and $M_2$, denoted $M_1\#M_2$, 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. Further information: homotopy type of connected sum depends on choice of gluing map

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 (this needs to be verified). In particular, the fundamental group of a connected sum of simply connected manifolds is simply connected.

Related notions

• Fiber sum
• Symplectic sum
• Knot sum