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 $M_{i}'$ 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 $M_{1}\sqcup M_{2}$ 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 $M_{i}'$s slightly, and using the fact that $M_{i}'$ 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

• Fundamental group of connected sum is free product of fundamental groups in dimension at least three: This fails in dimension two, because the circle $S^{1}$ has nontrivial fundamental group.
• Connected sum of simply connected manifolds is simply connected

Related notions

• Fiber sum
• Symplectic sum
• Knot sum