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 $f_{i} : R^{n} \to U_{i}$ 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 $R^{n}$ , under $f_{i}$ . Then the connected sum is the quotient of $M_{1} ⊔ M_{2}$ 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

Definition

Homology

Fundamental group

Related notions