# Connected sum of manifolds

## Definition

Let and be connected manifolds. A **connected sum** of and , denoted , is constructed as follows. Let be homeomorphisms where are open subsets of . Let denote the complement in of the image of the open unit ball in , under . Then the connected sum is the quotient of under the identification of the boundary s with each other, via the composite .

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 s slightly, and using the fact that is a strong deformation retract of minus a point.

The interesting phenomena occur at and , 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 has nontrivial fundamental group.
- Connected sum of simply connected manifolds is simply connected