# 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 $M_1 \sqcup M_2$ 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.