Contents

# Contents

## Idea

In differential topology the connected sum of two manifolds equipped with a common submanifold is the result of gluing them along their boundaries of the tubular neighbourhoods of this submanifold.

Often this concept is considered by default for the case that the given submanifold is the point space. In this case the correspinding neighbourhoods are balls with boundary n-spheres along which the two manifolds are being glued.

## Properties

###### Proposition

(Euler characteristic of connected sums)

Let $X$ and $Y$ be closed manifolds (topological or differentiable) of even dimension. Then the Euler characteristic of their connected sum is the sum of their separate Euler characteristics, minus 2:

$\chi \big( X \sharp Y \big) \;=\; \chi \big(X\big) + \chi \big(Y\big) -2$

(e.g. here)