nLab
connected sum

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 XX and YY 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:

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

(e.g. here)

References

See also

Last revised on March 25, 2019 at 06:07:31. See the history of this page for a list of all contributions to it.