Double (manifold) explained

In the subject of manifold theory in mathematics, if

is a topological manifold with boundary, its double is obtained by gluing two copies of

together along their common boundary. Precisely, the double is

M x \{0,1\}/\sim

where

(x,0)\sim(x,1)

for all

x\in\partialM

has a smooth structure, then its double can be endowed with a smooth structure thanks to a collar neighbourdhood.

Although the concept makes sense for any manifold, and even for some non-manifold sets such as the Alexander horned sphere, the notion of double tends to be used primarily in the context that

\partialM

is non-empty and

is compact.

Doubles bound

Given a manifold

, the double of

is the boundary of

M x [0,1]

. This gives doubles a special role in cobordism.

Examples

The n-sphere is the double of the n-ball. In this context, the two balls would be the upper and lower hemi-sphere respectively. More generally, if

is closed, the double of

M x D^k

M x S^k

. Even more generally, the double of a disc bundle over a manifold is a sphere bundle over the same manifold. More concretely, the double of the Möbius strip is the Klein bottle.

is a closed, oriented manifold and if

is obtained from

by removing an open ball, then the connected sum

Ml{\#}-M

is the double of

The double of a Mazur manifold is a homotopy 4-sphere.^[1]

Notes and References

. See in particular p. 24.