In the subject of

manifold theory In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, if

M

is a

manifold with boundary In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ne ...

, its double is obtained by gluing two copies of

M

together along their common boundary. Precisely, the double is

M \times \ / \sim

where

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

for all

x \in \partial M

. Although the concept makes sense for any manifold, and even for some non-manifold sets such as the

Alexander horned sphere The Alexander horned sphere is a pathological object in topology discovered by . Construction The Alexander horned sphere is the particular embedding of a sphere in 3-dimensional Euclidean space obtained by the following construction, starting ...

, the notion of double tends to be used primarily in the context that

\partial M

is non-empty and

M

compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...

Doubles bound

Given a manifold

M

, the double of

M

is the boundary of

M \times,1 /math>.  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

M

is closed, the double of

M \times D^k

M \times 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 In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a o ...

. If

M

is a closed, oriented manifold and if

M'

is obtained from

M

by removing an open ball, then the

connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...

M \mathrel -M

is the double of

M'

. The double of a Mazur manifold is a homotopy 4-sphere.. See in particula
p. 24

References

{{topology-stub Differential topology Manifolds