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In the subject of manifold theory in
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, if M is a topological
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 n ...
, 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. If M 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 Alexander horned sphere is a pathological object in topology discovered by . It is a particular topological embedding of a two-dimensional sphere in three-dimensional space. Together with its inside, it is a topological 3-ball, the Alexande ...
, the notion of double tends to be used primarily in the context that \partial M is non-empty and M is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact, a type of agreement used by U.S. states * Blood compact, an ancient ritual of the Philippines * Compact government, a t ...
.


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 In mathematics, cobordism is a fundamental equivalence relation on the class of compact space, compact manifolds of the same dimension, set up using the concept of the boundary (topology), boundary (French ''wikt:bord#French, bord'', giving ''cob ...
.


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 is 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 In mathematics, a Möbius strip, Möbius band, or Möbius loop is a Surface (topology), surface that can be formed by attaching the ends of a strip of paper together with a half-twist. As a mathematical object, it was discovered by Johann Bened ...
is the
Klein bottle In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
. 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
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References

{{topology-stub Differential topology Manifolds