Viscoplasticity

28 year-old Painting Investments Worker Truman from Regina, usually spends time with pastimes for instance interior design, property developers in new launch ec Singapore and writing. Last month just traveled to City of the Renaissance. In the subject of manifold theory in mathematics, if $M$ is a manifold with boundary, its double is obtained by gluing two copies of $M$ together along their common boundary. Precisely, the double is $M\times \{0,1\}/\sim$ 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 notion of double tends to be used primarily in the context that $\partial M$ is non-empty and $M$ is compact.

Doubles bound

Given a manifold $M$, the double of $M$ is the boundary of $M\times [0,1]$. 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 $M$ 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 is the Klein bottle.

If $M$ is a closed, oriented manifold and if $M^{\prime }$ is obtained from $M$ by removing an open ball, then the connected sum $M\#-M$ is the double of $M^{\prime }$.

The double of a Mazur manifold is a homotopy 4-sphere.