Double of a manifold with boundary

Let $M$ be a smooth manifold with boundary. Its double is the quotient

DM=(M\times\{+,-\})/{\sim}, \qquad (x,+)\sim(x,-)\ \text{for }x\in\partial M,

equipped with the smooth structure obtained from a collar of $\partial M$ . In collar coordinates, the two inward parameters $t\geq0$ are joined as the signed coordinate $t\in\mathbb R$ . The result is a smooth manifold without boundary containing two copies of $M$ , whose intersection is their common embedded hypersurface $\partial M$ . Reflection exchanges the two copies and fixes this hypersurface pointwise.

Smooth construction

Choose a collar

c:\partial M\times[0,\varepsilon)\longrightarrow M.

It gives a chart across the seam by

\widetilde c(x,t)= \begin{cases} [(c(x,t),+)],&t\geq0,\\ [(c(x,-t),-)],&t\leq0, \end{cases} \qquad (x,t)\in\partial M\times(-\varepsilon,\varepsilon).

Together with the original smooth charts away from the seam, these maps define the smooth structure on $DM$ . Different collar choices produce diffeomorphic smooth structures, so the smooth double is well defined up to diffeomorphism.

Examples and consequences

The double of $[0,1]$ is $S^1$ , and the double of the closed $n$ -ball is $S^n$ . The double of a cylinder $N\times[0,1]$ is $N\times S^1$ . If $M$ is compact, then $DM$ is compact; if $M$ is oriented, the two copies are given opposite orientations so that the orientation extends across the seam.

Scope

The construction does not require compactness or connected boundary. If $\partial M=\varnothing$ , the displayed quotient is the disjoint union of two copies of $M$ , although some authors reserve “double” for the nonempty-boundary case. Gluing two different manifolds along a boundary diffeomorphism is a broader construction and may require additional collar data.

References

John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Springer, 2012. Springer DOI record. Relevant: Chapter 9, collar neighborhoods and the double construction.
Morris W. Hirsch, Differential Topology, Springer, 1976. Springer DOI record. Relevant: Chapter 4, collars and boundary constructions.