Double of a manifold with boundary - Imported Knowlpedia Content

Definition

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 $c:\partial M\times[0,\varepsilon)\to M$ . Use $t$ on the positive copy and $-t$ on the negative copy to form charts across the seam. Different choices of collar produce diffeomorphic smooth doubles, so the diffeomorphism type depends only on $M$ . The collar is essential: the bare quotient describes the topology but does not by itself specify compatible smooth charts Lee, Chapter 9.

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.