Given two manifolds with boundaries \(M\) and \(N\) of dimension \(n\) and a homeomorphism \(h\colon \partial M \to \partial N\).

Then the adjunction space \(M\cup_h N\) is a manifold of dimension \(n\) without boundary. (Note, that according to (0x677bdca6) \(\partial B\) is closed).

There are also topological embeddings \(e\colon M\to M\cup_h N\) and \(f\colon N\to M\cup_h N\) with \(e(M)\cup f(N)=M\cup_h N\) and \(e(M)\cap f(N)=e(\partial M)=f(\partial N)\). [1, Theorem 3.79]

Proof

The proof is mostly concerned with proving the locally Euclidean property by gluing together \(M\) and \(N\) along the boundary.

According to (0x67a3c50c) \(M\cup_h N\) is second countable. It lasts to prove that the quotient space is Hausdorff.

See also Link to heading

References Link to heading

  1. J. Lee, Introduction to Topological Manifolds. New York, NY: Springer New York, 2011. doi:10.1007/978-1-4419-7940-7