Smooth collars of a manifold boundary

Depends on

• The boundary of a positive-dimensional manifold is a closed embedded smooth (n-1)-manifold
• Smooth maps between manifolds with boundary

Used by

• Homotopy spheres of dimension at least five bounding a contractible manifold are standard spheres Corollary
• Handle decomposition relative to the incoming boundary Definition
• Morse function adapted to a cobordism Definition
• Null-cobordant closed manifolds Definition
• p-surgery on a smooth m-manifold Definition
• Smooth cobordism triad for Morse theory Definition
• Surgery trace cobordism Definition
• The double of a smooth manifold with boundary Definition
• Unoriented smooth cobordism of closed manifolds Definition
• A circle is the boundary of a disk Example
• Signed points give the oriented zero-bordism invariant Example
• The pair of pants is a cobordism realizing addition of circles Example
• The standard collar of a closed ball Example
• Two unoriented points bound an interval Example
• A product collar deformation retracts onto its face Lemma
• Boundary connected sum with a disk does not change the diffeomorphism type Lemma
• Boundary product function on a collared cobordism Lemma
• Collar gluing and seam smoothing give transitivity Lemma
• Connected sum descends to oriented h-cobordism classes Lemma
• Decorated ribbon graphs lift through a marked relative surgery calculus Lemma
• Formal-immersion homotopies extend over a collar Lemma
• Isotopic attaching embeddings give diffeomorphic handle attachments Lemma
• Orientation reversal is the connected-sum inverse Lemma
• Parallelizable boundaries form a subgroup Lemma
• The outgoing boundary of a handle attachment trades the disk factors Lemma
• The surgery gluing has a canonical smooth structure up to diffeomorphism Lemma
• Collar neighborhood theorem Theorem

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Ioan Mărcuț, Manifolds (2017 lecture notes), §§14.5, 15.1 (standard reference, not scraped)
• Will Merry, Differential Geometry (2021), Lecture 24 (standard reference, not scraped)