Smooth charts, atlases, and structures with boundary

Depends on

• Topological manifolds with boundary
• Smooth functions on relatively open half-space sets
• Chain rule for smooth half-space maps

Used by

• Curvilinear face-to-face triangulation Definition
• Euler characteristic of a compact manifold Definition
• Framed embedded surgery sphere Definition
• Geodesic triangulation with prescribed boundary arcs Definition
• Interior and boundary of a manifold with boundary Definition
• p-surgery on a smooth m-manifold Definition
• Pointwise Borel nonnegative densities Definition
• Smooth cobordism triad for Morse theory Definition
• Smooth foliated concordance of codimension-one foliations Definition
• Smooth foliations tangent to the boundary Definition
• Smooth maps between manifolds with boundary Definition
• Smooth partition of unity on a 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 closed ball and its sphere boundary Example
• The closed half-space as a manifold with boundary Example
• The pair of pants is a cobordism realizing addition of circles Example
• Two unoriented points bound an interval Example
• A countable boundary coordinate cover has a locally finite shrinking Lemma
• Boundary connected sum with a disk does not change the diffeomorphism type Lemma
• Boundary of a compact 1-manifold has even cardinality Lemma
• Boundary smoothness is independent of charts and extensions Lemma
• Collar gluing and seam smoothing give transitivity Lemma
• Complex flag splitting with injective real pullback on smooth bases Lemma
• Disk bundles over compact bases are compact manifolds with boundary Lemma
• General-position isotopies of links Lemma
• Gluing manifolds with boundary along a boundary diffeomorphism Lemma
• Oppositely framed points cancel in pairs Lemma
• Oriented real two-plane splitting with real-cohomology injection Lemma
• Product boundary formula for oriented manifolds Lemma
• Regular sublevels are compact manifolds with boundary Lemma
• Relatively compact coordinate balls and half-balls form a boundary-manifold basis Lemma
• Restriction of a foliation transverse to the boundary Lemma
• Smooth manifolds have CW homotopy type Lemma
• The de Rham complex and pullback extend to manifolds with boundary Lemma
• The Reeb foliation of the solid torus has the boundary as a leaf Proposition
• Zero-dimensional bordism groups Proposition
• Closedness is needed for the Hopf degree classification Remark
• Gauss-Bonnet for compact oriented surface regions with boundary and corners Theorem

…and 7 more results.

Dependency tree · two levels

8 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)