Smooth maps between manifolds with boundary

Depends on

• Smooth charts, atlases, and structures with boundary
• Smooth functions on relatively open half-space sets

Used by

• Diffeomorphisms preserve interior and boundary Corollary
• An immersed disk in the four-ball cannot always be cleaned relative to its boundary Counterexample
• Boundary-defining functions Definition
• Gauge group of a smooth principal bundle Definition
• Immersions and embeddings for manifolds with boundary Definition
• Lie group Definition
• Regular homotopy of immersions Definition
• Smooth cobordism triad for Morse theory Definition
• Smooth collars of a manifold boundary Definition
• Smooth isotopies, diffeotopies and ambient isotopies Definition
• Smooth principal G-bundle Definition
• Smooth singular simplex Definition
• Smooth variation and variation field of a curve Definition
• The relative handle decomposition of a cylinder Example
• Bordant cycles have equal intersection numbers 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
• Formal-immersion homotopies extend over a collar Lemma
• Isotopy extension for a compact source with boundary Lemma
• Metastable approximation of maps by embeddings Lemma
• Oriented real two-plane splitting with real-cohomology injection Lemma
• Smoothing continuous families of genuine immersions Lemma
• The basepoint evaluation of the Stiefel section space is a fibration Lemma
• The boundary stable tangent bundle splits off a trivial line Lemma
• The de Rham complex and pullback extend to manifolds with boundary Lemma
• The velocity field of an isotopy is well defined along its image Lemma
• First variation of volume for a normal variation Proposition
• Proper smooth maps pull back compactly supported forms Proposition
• Smooth singular chains and cochains are functorial for smooth maps Proposition
• Connection independence and naturality of Chern–Weil classes Theorem
• The isotopy extension theorem Theorem
• Transverse preimages for maps from manifolds with boundary Theorem

Dependency tree · two levels

5 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)