Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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Every continuous map between smooth manifolds is homotopic to a smooth map
Statement
Assume countable choice . Every continuous map between smooth manifolds is homotopic to a smooth map.
Facts & Assumptions
Given: Countable choice and a continuous map between smooth manifolds.
Countable choice is The Axiom of Countable Choice ().
Under [A1], every continuous manifold-valued map admits a smooth approximation that is homotopic to it (Whitney approximation for manifold-valued maps).
Proof
Under [A1], apply [L1] to the given continuous map and obtain a smooth map homotopic to it.
The map is smooth and lies in the homotopy class of the original map, so the claim follows.
Depends on
Used by
- The smooth and continuous homotopy categories of smooth manifolds have the same morphism sets Corollary
- Degree-one normal map for the surgery program Definition
- Kernel classes are represented by embedded spheres below the middle dimension Lemma
- The mod-two degree is well defined and homotopy invariant Lemma
- Zero- and one-handles are eliminated in a simply connected h-cobordism Lemma
- Surgery on a normal map preserves its normal bordism class Proposition
- The Hopf degree theorem for oriented domains Theorem
- The Hopf mod-two degree theorem for nonorientable domains Theorem
- The Pontryagin-Thom correspondence in fixed codimension Theorem
Dependency tree · two levels
18 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
- John M. Lee, Introduction to Smooth Manifolds, 2nd ed., Smooth Approximation of Maps Between Manifolds (standard reference, not scraped)