DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generated

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The derivative map from immersions to formal immersions

Definition

Assume ${A C}_{ω}$ (The Axiom of Countable Choice ( ${A C}_{ω}$ )) for the canonical smooth tangent bundles and their total-space mapping topology. The derivative map $D : Imm (M, N) \to FImm (M, N)$ is $D (f) : = (f, d f)$ , where $d f : T M \to T N$ is the differential of $f$ , which is a smooth bundle map over $f$ and fibrewise injective exactly because $f$ is an immersion. Thus $D$ is the inclusion of the holonomic (genuine) formal immersions into all formal immersions; both its source and target are the spaces of the preceding items, with the weak compact-open $C^{\infty}$ topology.

Depends on

Used by

Dependency tree · two levels

23 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

Andrew Ranicki, Algebraic and Geometric Surgery, Ch. 7 §7.4 “The Smale–Hirsch classification of immersions”, printed pp. 142–146 (Theorem 7.35, Proposition 7.39) (standard reference, not scraped)
John Francis, The h-Principle, Lecture 3: Immersion theory (notes by O. Gwilliam), PDF pp. 1–4: Proposition 2.2 (disk), Definition 2.5 (Serre fibration), Definition 2.6 and Proposition 2.7 (flexible sheaves) (standard reference, not scraped)
Ralph L. Cohen, Bundles, Manifolds, and Homotopy, Ch. 7 §2 “Obstructions to the existence of embeddings and immersions, the Hirsch–Smale theorem”, printed pp. 226–232 (Theorem 7.5, Corollary 7.6) (standard reference, not scraped)