Embeddings and ImmersionsThis book covers fundamental techniques in the theory of $C^{\infty}$-imbeddings and $C^{\infty}$-immersions, emphasizing clear intuitive understanding and containing many figures and diagrams. Adachi starts with an introduction to the work of Whitney and of Haefliger on $C^{\infty}$-imbeddings and $C^{\infty}$-manifolds. The Smale-Hirsch theorem is presented as a generalization of the classification of $C^{\infty}$-imbeddings by isotopy and is extended by Gromov's work on the subject, including Gromov's convex integration theory. Finally, as an application of Gromov's work, the author introduces Haefliger's classification theorem of foliations on open manifolds. Also described here is the Adachi's work with Landweber on the integrability of almost complex structures on open manifolds. This book would be an excellent text for upper-division undergraduate or graduate courses. |
Contents
III | 1 |
IV | 7 |
V | 15 |
VI | 32 |
VII | 37 |
VIII | 42 |
IX | 45 |
X | 47 |
XXX | 108 |
XXXI | 111 |
XXXII | 114 |
XXXIII | 116 |
XXXIV | 123 |
XXXV | 127 |
XXXVI | 128 |
XXXVIII | 129 |
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Common terms and phrases
A₁ assume B₁ B₂ C₂ Chapter classification codimension compact compact-open topology completely regular immersion complex structure construct continuous map convex coordinate bundle COROLLARY cross sections DEFINITION denote diffeomorphism differential dimensional Dm-k equivalence class equivalence relation equivariant Euclidean space exists f₁ fiber bundle Figure function ƒ and g GL(n Gromov's theorem groupoid H₁ Haefliger's Hence homeomorphism homotopy classes homotopy connecting intersection isotopies jet bundle Lemma Let f locally finite M₁ M₂ manifold and let map f Math measure zero n-dimensional neighborhood open covering open manifolds open subset P₁ PROOF OF THEOREM PROPOSITION regular closed curve regular homotopy regularly homotopic satisfies the following say that ƒ Sect(X self-intersections Smale-Hirsch theorem structural group submanifold surjection T-foliation T-structure tangent bundle topological groupoid topological space transverse U₁ vector bundle weak homotopy equivalence Whitney's x₁