Embeddings and Immersions
This work covers fundamental techniques in the theory of C (alt 133) embeddings and C (alt 133) - 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 (alt 133)- embeddings and C (alt 133) manifolds. The Smale-Hirsch theorem is presented as a generalization of the classification of C (alt 133) embeddings by isotopy and is extended by Gromov's convex integration theory. Finally, as an application of Gromov's work, Adachi introduces Haefliger's classification theorem of foliations on open manifolds. Also described here is Adachi's work with Landweber on the integrability of almost complex structures on open manifolds.
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assume bundle 38 bundle map Chapter classifying space codimension codimension q compact compact-open topology completely regular immersion complex structure consider construct continuous map coordinate bundle Corollary Cr manifold Cr maps cross sections Define a map Definition denote diffeomorphism differential dimensional equivalence class equivalence relation equivariant Euclidean space f and g fiber bundle fibration Figure function Gromov's theorem groupoid Haefliger's Hence homeomorphism homotopy classes homotopy connecting intersection isomorphism isotopies isovariant jet bundle Lemma Let f locally finite manifold and let manifolds of dimensions map f Math measure zero n-dimensional normal bundle open covering open manifolds open subset Proof of Theorem Proposition regular closed curve regular homotopy regularly homotopic satisfies the following say that f self-intersections Smale-Hirsch theorem smooth fiber bundle structural group submanifold subspace Suppose surjection T-foliation T-structure tangent bundle theory topological groupoid topological space transverse Tx(M vector bundle weak homotopy equivalence Whitney