Topological Library - Part 2

Characteristic Classes And Smooth Structures On Manifolds

By: Iskander A Taymanov (Editor), Vassily Olegovich Manturov (Translator), Serguei Petrovich Novikov (Editor)

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Hardcover | 12 October 2009

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278 Pages

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This is the second of a three-volume set collecting the original and now-classic works in topology written during the 1950s-1960s. The original methods and constructions from these works are properly documented for the first time in this book. No existing book covers the beautiful ensemble of methods created in topology starting from approximately 1950, that is, from Serre's celebrated "singular homologies of fiber spaces."

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Non-Fiction Mathematics History of Mathematics Topology Algebraic Topology Booktopia Publisher Services World Scientific Publishing

Preface	p. xi
On manifolds homeomorphic to the 7-sphere	p. 1
The invariant ¿(M⁷)	p. 2
A partial characterization of the n-sphere	p. 4
Examples of 7-manifolds	p. 6
Miscellaneous results	p. 8
References	p. 9
Groups of homotopy spheres	p. 111
Introduction	p. 11
Construction of the group ¿_n	p. 12
Homotopy spheres are s-parallelizable	p. 15
Which homotopy spheres bound parallelizable manifolds?	p. 17
Spherical modifications	p. 20
Framed spherical modifications	p. 27
The groups bP_2k	p. 35
A cohomology operation	p. 40
References	p. 47
Homotopically equivalent smooth manifolds	p. 49
Introduction	p. 49
The fundamental construction	p. 53
Morse's surgery	p. 53
Relative ¿-manifolds	p. 56
The general construction	p. 60
Realization of classes	p. 62
The manifolds in one class	p. 84
One manifold in different classes	p. 88
Processing the results	p. 103
The Thorn space of a normal bundle. Its homotopy structure	p. 103
Obstructions to a diffeomorphism of manifolds having the same homotopy type and a stable normal bundle	p. 111
Variation of a smooth structure keeping triangulation preserved	p. 115
Varying smooth structure and keeping the triangulation preserved. Morse surgery	p. 132
Corollaries and applications	p. 150
Smooth structures on Cartesian product of spheres	p. 150
Low-dimensional manifolds. Cases n = 4,5,6,7	p. 159
Connected sum of a manifold with Milnor's sphere	p. 164
Normal bundles of smooth manifolds	p. 167
Homotopy type and Pontrjagin classes	p. 168
Combinatorial equivalence and Milnor's microbundle theory	p. 171
On groups ¿^4k-1(&partial;¿)	p. 175
Embedding of homotopy spheres into Euclidean space and the suspension stable homomorphism	p. 178
References	p. 181
Rational Pontrjagin classes. Homeomorphism and homotopy type of closed manifolds	p. 185
Introduction	p. 186
Signature of a cycle and its properties	p. 187
The basic lemma	p. 189
Theorems on homotopy invariance. Generalized signature theorem	p. 192
The topologicalinvariance theorem	p. 197
Consequences of the topological invariance theorem	p. 199
Appendix (V. A. Rokhlin). Diffeomorphisms of the manifold S² × S³	p. 201
References	p. 202
On manifolds with free abelian fundamental group and their applications (Pontrjagin classes, smooth structures, high-dimensional knots)	p. 205
Introduction	p. 205
Formulation of results	p. 208
The proof scheme of main theorems	p. 210
A geometrical lemma	p. 213
An analog of the Hurewicz theorem	p. 217
The functor P = Hom_c and its application to the study of homology properties of degree one maps	p. 221
Stably freeness of kernel modules under the assumptions of Theorem 3	p. 227
The Homology effect of a Morse surgery	p. 230
Proof of Theorem 3	p. 232
Proof of Theorem 6	p. 233
One generalization of Theorem 5	p. 236
On the signature formul	p. 237
Unsolved questions concerning characteristic class theory	p. 242
Algebraic remarks about the functor P = Hom_c	p. 248
References	p. 251
Stable homeomorphisms and the annulus conjecture	p. 253
References	p. 260
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Topological Library - Part 2

Characteristic Classes And Smooth Structures On Manifolds

At a Glance

Hardcover

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