High-dimensional Knot Theory

Algebraic Surgery in Codimension 2

Andrew Ranicki (Autor)

Buch | Softcover

XXXVI, 646 Seiten

2010
Springer Berlin (Verlag)
978-3-642-08329-7 (ISBN)

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High-dimensional knot theory is the study of the embeddings of n-dimensional manifolds in (n+2)-dimensional manifolds, generalizing the traditional study of knots in the case n=1. The main theme is the application of the author's algebraic theory of surgery to provide a unified treatment of the invariants of codimension 2 embeddings, generalizing the Alexander polynomials and Seifert forms of classical knot theory. Many results in the research literature are thus brought into a single framework, and new results are obtained. The treatment is particularly effective in dealing with open books, which are manifolds with codimension 2 submanifolds such that the complement fibres over a circle. The book concludes with an appendix by E. Winkelnkemper on the history of open books.

Algebraic K-theory.- Finite structures.- Geometric bands.- Algebraic bands.- Localization and completion in K-theory.- K-theory of polynomial extensions.- K-theory of formal power series.- Algebraic transversality.- Finite domination and Novikov homology.- Noncommutative localization.- Endomorphism K-theory.- The characteristic polynomial.- Primary K-theory.- Automorphism K-theory.- Witt vectors.- The fibering obstruction.- Reidemeister torsion.- Alexander polynomials.- K-theory of Dedekind rings.- K-theory of function fields.- Algebraic L-theory.- Algebraic Poincaré complexes.- Codimension q surgery.- Codimension 2 surgery.- Manifold and geometric Poincaré bordism of X × S 1.- L-theory of Laurent extensions.- Localization and completion in L-theory.- Asymmetric L-theory.- Framed codimension 2 surgery.- Automorphism L-theory.- Open books.- Twisted doubles.- Isometric L-theory.- Seifert and Blanchfield complexes.- Knot theory.- Endomorphism L-theory.- Primary L-theory.- Almost symmetric L-theory.- L-theory of fields and rational localization.- L-theory of Dedekind rings.- L-theory of function fields.- The multisignature.- Coupling invariants.- The knot cobordism groups.

Erscheint lt. Verlag	15.12.2010
Reihe/Serie	Springer Monographs in Mathematics
Mitarbeit	Anhang von: E. Winkelnkemper
Zusatzinfo	XXXVI, 646 p.
Verlagsort	Berlin
Sprache	englisch
Maße	155 x 235 mm
Gewicht	1017 g
Themenwelt	Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
Schlagworte	Homology • knots • K-theory • Manifolds • open books • Surgery
ISBN-10	3-642-08329-3 / 3642083293
ISBN-13	978-3-642-08329-7 / 9783642083297
Zustand	Neuware