Author | Ranicki, Andrew. author |
---|---|

Title | High-dimensional Knot Theory [electronic resource] : Algebraic Surgery in Codimension 2 / by Andrew Ranicki |

Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 1998 |

Connect to | http://dx.doi.org/10.1007/978-3-662-12011-8 |

Descript | XXXVI, 646 p. online resource |

SUMMARY

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. This is the first book entirely devoted to high-dimensional knots. 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

CONTENT

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 Poincarรฉ complexes -- Codimension q surgery -- Codimension 2 surgery -- Manifold and geometric Poincarรฉ bordism of X ร{151} 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

Mathematics
Algebraic topology
Mathematics
Algebraic Topology