High-dimensional knot theory. Algebraic surgery in by Andrew Ranicki, E. Winkelnkemper

By Andrew Ranicki, E. Winkelnkemper

High-dimensional knot thought is the research of the embeddings of n-dimensional manifolds in (n+2)-dimensional manifolds, generalizing the conventional learn of knots within the case n=1. the most topic is the applying of the author's algebraic concept of surgical procedure to supply a unified therapy of the invariants of codimension 2 embeddings, generalizing the Alexander polynomials and Seifert types of classical knot conception. Many leads to the study literature are hence introduced right into a unmarried framework, and new effects are bought. The therapy is very potent in facing open books, that are manifolds with codimension 2 submanifolds such that the supplement fibres over a circle. The publication concludes with an appendix by means of E. Winkelnkemper at the historical past of open books.

Show description

Read Online or Download High-dimensional knot theory. Algebraic surgery in codimension 2. With errata PDF

Best algebra books

Lie Algebras: Finite and Infinite Dimensional Lie Algebras and Applications in Physics

This can be the lengthy awaited follow-up to Lie Algebras, half I which lined a massive a part of the idea of Kac-Moody algebras, stressing basically their mathematical constitution. half II offers normally with the representations and purposes of Lie Algebras and comprises many go references to half I. The theoretical half principally offers with the illustration thought of Lie algebras with a triangular decomposition, of which Kac-Moody algebras and the Virasoro algebra are top examples.

Work and Health: Risk Groups and Trends Scenario Report Commissioned by the Steering Committee on Future Health Scenarios

Will the current excessive paintings velocity and the powerful time strain survive within the coming two decades? within the 12 months 2010 will there be much more staff operating lower than their point of schooling and struggling with illnesses because of tension at paintings than is the case in the meanwhile?

Extra info for High-dimensional knot theory. Algebraic surgery in codimension 2. With errata

Sample text

For n ≥ 3 the knot k is unknotted if and only if (g, c) is a homotopy equivalence, and k is null-cobordant if and only if (g, c) is normal bordant by a homology equivalence to a homotopy equivalence. The high-dimensional knot cobordism groups C2∗−1 were expressed by Levine [156] as the Witt groups of Seifert matrices in Z, and by Cappell and Shaneson [40] as certain types of algebraic Γ -groups. Pardon [221], Smith [271] and Ranicki [237, Chap. 9] expressed the knot cobordism groups as torsion L-groups.

I) The relative algebraic K-groups are such that K∗ (A, ST ) = K∗ (A, S) ⊕ K∗ (A, T ) and there is defined a Mayer–Vietoris exact sequence . . −−→ Kn (A) −−→ Kn (S −1 A) ⊕ Kn (T −1 A) −−→ Kn ((ST )−1 A) −−→ Kn−1 (A) −−→ . . (ii) A finitely dominated A-module chain complex C is (ST )−1 A-contractible if and only if it is chain equivalent to the sum C ⊕ C of a finitely dominated S −1 A-contractible A-module chain complex C and a finitely dominated T −1 A-contractible A-module chain complex C , in which case C T −1 C −1 and C S C.

Iii) Immediate from (ii). 15 Let A be a ring with a multiplicative subset S ⊂ A. The S-adic completion of A is the ring defined by the inverse limit AS = ← lim − A/(s) . s∈S Let S ⊂ AS be the multiplicative subset defined by the image of S ⊂ A under the canonical inclusion A−−→AS . 3 ∞ ∞ k AS = As = ker(I − T : A/(sk )) A/(s )−−→ k=1 k=1 with ∞ ∞ A/(sk ) −−→ I −T : k=1 ∞ A/(sk ) ; k=1 ∞ ak −−→ k=1 (ak − ak−1 ) . 17 For any ring with multiplicative subset (A, S) the localization S −1 A and completion AS are such that the inclusion A−−→AS defines a cartesian morphism (A, S) −−→ (AS , S) , inducing a cartesian square of rings A / S −1 A  AS  / S −1 A S and a Mayer–Vietoris exact sequence .

Download PDF sample

Rated 4.28 of 5 – based on 16 votes