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.

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**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 .