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

By Andrew Ranicki, E. Winkelnkemper

High-dimensional knot thought is the learn of the embeddings of n-dimensional manifolds in (n+2)-dimensional manifolds, generalizing the conventional examine 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 remedy of the invariants of codimension 2 embeddings, generalizing the Alexander polynomials and Seifert kinds of classical knot conception. Many ends up in the study literature are hence introduced right into a unmarried framework, and new effects are received. The therapy is especially powerful 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 way of E. Winkelnkemper at the background of open books.

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Kn (A[z]) −−→ Kn (A[z, z −1 ]) −−→ Kn (A[z], Z) −−→ Kn−1 (A[z]) −−→ . . are the algebraic K-groups K∗ (A[z], Z) = K∗−1 (H (A[z], Z)) of the exact category H (A[z], Z) of (A[z], Z)-modules. Use the isomorphism of exact categories Nil(A) −−→ H (A[z], Z) ; (P, ν) −−→ coker(z − ν : P [z]−−→P [z]) to identify K∗ (A[z], Z) = K∗−1 (H (A[z], Z)) = Nil∗−1 (A) = K∗−1 (A) ⊕ Nil∗−1 (A) . (ii) The localization exact sequence breaks up into split exact sequences i+ ∂+ 0 −−→ Kn (A[z]) −−→ Kn (A[z, z −1 ]) −−→ Niln−1 (A) −−→ 0 , so that Kn (A[z, z −1 ]) = Kn (A[z]) ⊕ Niln−1 (A) (n ∈ Z) .

G. free Amodule chain complex D with chain maps f : C−−→D, g : D−−→C and a chain homotopy gf 1 : C−−→C. An A-module chain complex C is finitely dominated if it admits a finite domination. g. projective A-modules P : . . −−→ 0 −−→ . . −−→ 0 −−→ Pn −−→ Pn−1 −−→ . . −−→ P0 . The projective class of a finitely dominated complex C is defined by ∞ (−)i [Pi ] ∈ K0 (A) [C] = [P ] = i=0 for any such P . g. free A-module chain complex. The reduced projective class [C] ∈ K0 (A) of a finitely dominated A-module chain complex is such that [C] = 0 if and only if C is chain homotopy finite.

With relative K-groups K∗ (f ). g. projective A-modules P, Q and a B-module isomorphism g : f! P = B ⊗A P −−→ f! g. projective A-module R and an A-module isomorphism ∼ P ⊕Q⊕R h : P ⊕Q ⊕R = such that τ ((g −1 ⊕ g ⊕ 1f! R )h : f! (P ⊕ Q ⊕ R) −−→ f! (P ⊕ Q ⊕ R)) = 0 ∈ K1 (B) . In the special case when f : A−−→B = S −1 A is the inclusion of A in the localization inverting a multiplicative subset S ⊂ A the relative K-groups K∗ (f ) are identified with the K-groups of the exact category of homological dimension 1 S-torsion A-modules.

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