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**High-dimensional knot theory. Algebraic surgery in codimension 2. With appendix by Elmar Winkelnkemper.**
*(English)*
Zbl 0910.57001

Springer Monographs in Mathematics. Berlin: Springer. xxxvi, 646 p. (1998).

For the reviewer, a geometric topologist, one of the delights of knot theory is the fact that there are so many connections with other branches of mathematics. Algebraic topology, quadratic forms, algebraic number theory and singularity theory are but a few, and the coming of the Jones and HOMFLY polynomials opened up yet more. All the books on knot theory with which I am familiar take a similar view, that of a geometric topologist using algebraic tools. This book is different. Part One is entitled ‘Algebraic \(K\)-theory’, and that’s exactly what it is. But unlike some accounts of this subject, the author is at pains to explain the geometric motivation for what he is doing; for example, the connection between algebraic and geometric transversality. Part Two is entitled ‘Algebraic \(L\)-theory’, and it is here that geometry begins to appear (although algebra continues to dominate). There are chapters on codimension \(q\) surgery, codimension 2 surgery, framed codimension 2 surgery, and open books before we reach the chapter on knot theory. It is at this point that the thorough and detailed preparation pays off with applications to high-dimensional knots. After some more chapters on L-theory and multi-signatures, the book ends with a chapter on the knot cobordism groups. Finally there is an appendix by H. E. Winkelnkemper on open books. This is not a book from which to learn knot theory; it is a book from which to learn algebraic surgery, written with the geometric applications always in mind.

Reviewer: C.Kearton (Durham)

### MSC:

57-02 | Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

57R67 | Surgery obstructions, Wall groups |

57Q45 | Knots and links in high dimensions (PL-topology) (MSC2010) |