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2-knots and their groups. (English) Zbl 0669.57008

Australian Mathematical Society Lecture Series, 5. Cambridge etc.: Cambridge University Press. x, 164 p. (1989).
After a review in Chapter 1 of the main concepts of knot theory, Chapter 2 begins with a careful treatment of Kervaire’s results on n-knot groups for \(n\geq 2\). Various other results on these groups are then presented, including Farber’s example of a 3-knot group which is not a 2-knot group and Eckmann’s proof that the exterior of a non-trivial n-knot with \(n>1\) is never aspherical.
The third chapter contains what the author describes as his key result, that there are many instances where the closed 4-manifold M(K) obtained by surgery on a 2-knot K is aspherical. He uses this to investigate the knot group \(\pi\), or \(\pi\) /T where T is the maximal locally-finite normal subgroup of \(\pi\), which in many cases turns out to be an orientable Poincaré duality group of formal dimension 4. These are investigated in great detail, particularly with regard to the case where \(\pi\) or \(\pi\) /T contains a torsion-free abelian normal subgroup A. The rank of A is necessarily at most 4, and the various possibilities are carefully distinguished in subsequent chapters.
The final two chapters consider to what extent the knot K is determined by algebraic data associated with its group. For this purpose surgery is needed, and it is because of this that the author works throughout in the TOP category. In conclusion, a list of some thirty open questions is given.
This is a book which belongs on the shelves of every knot theorist, indeed of everyone interested in the interplay between algebra and geometric topology. The subject is hard but the treatment is masterly.
Reviewer: Ch.Kearton

MSC:

57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57N65 Algebraic topology of manifolds
57N70 Cobordism and concordance in topological manifolds
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
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