Homotopy Type Invariants of Four-Dimensional Knot Complements

Thesis (PDF Available) · May 1984with 86 Reads
Thesis for: PhD
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Abstract
This thesis studies the homotopy type of smooth four-dimensional knot complements. In contrast with the classical case, high-dimensional knot complements with fundamental group different from are never aspherical. The second homotopy group already provides examples of the way in which a knot in S^4 can fail to be determined by its fundamental group (C.McA. Gordon, S.P. Plotnick). A natural class of knots to investigate is ribbon knots. They bound immersed disks with ribbon singularities. A method is given for computing π_2 of such knot complements. I show that there are infinitely many ribbon knots in S^4 with isomorphic π_1 but distinct π_2 (viewed as π_1-modules). They appear as boundaries of distinct ribbon disk pairs with the same exterior. These knots have the fundamental group of the spun trefoil, but none in a spun knot. To a four-dimensional knot complement, one can associate a certain cohomology class, the first k-invariant of Eilenberg, MacLane and Whitehead. In a joint paper, Plotnick and I showed that there are arbitrarily many knots in S^4 whose complements have isomorphic π_1 and π_2 (as π_1-modules), but distinct k-invariants. Here I prove this result using examples which are somewhat more natural and easier to produce. They are constructed from a fibered knot with fiber a punctured lens space and a ribbon knot by surgery. The proofs involve writing down explicit cell complexes, computing twisted cohomology groups, combinatorial group theory and calculations in group rings.
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  • Article
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    This paper studies some questions concerning homotopy type invariants of smooth four-dimensional knot complements. Higher-dimensional knot theory diverges sharply from classical knot theory in this respect. A knot complement S<sup>4</sup> \ S<sup>2</sup> has the homotopy type of a 3-complex, so a natural question is whether the homotopy theory of knot complements in S<sup>4</sup> can be as complicated as that of arbitrary 3-complexes. The main result of this paper indicates that the answer is yes.
  • Article
    A knot K = (S^{n+2}, S^n) is a ribbon knot if S^n bounds an immersed disc D^{n+1} in S^{n+2} with no triple points and such that the components of the singular set are n-discs whose boundary (n-1)-spheres either lie on S^n or are disjoint from S^n. Pushing the disc D^{n+1} into D^{n+3} produces a ribbon disc pair D = (D^{n+3}, D^{n+1}), with the ribbon knot (S^{n+2}, S^n) on its boundary. The double of a ribbon (n+1)-disc pair is an (n+1)-ribbon knot. Every (n+1)-ribbon knot is obtained in this manner.