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# Homotopy Type Invariants of Four-Dimensional Knot Complements

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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Content uploaded by Alex Suciu

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All content in this area was uploaded by Alex Suciu on Mar 09, 2017

Content may be subject to copyright.