HOMOTOPY TYPE INVARIANTS OF FOUR-DIMENSIONAL KNOT COMPLEMENTS

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 (pi)(,2) of such knot complements. I show that there are infinitely many ribbon knots in S('4) with isomorphic (pi)(,1) but distinct (pi)(,2) (viewed as (pi)(,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 (pi)(,1) and (pi)(,2) (as (pi)(,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.