Let K be a knotted sphere in S^{n+1}, with closed fiber a spherical space form, S^n/pi. If n=3, and the monodromy has odd order, then K and its Gluck reconstruction, K^*, are inequivalent. On the other hand, if n>3, then pi is cyclic, and K is equivalent to K^*.

We exhibit integral-homology 4-spheres with isomorphic pi_1 and pi_2 (as pi_1-modules), but with distinct k-invariants.

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 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

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 $S^4setminus S^2$ has the homotopy type of a $3$-complex, so a natural question is whether the homotopy theory of knot complements in $S^4$ can be as complicated as that of arbitrary $3$-complexes. The main result of this paper

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