Cyclic bordism and rack spaces
This thesis falls into two parts, the first explores a cyclic version of bordism and the second studies the homotopy groups of rack spaces. In chapters 2-5 we begin by reviewing some theory of cyclic homology but we present it in a topological framework. Then cyclic bordism is introduced as a parallel theory. In particular we prove the equivalence of cyclic and equivariant theories. This enables us to reduce the question of representation of cyclic homology by cyclic bordism to that of representation of ordinary homology by bordism. Finally, we state a fixed point theorem of periodic bordism. In chapters 6-10 we study rack spaces, or the classifying spaces of racks. The homotopy groups of rack spaces are invariants of the rack up to rack isomorphism, and give invariants of semiframednon-split (irreducible) links ill the three-sphere. We describe methods for calculating the second homotopy group in chapter 7 and in the next chapter we apply one of the methods to find generators for the second homotopy of a class of racks, the finite Alexander quotients. Chapter 9 discusses topological racks. The classifying spaces of racks with a non-discrete topology have a cell structure and, although it fails to be a CVV cell structure, it can be used to calculate homotopy groups. The third homotopy group of a rack space is seen to be in one-to-one correspondence with bordism classes of framed labelled immersed surfaces in the three-sphere. We finish in chapter 10 by simplifying such surfaces within bordism to calculate the third homotopy group of the trivial rack and the cyclic racks, [pie]3(B(Cn))=Z2.