Date of Award

1996

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

First Advisor

Patrick M. Gilmer

Abstract

We consider the sequence of finite branched cyclic covers of $S\sp3$ branched along a tame knot $K : S\sp1\to S\sp3$ and prove several results about the homology of these manifolds. We show that the sequence of cyclic resultants of the Alexander polynomial of K satisfies a linear recursion formula with integral coefficients. This means that the orders of the first homology groups of the branched cyclic covers of K can be computed recursively. We further establish the existence of a recursion formula that generates sequences which contain the square roots of the orders of the odd-fold covers and that contain the square roots of the orders of the even-fold covers quotiented by the order of the 2-fold cover (that these numbers are all integers follows from a theorem of Plans (P)). We also show that the $\doubz/p\sp{r}$-homology of this sequence of manifolds is periodic in every dimension, and we investigate these periods for the one-dimensional homology. Additionally, we give a new proof of Plans' theorem in the even-fold case (that the kernel of the map induced on homology by the covering projection $M\sb{k}\to M\sb2$ is a direct double for k even) in the style of Gordon's proof of Plans' theorem in the odd-fold case (that $H\sb1(M\sb{k}$) is a direct double for k odd).

ISBN

9780591133806

Pages

40

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