## LSU Historical Dissertations and Theses

1995

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

Mathematics

Patrick Gilmer

#### Abstract

Blanchet, Habegger, Masbaum and Vogel defined a quantization functor on a category whose objects are oriented closed surfaces together with a collection of colored banded points and $p\sb1$-structure. The functor assigns a module $V\sb{p}(\Sigma)$ to each surface $\Sigma$. This assignment satisfies certain axioms. For p even, it satisfies the tensor product axiom, which gives the modules associated to a disconnected surface as the tensor-product of the modules associated to its components. In this dissertation we show that the p odd case satisfies a generalized tensor product formula. The notion of a generalized tensor product formula is due to Blanchet, and Masbaum. We let $\ V\sb{p}(\Sigma)$ denote $V\sb{p}(\Sigma{\rm II}\widehat{S\sp2}),$ where $\widehat{S\sp2}$ is a sphere with one banded point colored p-2. The generalized tensor product formula expresses $V\sb{p}(\Sigma\sb1{\rm II}\Sigma\sb2)$ in terms of $V\sb{p}(\Sigma\sb1),\ V\sb{p}(\Sigma\sb2),\ \ V\sb{p}(\Sigma\sb1),$ and $\ V\sb{p}(\Sigma\sb2).$ We reduce the calculation of $\ V\sb{p}(\Sigma)$ to known results, and calculate $\ V\sb{p}(\Sigma)$ explicitly in many cases. We consider the application of this theory to links of odd wrapping number in $S\sp1 \times S\sp2.$.

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