## LSU Historical Dissertations and Theses

#### Title

Pure Framed Braids and 3-Manifolds.

1997

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

Lawrence Smolinsky

#### Abstract

This dissertation looks at representations of framed pure braids and compact orientable three manifolds. A representation, $\Phi:Z\sp{n}\oplus P\sb{n}\to \Gamma\sb{n},$ is constructed from the framed pure braid group on n strands to the mapping class group on a surface of genus n. The representation is used to obtain a presentation of the fundamental group. The representation, like that of (M-P), is compatible with Heegaard and Surgery descriptions. An algorithm is presented for transforming mapping class group elements to a stably equivalent pure framed braid, under the correspondence given by the representation. A geometric description, using the representation, is given for multiplication in a subgroup of a central extension to the mapping class group coming from (A). A question of providing a group representation development for Witten's three manifold invariant is explored. The result is negative, except for a restricted case of pure framed braids.

9780591458381

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