#### Date of Award

1997

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### First Advisor

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.

#### Recommended Citation

Natov, Jonathan, "Pure Framed Braids and 3-Manifolds." (1997). *LSU Historical Dissertations and Theses*. 6436.

http://digitalcommons.lsu.edu/gradschool_disstheses/6436

#### ISBN

9780591458381

#### Pages

63