The PATCH seminar, a joint venture with Bryn Mawr College, Haverford College, and the University of Pennsylvania, has its last meeting of Spring 2012 on Friday, April 20, at Haverford. The two speakers are Liam Watson of UCLA and David Gay of the University of Georgia.

The talks will take place at 3:00 and 4:30, in room KINS H108 at Haverford College.

Watson is speaking first, at 3:00pm.

**Title:** *L-spaces and Left-orderability*

**Abstract:** A group is left-orderable if it admits a strict total order of its elements that is invariant under multiplication on the left. As an immediate consequence (exercise!), left-orderable groups are torsion free. For example, a finite cyclic group cannot be left-ordered; hence the fundamental group of a lens space is not left-orderable. L-spaces provide a generalizations of lens spaces in the context of Heegaard Floer homology. These manifolds have simplest possible Heegaard Floer homology, though they need not have cyclic fundamental group. This talk will describe some evidence supporting the conjecture that L-spaces are equivalent to 3-manifolds with non-left-orderable fundamental group.

Gay is speaking second, at 4:30pm.

**Title:** *Using Morse 2-Functions to trisect 4-manifolds*

**Abstract:** Morse 2-functions are generic smooth maps to 2-manifolds, just as ordinary Morse functions are generic smooth maps to 1-manifolds. The goal of this talk is to get the audience to feel comfortable thinking about Morse 2-functions and to make the case that they are worth thinking about. As a vehicle for this agenda I’ll show how to construct a Morse 2-function on an arbitrary closed oriented 4-manifold X which yields a very natural decomposition of X into three diffeomorphic pieces. This seems to be the correct 4-dimensional analog of a Heegaard splitting; each piece is a 4-dimensional 1-handlebody, and the pieces are glued together along naturally arising 3-dimensional handlebodies. I will also throw in some other tidbits and hints about our larger Morse 2-function vision. This is a report on joint work with Rob Kirby.

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