L-space

In mathematics , L-spaces are certain 3-manifolds whose Heegaard-Floer homology is the simplest possible. The L-space conjecture suggests a connection with the (non) arrangability of fundamental groups and the (non) existence of tight foliage.

definition

A closed, orientable 3-manifold is an L-space if it is a rational sphere of homology and its Heegaard-Floer homology is a free Abelian group with generators. ${\displaystyle {\widehat {HF}}(Y)}$ ${\displaystyle \sharp H_{1}(Y)}$

Examples

Spherical 3-manifolds are L-spaces, this is especially the case for lens spaces . Also coherent sums of spherical 3-manifolds are L-spaces.

L-space conjecture

The L-space conjecture says that the following conditions should be equivalent for irreducible rational homology spheres: ${\displaystyle Y}$

• ${\displaystyle Y}$ is not an L-space,
• the fundamental group is an ordered group , ${\displaystyle \pi _{1}Y}$
• ${\displaystyle Y}$has tight foliage .

literature

• Boyer- Gordon- Watson: On L-spaces and left-orderable fundamental groups , Math. Ann. 356: 1213-1245 (2013).