19 research outputs found
Groupoid Extensions of Mapping Class Representations for Bordered Surfaces
The mapping class group of a surface with one boundary component admits
numerous interesting representations including as a group of automorphisms of a
free group and as a group of symplectic transformations. Insofar as the mapping
class group can be identified with the fundamental group of Riemann's moduli
space, it is furthermore identified with a subgroup of the fundamental path
groupoid upon choosing a basepoint. A combinatorial model for this, the mapping
class groupoid, arises from the invariant cell decomposition of Teichm\"uller
space, whose fundamental path groupoid is called the Ptolemy groupoid. It is
natural to try to extend representations of the mapping class group to the
mapping class groupoid, i.e., construct a homomorphism from the mapping class
groupoid to the same target that extends the given representations arising from
various choices of basepoint.
Among others, we extend both aforementioned representations to the groupoid
level in this sense, where the symplectic representation is lifted both
rationally and integrally. The techniques of proof include several algorithms
involving fatgraphs and chord diagrams. The former extension is given by
explicit formulae depending upon six essential cases, and the kernel and image
of the groupoid representation are computed. Furthermore, this provides
groupoid extensions of any representation of the mapping class group that
factors through its action on the fundamental group of the surface including,
for instance, the Magnus representation and representations on the moduli
spaces of flat connections.Comment: 24 pages, 4 figures Theorem 3.6 has been strengthened, and Theorems
8.1 and 8.2 have been adde
Finite type invariants and fatgraphs
We define an invariant of pairs M,G, where M is a 3-manifold
obtained by surgery on some framed link in the cylinder , S is a
connected surface with at least one boundary component, and G is a fatgraph
spine of S. In effect, is the composition with the maps of
Le-Murakami-Ohtsuki of the link invariant of Andersen-Mattes-Reshetikhin
computed relative to choices determined by the fatgraph G; this provides a
basic connection between 2d geometry and 3d quantum topology. For each fixed G,
this invariant is shown to be universal for homology cylinders, i.e.,
establishes an isomorphism from an appropriate vector space
of homology cylinders to a certain algebra of Jacobi diagrams. Via
composition for any pair of fatgraph spines
G,G' of S, we derive a representation of the Ptolemy groupoid, i.e., the
combinatorial model for the fundamental path groupoid of Teichmuller space, as
a group of automorphisms of this algebra. The space comes equipped
with a geometrically natural product induced by stacking cylinders on top of
one another and furthermore supports related operations which arise by gluing a
homology handlebody to one end of a cylinder or to another homology handlebody.
We compute how interacts with all three operations explicitly in
terms of natural products on Jacobi diagrams and certain diagrammatic
constants. Our main result gives an explicit extension of the LMO invariant of
3-manifolds to the Ptolemy groupoid in terms of these operations, and this
groupoid extension nearly fits the paradigm of a TQFT. We finally re-derive the
Morita-Penner cocycle representing the first Johnson homomorphism using a
variant/generalization of .Comment: 39 page
Age at first birth in women is genetically associated with increased risk of schizophrenia
Prof. Paunio on PGC:n jäsenPrevious studies have shown an increased risk for mental health problems in children born to both younger and older parents compared to children of average-aged parents. We previously used a novel design to reveal a latent mechanism of genetic association between schizophrenia and age at first birth in women (AFB). Here, we use independent data from the UK Biobank (N = 38,892) to replicate the finding of an association between predicted genetic risk of schizophrenia and AFB in women, and to estimate the genetic correlation between schizophrenia and AFB in women stratified into younger and older groups. We find evidence for an association between predicted genetic risk of schizophrenia and AFB in women (P-value = 1.12E-05), and we show genetic heterogeneity between younger and older AFB groups (P-value = 3.45E-03). The genetic correlation between schizophrenia and AFB in the younger AFB group is -0.16 (SE = 0.04) while that between schizophrenia and AFB in the older AFB group is 0.14 (SE = 0.08). Our results suggest that early, and perhaps also late, age at first birth in women is associated with increased genetic risk for schizophrenia in the UK Biobank sample. These findings contribute new insights into factors contributing to the complex bio-social risk architecture underpinning the association between parental age and offspring mental health.Peer reviewe