1,565 research outputs found

### Reidemeister-Turaev torsion modulo one of rational homology three-spheres

Given an oriented rational homology 3-sphere M, it is known how to associate
to any Spin^c-structure \sigma on M two quadratic functions over the linking
pairing. One quadratic function is derived from the reduction modulo 1 of the
Reidemeister-Turaev torsion of (M,\sigma), while the other one can be defined
using the intersection pairing of an appropriate compact oriented 4-manifold
with boundary M. In this paper, using surgery presentations of the manifold M,
we prove that those two quadratic functions coincide. Our proof relies on the
comparison between two distinct combinatorial descriptions of Spin^c-structures
on M Turaev's charges vs Chern vectors.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper22.abs.htm

### Loops on surfaces, Feynman diagrams, and trees

We relate the author's Lie cobracket in the module additively generated by
loops on a surface with the Connes-Kreimer Lie bracket in the module additively
generated by trees. To this end we introduce a pre-Lie coalgebra and a
(commutative) Hopf algebra of pointed loops on a surface. In the last version I
added sections on Wilson loops and knot diagrams.Comment: 13 pages, no figures. Added sections on Hopf algebras, Wilson loops
on surfaces and knot diagram

### A function on the homology of 3-manifolds

In analogy with the Thurston norm, we define for an orientable 3-manifold $M$
a numerical function on $H_2(M;Q/Z)$. This function measures the minimal
complexity of folded surfaces representing a given homology class. A similar
function is defined on the torsion subgroup of $H_1(M)$. These functions are
estimated from below in terms of abelian torsions of $M$.Comment: 16 pages, no figures. The second version incorporates several minor
correction

### A norm for the cohomology of 2-complexes

We introduce a norm on the real 1-cohomology of finite 2-complexes determined
by the Euler characteristics of graphs on these complexes. We also introduce
twisted Alexander-Fox polynomials of groups and show that they give rise to
norms on the real 1-cohomology of groups. Our main theorem states that for a
finite 2-complex X, the norm on H^1(X; R) determined by graphs on X majorates
the Alexander-Fox norms derived from \pi_1(X).Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol2/agt-2-7.abs.htm

### Unoriented HQFT and its underlying algebra

Turaev and Turner introduced a bijection between unoriented topological
quantum field theories and extended Frobenius algebras. In this paper, we will
show that there exists a bijective correspondence between unoriented (1 +
1)-dimensional homotopy quantum field theories and extended crossed group
algebras.Comment: 23 pages, 29 figures, I rearrange the main theorem and correct some
typo

### Asymptotic formulas for curve operators in TQFT

Topological quantum field theories with gauge group $\textrm{SU}_2$ associate
to each surface with marked points $\Sigma$ and each integer $r>0$ a vector
space $V_r (\Sigma)$ and to each simple closed curve $\gamma$ in $\Sigma$ an
Hermitian operator $T_r^{\gamma}$ acting on that space. We show that the matrix
elements of the operators $T_r^{\gamma}$ have an asymptotic expansion in orders
of $\frac{1}{r}$, and give a formula to compute the first two terms in terms of
trace functions, generalizing results of March\'e and Paul

- â€¦