Expand all Collapse all | Results 1 - 1 of 1 |
1. CJM 2014 (vol 67 pp. 152)
On Homotopy Invariants of Combings of Three-manifolds Combings of compact, oriented $3$-dimensional manifolds $M$ are
homotopy classes of nowhere vanishing vector fields.
The Euler class of the normal bundle is an invariant of the combing,
and it only depends on the underlying Spin$^c$-structure. A combing
is called torsion
if this Euler class is a torsion element of $H^2(M;\mathbb Z)$. Gompf
introduced a $\mathbb Q$-valued invariant $\theta_G$ of torsion combings
on closed $3$-manifolds, and he showed that $\theta_G$ distinguishes
all torsion combings with the same Spin$^c$-structure.
We give an alternative definition for $\theta_G$ and we express
its variation as a linking number. We define a similar invariant
$p_1$ of combings for manifolds bounded by $S^2$. We relate $p_1$
to the $\Theta$-invariant, which is the simplest configuration
space integral invariant of rational homology $3$-balls, by the
formula $\Theta=\frac14p_1 + 6 \lambda(\hat{M})$ where $\lambda$
is the Casson-Walker invariant.
The article also includes a self-contained presentation of combings
for $3$-manifolds.
Keywords:Spin$^c$-structure, nowhere zero vector fields, first Pontrjagin class, Euler class, Heegaard Floer homology grading, Gompf invariant, Theta invariant, Casson-Walker invariant, perturbative expansion of Chern-Simons theory, configuration space integrals Categories:57M27, 57R20, 57N10 |