It has been known for a long time that the fundamental group of the quotient of $RR ^3$ by the Case-Chamberlin continuum is nontrivial. In the present paper we prove that this group is in fact, uncountable.
We show that the homotopy type of a finite oriented Poincar{e} 4-complex is determined by its quadratic 2-type provided its fundamental group is finite and has a dihedral Sylow 2-subgroup. By combining with results of Hambleton-Kreck and Bauer, this
applies in the case of smooth oriented 4-manifolds whose fundamental group is a finite subgroup of SO(3). An important class of examples are elliptic surfaces with finite fundamental group.
A fibration of $mathbb{R}^3$ by oriented lines is given by a unit vector field $V : mathbb{R}^3 to S^2$, for which all of the integral curves are oriented lines. A line fibration is called skew if no two fibers are parallel. Skew fibrations have been
the focus of recent study, in part due to their close relationships with great circle fibrations of $S^3$ and with tight contact structures on $mathbb{R}^3$. Both geometric and topological classifications of the space of skew fibrations have appeared; these classifications rely on certain rigid geometric properties exhibited by skew fibrations. Here we study these properties for line fibrations which are not necessarily skew, and we offer some partial answers to the question: in what sense do nonskew fibrations look and behave like skew fibrations? We develop and utilize a technique, called the parallel plane pushoff, for studying nonskew fibrations. In addition, we summarize the known relationship between line fibrations and contact structures, and we extend these results to give a complete correspondence. Finally, we develop a technique for generating nonskew fibrations and offer a number of examples.
Garoufalidis and Levine defined a filtration for 3-manifolds equipped with some degree 1 map ($mathbb{Z}pi$-homology equivalence) to a fixed 3-manifold $N$ and showed that there is a natural surjection from a space of $pi=pi_1N$-decorated graphs to t
he graded quotient of the filtration over $mathbb{Z}[frac{1}{2}]$. In this paper, we show that in the case of $N=T^3$ the surjection of Garoufalidis--Levine is actually an isomorphism over $mathbb{Q}$. For the proof, we construct a perturbative invariant by applying Fukayas Morse homotopy theoretic construction to a local system of the quotient field of $mathbb{Q}pi$. The first invariant is an extension of the Casson invariant to $mathbb{Z}pi$-homology equivalences to the 3-torus. The results of this paper suggest that there is a highly nontrivial equivariant quantum invariants for 3-manifolds with $b_1=3$. We also discuss some generalizations of the perturbative invariant for other target spaces $N$.
We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an $Sp(2g,mathbb{Z})$-module.
We generalize Bonahon and Wongs $mathrm{SL}_2(mathbb{C})$-quantum trace map to the setting of $mathrm{SL}_3(mathbb{C})$. More precisely, for each non-zero complex number $q$, we associate to every isotopy class of framed oriented links $K$ in a thick
ened punctured surface $mathfrak{S} times (0, 1)$ a Laurent polynomial $mathrm{Tr}_lambda^q(K) = mathrm{Tr}_lambda^q(K)(X_i^q)$ in $q$-deformations $X_i^q$ of the Fock-Goncharov coordinates $X_i$ for a higher Teichm{u}ller space, depending on the choice of an ideal triangulation $lambda$ of the surface $mathfrak{S}$. Along the way, we propose a definition for a $mathrm{SL}_n(mathbb{C})$-version of this invariant.
K. Eda
,U. H. Karimov
,D. Repovv{s}
.
(2007)
.
"On the fundamental group of $mathbb R^3$ modulo the Case-Chamberlin continuum"
.
Du\\v{s}an Repov\\v{s}
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