arXiv subjects
Publications and source records attributed to Jonathan A. Hillman.
If the group of a 2-knot group $K$ has an abelian normal subgroup of rank $\geq1$ which is not finitely generated then either $K$ has no minimal Seifert hypersurface or $K$ is topologically equivalent to Example 10 of Ralph Fox's``{\it A quick trip through knot theory}".
We characterize the groups of branched twist spins of classical knots in terms of 3-manifold groups, and also give a purely algebraic, conjectural characterization in terms of $PD_3$-groups. We show also that each group is the group of at most finitely many branched twist spins.
Let $X$ be a $PD_4$-complex and let $π=π_1(X)$. If $π$ is torsion-free and $π_2(X)$ is a finitely generated projective $\mathbb{Z}[π]$-module then either $π$ is free or $π$ is $FP$ and $c.d.π=4$. If, moreover, $H^3(π;\mathbb{Z}[π])=0$ then $π$ is a free product of $PD_4$-groups and a free group.
We extend two results known for aspherical 3-manifolds to $PD_3$-pairs $(P,\partial{P})$ with aspherical ambient space $P$. Every such $PD_3$-pair may be assembled by attaching 1-handles to $PD_3$-pairs with aspherical; ambient space and $π_1$-injective boundary. (Thus the study of such pairs reduces to the study of $PD_3$-pairs of groups.) If $π$ is a group of type $FP$ whose indecomposable factors $G$ each have $χ(G_i)=0$ then there are only finitely many such $PD_3$-pairs with $π_1(P)\congπ$.
We address the question of existence of sections of fibrations in two settings. First, we show that a bundle with base a finite 2-complex admits a section if and only if the inclusion of the fiber is $π_1$-injective and the associated short exact sequence of fundamental groups splits. Second, for Lefschetz fibrations over the disk we provide a complete algebraic criterion characterizing which loops in the boundary mapping torus extend to continuous or smooth sections over the disk. Finally, we apply our results to achiral Lefschetz fibrations over the sphere obtained by doubling along the vertical boundary, and give a criterion ensuring the existence of at least two homologically distinct sections.
There are perfect Bieberbach groups of Hirsch length 15, but none in lower dimensions. We shall show that a nonsolvable, torsion free, virtually solvable group $S$ must have Hirsch length $h(S)\geq10$. If $h(S)\leq13$ then we may assume that $A_5$ is the only simple factor, but $PSL(2,7)$ and $SL(2,8)$ may occur when $h(S)\geq14$. There are no known examples with $h(S)<15$.
We construct 3-manifolds which have at least two inequivalent embeddings such that both complementary regions have abelian fundamental group.
We give a new argument for the characterization of the cohomology rings of closed 3-manifolds with coefficients $\mathbb{F}_2$, first given by M. M. Postnikov (in terms of intersection rings) in 1948
We show that the homotopy type of a 4-manifold $M$ whose fundamental group is a finitely presentable $PD_3$-group $π$ and with $w_1(M)=w_1(π)$ is determined by $π$, $π_2(M)$, $k_1(M)$ and the equivariant intersection pairing $λ_M$.
We consider closed topological 4-manifolds $M$ with universal cover ${S^2\times{S^2}}$ and Euler characteristic $χ(M) = 1$. All such manifolds with $π=π_1(M)\cong {\mathbb Z}/4$ are homotopy equivalent. In this case, we show that there are four homeomorphism types, and propose a candidate for a smooth example which is not homeomorphic to the geometric quotient. If $π\cong {\mathbb Z}/2 \times {\mathbb Z}/2$, we show that there are three homotopy types (and between 6 and 24 homeomorphism types).
We show that every $PD_3$-complex $P$ bounds a $PD_4$-pair $(Z,P)$. If $P$ is orientable we may assume that $π_1(Z)=1$. We show also that if $P$ has a manifold 1-skeleton then it is homotopy equivalent to a closed 3-manifold, and that if the inclusion of $Z$ into $P$ induces an isomorphism on fundamental groups then $π_1(Z)$ is a free group.
We extend work of Turaev and Bleile to relax the $π_1$-injectivity hypothesis in the characterization of the fundamental triples of $PD_3$-pairs with aspherical boundary components. This is further extended to pairs $(P,\partial{P})$ which also have spherical boundary components and with $c.d.π_1(P)\leq2$.
We show that torsion-free elementary amenable groups of Hirsch length $\leq3$ are solvable, of derived length $\leq3$. This class includes all solvable groups of cohomological dimension 3. We show also that groups in the latter subclass are either polycyclic, semidirect products $BS(1,n)\rtimes\mathbb{Z}$ or properly ascending HNN extensions with base $\mathbb{Z}^2$ or $π_1(Kb)$.
We consider aspects of the question "when is an orientable closed 4-manifold $Y$ dominated by another such manifold $X$", focusing on the cases when $X$ is geometric or fibres non-trivially over an orientable surface.
We show that if $P$ is a $PD_3$-complex and $g\inπ_1(P)$ has finite order $>1$ and infinite centraliser then $π_1(P)$ retracts onto $Z/2Z\oplus\mathbb{Z}$. If $P$ is an irreducible closed 3-manifold then it follows from the Projective Plane Theorem that $P\cong{RP^2}\times{S^1}$.
We show that if a $PD_3$-group $G$ splits as an HNN extension $A*_Cφ$ where $C$ is a $PD_3$-group then the Poincaré dual in $H^1(G;\mathbb{Z})=Hom(G,\mathbb{Z})$ of the homology class $[C]$ is the epimorphism $f:G\to\mathbb{Z}$ with kernel the normal closure of $A$. We also make several other observations about $PD_3$-groups which split over $PD_2$-groups.
We show that if $B$ is an aspherical 2-orbifold in one of the families known to have orbifold fundamental groups of weight 1 then $B$ is the base of a Seifert fibration of a 2-knot manifold $M(K)$.
We show that every $\mathbb{Z}$-torsion free knot module is realized by a ribbon 2-knot with group of geometric dimension at most 2, and give some partial results on the characterization of the knot modules of fibred ribbon 2-knots.