arXiv subjects
Publications and source records attributed to Jonathan A. Hillman.
This paper is a synthesis and extension of three earlier papers on $PD_4$-complexes $X$ with fundamental group $π$ such that $c.d.π=2$ and $π$ has one end. Our goal is to show that the homotopy types of such complexes are determined by $π$, the Stiefel-Whitney classes and the equivariant intersection pairing on $π_2(X)$. We achieve this under further conditions on $π$.
We consider the possible Euler characteristics and fundamental groups of the complementary components $X$ and $Y$ of an embedding of a connected closed 3-manifold $M$ in $S^4$. We use a 2-knot satellite construction to change the fundamental groups, and Massey products to limit the values of $χ(X)$ and $χ(Y)$ when $M$ is the total space of an $S^1$-bundle with orientable base and Euler number 1.
Baues and Bleille showed that, up to oriented homotopy equivalence, a Poincare duality complex of dimension $n \ge 3$ with $(n-2)$-connected universal cover, is classified by its fundamental group, orientation class and the image of its fundamental class in the homology of the fundamental group. We generalise Turaev's results for the case $n=3$, by providing necessary and sufficient conditions for a triple $(G, ω, μ)$, comprising a group, $G$, $ω\in H^{1}(G;\mathbb{Z}/2\mathbb{Z})$ and $μ\in H_{3}(G, \mathbb{Z}^ω)$ to be realised by a Poincaré duality complex of dimension $n$ with $(n-2)$-connected universal cover, and by showing that such a complex is a connected sum of two such complexes if and only if its fundamental group is a free product of groups.
Let $K$ be a simple $2q$-knot with exterior $X$. We show directly how the Farber quintuple $(A,Π,α,\ell,ψ)$ determines the homotopy type of $X$ if the torsion subgroup of $A=π_q(X)$ has odd order. We comment briefly on the possible role of the EHP sequence in recovering the boundary inclusion from the duality pairings $\ell $ and $ψ$. Finally we reformulate the Farber quintuple as an hermitian self-duality of an object in an additive category with involution.
A closed 4-manifold (or, more generally, a finite $PD_4$-space) has a finitely dominated infinite regular covering space if and only if either its universal covering space is finitely dominated or it is finitely covered by the mapping torus of a self homotopy equivalence of a $PD_3$-complex.
A bundle with base $B$ and fibre $F$ aspherical closed surfaces has a section if and only if the action $:π_1(B)\to{Out}(π_1(F))$ factors through $Aut(π_1(F))$ and a cohomology class is 0. We simplify and make more explicit the latter condition. We also show that the transgression $d^2_{2,0}$ in the homology LHS spectral sequence of a central extension is evaluation of the extension class. Examples with hyperbolic fibre and no section (based on ideas of Endo) added.
We compute the rings $H^*(N;\mathbb{F}_2)$ for $N$ a closed $\mathbb{S}ol^3$-manifold and then determine the Borsuk-Ulam indices $BU(N,ϕ)$ with $ϕ\not=0$ in $H^1(N;\mathbb{F}_2)$.
A group $Γ$ is defined to be cofinitely Hopfian if every homomorphism $Γ\toΓ$ whose image is of finite index is an automorphism. Geometrically significant groups enjoying this property include certain relatively hyperbolic groups and many lattices. A knot group is cofinitely Hopfian if and only if the knot is not a torus knot. A free-by-cyclic group is cofinitely Hopfian if and only if it has trivial centre. Applications to the theory of open mappings between manifolds are presented.
Let $M$ be a closed 4-manifold with $π_2(M)\cong{Z}$. Then $M$ is homotopy equivalent to either $CP^2$, or the total space of an orbifold bundle with general fibre $S^2$ over a 2-orbifold $B$, or the total space of an $RP^2$-bundle over an aspherical surface. If $π=π_1(M)\not=1$ there are at most two such bundle spaces with given action $u:π\to{Aut}(π_2(M))$. The bundle space has the geometry $\mathbb{S}^2\times\mathbb{E}^2$ (if $χ(M)=0$) or $\mathbb{S}^2\times\mathbb{H}^2$ (if $χ(M)<0$), except when $B$ is orientable and $π$ is generated by involutions, in which case the action is unique and there is one non-geometric orbifold bundle.
We compute the $p$-primary components of the linking pairings of orientable 3-manifolds admitting a fixed-point free $S^1$-action. Using this, we show that any non-singular linking pairing on a finite abelian group with homogeneous 2-primary summand is realized by such a manifold. However, some pairings on inhomogeneous 2-groups are not realizable by any Seifert fibred 3-manifold.
We explore algebraic characterizations of 2-knots whose associated knot manifolds fibre over lower-dimensional orbifolds, and consider also some issues related to the groups of higher-dimensional fibred knots.
We complete the TOP classification of 2-knots with torsion-free, solvable knot group by showing that fibred 2-knots with closed fibre the Hantzsche-Wendt flat 3-manifold $HW$ are not reflexive, while every fibred 2-knot with closed fibre a $\mathbb{N}il^3$-manifold with base orbifold $S^2(3,3,3)$ is reflexive, and by giving explicit normal forms for the strict weight orbits of normal generators for the groups of all knots in either class. We also determine when the knots are amphicheiral or invertible, and show that the only non-trivial doubly null-concordant knots with such groups are those with group $πτ_29_{46}$.
Given a knot and an SL(n,C) representation of its group that is conjugate to its dual, the representation that replaces each matrix with its inverse-transpose, the associated twisted Reidemeister torsion is reciprocal. An example is given of a knot group and SL(3,Z) representation that is not conjugate to its dual for which the twisted Reidemeister torsion is not reciprocal.
Let Γbe a finitely presentable pro-p group with a nontrivial finitely generated closed normal subgroup N of infinite index. Then def(Γ)\leq 1, and if def(Γ)=1 then Γis a pro-p duality group of dimension 2, N is a free pro-p group and Γ/N is virtually free. In particular, if the centre of Γis nontrivial and def(Γ)\geq 1, then def(Γ)=1, cd G \leq 2 and Γis virtually a direct product F \times Z_p, with F a finitely generated free pro-p group.
We consider the homotopy types of $PD_4$-complexes $X$ with fundamental group $π$ such that $c.d.π=2$ and $π$ has one end. Let $β=β_2(π;F_2)$ and $w=w_1(X)$. Our main result is that (modulo two technical conditions on $(π,w)$) there are at most $2^β$ orbits of $k$-invariants determining "strongly minimal" complexes (i.e., those with homotopy intersection pairing $λ_X$ trivial). The homotopy type of a $PD_4$-complex $X$ with $π$ a $PD_2$-group is determined by $π$, $w$, $λ_X$ and the $v_2$-type of $X$. Our result also implies that Fox's 2-knot with metabelian group is determined up to TOP isotopy and reflection by its group.
We show that there are two homotopy types of PD_3-complexes with fundamental group S_3*_{Z/2Z}S_3, and give explicit constructions for each, which differ only in the attachment of the top cell.