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J. A. Hillman

Publications and source records attributed to J. A. Hillman.

At least 19 recordsLinked to original sources
arXiv2026

Aspherical manifolds with boundary

We undertake a systematic investigation of compact aspherical manifolds with boundary; motivated by the plethora of examples in the bounded case and by the beauty of the theory in the closed case. Our main theorems give a homological criterion for when a closed manifold, together with maps from the fundamental groups of its components to a fixed group, can be realized as the boundary of a compact aspherical manifold. This is done in two steps: we first produce a Poincaré pair and then apply surgery theory to obtain a manifold. We illustrate this in the case of abelian fundamental group. The results of this paper will be applied in a sequel where we classify compact aspherical 4-manifolds with elementary amenable fundamental group.

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arXiv2026

Aspherical 4-manifolds with elementary amenable fundamental group

We classify the possible elementary amenable fundamental groups of compact aspherical 4-manifolds with boundary and conclude that they are either polycyclic or solvable Baumslag- Solitar. Since these groups are good and satisfy the Farrell-Jones Conjecture, one concludes that such manifolds satisfy topological rigidity: a homotopy equivalence which is a homeomorphism on the boundary is homotopic, relative to the boundary, to a homeomorphism. We classify the closed 3-manifolds which arise as the boundary of an compact aspherical 4-manifold with elementary amenable fundamental group, generalizing results of Freedman and Quinn in the cases of trivial and infinite cyclic fundamental groups. Moreover, two such 4-manifolds are homeomorphic if and only if their "enhanced" peripheral group systems are equivalent, and each such manifold is the boundary connected sum of a compact aspherical 4-manifold with prime boundary and a contractible 4-manifold.

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arXiv2024

Locally flat embeddings of 3-manifolds in $S^4$

M. Freedman showed that every homology 3-sphere embeds as a locally flat submanifold of $S^4$. This is in striking contrast to the state of our knowledge of smooth embeddings of homology spheres. This book surveys what is presently known about the topic of its title, with emphasis on embeddings of Seifert fibred 3-manifolds and on distinguishing embeddings by properties of the complementary regions. In the final chapters 4-dimension surgery is used to study embeddings in which the complementary regions have abelian or nilpotent fundamental groups, and the final appendix contains a number of open questions. Work on smooth embeddings is only described briefly, as current techniques in this area are beyond the author's expertise.

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arXiv2024

Nilpotent groups with balanced presentations. II

If $G$ is a nilpotent group with a balanced presentation and $G\not\cong\mathbb{Z}^3$ then $β_1(G;\mathbb{Q})\leq2$ \cite{Hi22}. We show that if such a group $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}^2$ then $G$ is torsion-free and has Hirsch length $h(G)\leq4$. On the other hand, if $β_1(G;\mathbb{Q})=1$ and $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}$ then $G\cong\mathbb{Z}/m\mathbb{Z}\rtimes_n\mathbb{Z}$, for some $m,n\not=0$ such that $m$ divides a power of $n-1$.

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arXiv2023

Subnormality in $PD_3$-groups and $L^2$-Betti numbers

We reconsider work of Elkalla on subnormal subgroups of 3-manifold groups, giving essentially algebraic arguments that extend to the case of $PD_3$-groups and group pairs. However the argument relies on an $L^2$-Betti number hypothesis which has not yet been shown to hold in general.

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arXiv2021

Nilpotent groups with balanced presentations

We show that if a torsion free nilpotent group $G$ has a balanced presentations and Hirsch length $h(G)>3$ then $β_1(G;\mathbb{Q})=2$. There is just one such group which is torsion-free and of Hirsch length $h=4$, and none with $h=5$. We also construct a torsion-free nilpotent group $G$ with $h=6$ and such that $β_2(G;F)=β_1(G;F)$ for all fields $F$.

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arXiv2020

3-Manifolds with nilpotent embeddings in $S^4$

We consider embeddings of 3-manifolds $M$ in $S^4$ such that the two complementary regions $X$ and $Y$ each have nilpotent fundamental group. If $β=β_1(M)$ is odd then these groups are abelian and $β\leq3$. In general, $π_1(X)$ and $π_1(Y)$ have 3-generator presentations, and $β\leq6$. We determine all such nilpotent groups which are torsion-free and have Hirsch length $\leq5$.

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arXiv2018

3-Manifolds with abelian embeddings in $S^4$

We consider embeddings of 3-manifolds in $S^4$ such that each of the two complementary regions has an abelian fundamental group. In particular, we show that an homology handle $M$ has such an embedding if and only if $π_1(M)'$ is perfect, and that the embedding is then essentially unique.

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arXiv2017

Flat 2-orbifolds and Seifert fibred 4-manifolds

This is a summary of some of the basic facts about flat 2-orbifold groups, otherwise known as 2-dimensional crystallographic groups. We relate the geometric and topological presentations of these groups, and consider structures corresponding to decompositions of the orbifolds as fibrations or as unions. We also consider covering relations, and record the bases of Seifert fibrations of 4-manifolds with geometries of solvable Lie type.

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arXiv2017

Deficiency, commensurators and 4-dimensional infrasolvmanifolds

We show that if $π$ is the fundamental group of a 4-dimensional infrasolvmanifold then $-2\leq{def(π)}\leq0$, and give examples realizing each of these values. We also determine the abstract commensurators of such groups. Finally we show that if $G$ is a finitely generated group the kernel of the natural homomorphism from $G$ to its abstract commensurator $Comm(G)$ is locally nilpotent by locally finite, and is finite if $\mathrm{def}(G)>1$.

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arXiv2017

Solvable normal subgroups of 2-knot groups

If $X$ is an orientable, strongly minimal $PD_4$-complex and $π_1(X)$ has one end then it has no nontrivial locally-finite normal subgroup. Hence if $π$ is a 2-knot group then (a) if $π$ is virtually solvable then either $π$ has two ends or $π\congΦ$, with presentation $\langle{a,t}|ta=a^2t\rangle$, or $π$ is torsion-free and polycyclic of Hirsch length 4; (b) either $π$ has two ends, or $π$ has one end and the centre $ζπ$ is torsion-free, or $π$ has infinitely many ends and $ζπ$ is finite; and (c) the Hirsch-Plotkin radical $\sqrtπ$ is nilpotent.

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arXiv2015

$\mathbb{S}ol^3\times\mathbb{E}^1$-manifolds

We show that $\mathbb{S}ol^3\times\mathbb{E}^1$-manifolds are Seifert fibred, with general fibre the torus, and base one of the seven flat 2-orbifolds $T, Kb, \mathbb{A}, \mathbb{M}b, S(2,2,2,2), P(2,2)$ or $\mathbb{D}(2,2)$, and outline a classification of such 4-manifolds.

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arXiv2013

An aspherical 5-manifold with perfect fundamental group

We construct aspherical closed orientable 5-manifolds with perfect fundamental group. This completes part of our study (with D.H.Kochloukova and I.Lima) of $PD_n$-groups with pro-$p$ completion a pro-$p$ Poincaré duality group of dimension $\leq{n-2}$. We also consider the question of whether there are any examples with "dimension drop" 1.

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arXiv2013

Seifert fibred knot manifolds

We consider the question of when is the closed manifold obtained by elementary surgery on an $n$-knot Seifert fibred over a 2-orbifold. After some observations on the classical case, we concentrate on the cases n=2 and 3. We have found a new family of 2-knots with torsion-free, solvable group, overlooked in earlier work. We know of no higher dimensional examples.

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arXiv2013

Parallelizability of 4-dimensional infrasolvmanifolds

We show that if $M$ is an orientable 4-dimensional infrasolvmanifold and either $β=β_1(M;\mathbb{Q})\geq2$ or $M$ is a $\mathbb{S}ol_0^4$- or a $\mathbb{S}ol_{m,n}^4$-manifold (with $m\not=n$) then $M$ is parallelizable. There are non-parallelizable examples with $β=1$ for each of the other solvable Lie geometries $\mathbb{E}^4$, $\mathbb{N}il^4$, $\mathbb{N}il^3\times\mathbb{E}^1$ and $\mathbb{S}ol^3\times\mathbb{E}^1$. We also determine which non-orientable flat 4-manifolds have a $Pin^+$- or $Pin^-$-structure, and consider briefly this question for the other cases.

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