arXiv subjects

J. A. Hillman

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

21 records · Page 2Linked to original sources

arXiv2013

An infrasolvmanifold which does not bound

Every 4-dimensional infrasolvmanifold $M$ with $β_1(M;\mathbb{Q})>0$ or which is flat or has one of the geometries $\mathbb{N}il^4$, $\mathbb{S}ol_{m,n}^4$, or $\mathbb{S}ol_0^4$ bounds. However there are non-orientable $\mathbb{S}ol_1^4$-manifolds which do not bound. The question remains open for $\mathbb{N}il^3\times\mathbb{E}^1$-manifolds. We also give simple cobounding 5-manifolds for all but five of the 74 flat 4-manifolds, and investigate which orientable flat 4-manifolds embed in $\mathbb{R}^5$.

math.GT↗

arXiv2011

Indecomposable $PD_3$-complexes

We show that if $X$ is an indecomposable $PD_3$-complex and $π_1(X) is the fundamental group of a reduced finite graph of finite groups but is not virtually cyclic then $X$ is orientable, the underlying graph is a tree, all the edge groups are $Z/2Z$ and all but at most one of the vertex groups is dihedral of order $2m$ with $m$ odd. Every such group is realized by some $PD_3$-complex. We also propose a strategy for tackling the question of whether every $PD_3$-complex has a finite covering space which is homotopy equivalent to a closed orientable 3-manifold.

math.GT↗

arXiv2010

Surgery on $\widetilde{\Bbb{SL}}\times\Bbb{E}^n$-manifolds

We show that although closed $\widetilde{\Bbb{SL}}\times\Bbb{E}^n$-manifolds do not admit metrics of nonpositive sectional curvature, the arguments of Farrell and Jones can be extended to show that such manifolds are topologically rigid, if $n\geq2$.

math.GT↗