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Michael P. Cohen

Publications and source records attributed to Michael P. Cohen.

8 recordsLinked to original sources
arXiv2020

A Kuratowski closure-complement variant whose solution is independent of ZF

We pose the following new variant of the Kuratowski closure-complement problem: How many distinct sets may be obtained by starting with a set $A$ of a Polish space $X$, and applying only closure, complementation, and the $d$ operator, as often as desired, in any order? The set operator $d$ was studied by Kuratowski in his foundational text \textit{Topology: Volume I}; it assigns to $A$ the collection $dA$ of all points of second category for $A$. We show that in ZFC set theory, the answer to this variant problem is $22$. In a distinct system equiconsistent with ZFC, namely ZF+DC+PB, the answer is only $18$.

math.GN↗
arXiv2020

The closure-complement-frontier problem in saturated polytopological spaces

Let $X$ be a space equipped with $n$ topologies $τ_1,...,τ_n$ which are pairwise comparable and saturated, and for each $1\leq i\leq n$ let $k_i$ and $f_i$ be the associated topological closure and frontier operators, respectively. Inspired by the closure-complement theorem of Kuratowski, we prove that the monoid of set operators $\mathcal{KF}_n$ generated by $\{k_i,f_i:1\leq i\leq n\}\cup\{c\}$ (where $c$ denotes the set complement operator) has cardinality no more than $2p(n)$ where $p(n)=\frac{5}{24}n^4+\frac{37}{12}n^3+\frac{79}{24}n^2+\frac{101}{12}n+2$. The bound is sharp in the following sense: for each $n$ there exists a saturated polytopological space $(X,τ_1,...,τ_n)$ and a subset $A\subseteq X$ such that repeated application of the operators $k_i, f_i, c$ to $A$ will yield exactly $2p(n)$ distinct sets. In particular, following the tradition for Kuratowski-type problems, we exhibit an explicit initial set in $\mathbb{R}$, equipped with the usual and Sorgenfrey topologies, which yields $2p(2)=120$ distinct sets under the action of the monoid $\mathcal{KF}_2$.

math.GN↗
arXiv2020

Maximal pseudometrics and distortion of circle diffeomorphisms

We initiate a study of distortion elements in the Polish groups $\mbox{Diff}_+^k(\mathbb{S}^1)$ ($1\leq k<\infty$), as well as $\mbox{Diff}_+^{1+AC}(\mathbb{S}^1)$, in terms of maximal metrics on these groups. We classify distortion in the $k=1$ case: a $C^1$ circle diffeomorphism is $C^1$-undistorted if and only if it has a hyperbolic periodic point. On the other hand, answering a question of Navas, we exhibit analytic circle diffeomorphisms with only non-hyperbolic fixed points which are $C^{1+AC}$-undistorted, and hence $C^k$-undistorted for all $k\geq 2$. In the appendix, we exhibit a maximal metric on $\mbox{Diff}_+^{1+AC}(\mathbb{S}^1)$, and observe that this group is quasi-isometric to a hyperplane of $L^1(I)$.

math.GR↗
arXiv2017

Polishability of some groups of interval and circle diffeomorphisms

Let $M=I$ or $M=\mathbb{S}^1$ and let $k\geq 1$. We exhibit a new infinite class of Polish groups by showing that each group $\mathop{\rm Diff}_+^{k+AC}(M)$, consisting of those $C^k$ diffeomorphisms whose $k$-th derivative is absolutely continuous, admits a natural Polish group topology which refines the subspace topology inherited from $\mathop{\rm Diff}_+^k(M)$. By contrast, the group $\mathop{\rm Diff}_+^{1+BV}(M)$, consisting of $C^1$ diffeomorphisms whose derivative has bounded variation, admits no Polish group topology whatsoever.

math.GR↗
arXiv2017

On the large-scale geometry of diffeomorphism groups of $1$-manifolds

We apply the framework of Rosendal to study the large-scale geometry of the topological groups $\Diff_+^k(M^1)$, consisting of orientation-preserving $C^k$-diffeomorphisms (for $1\leq k\leq\infty$) of a compact $1$-manifold $M^1$ ($=I$ or $\mathbb{S}^1$). We characterize the relative property (OB) in such groups: $A\subseteq\Diff_+^k(M^1)$ has property (OB) relative to $\Diff_+^k(M^1)$ if and only if $\displaystyle\sup_{f\in A}\sup_{x\in M^1}|\log f'(x)|<\infty$ and $\displaystyle\sup_{f\in A}\sup_{x\in M^1}|f^{(j)}(x)|<\infty$ for every integer $2\leq j\leq k$. We deduce that $\Diff_+^k(M^1)$ has the local property (OB), and consequently a well-defined non-trivial quasi-isometry class, if and only if $k<\infty$. We show that the groups $\Diff_+^1(I)$ and $\Diff_+^1(\mathbb{S}^1)$ are quasi-isometric to the infinite-dimensional Banach space $C[0,1]$.

math.GR↗
arXiv2015

Existence and genericity of finite topological generating sets for homeomorphism groups

We show that the topological groups $Diff_{+}^{1}(I)$ and $Diff_{+}^{1}(\mathbb{S}^1)$ of orientation-preserving $C^1$-diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite topological generating sets in related groups. We show that the generic pair of elements in the homeomorphism group $Homeo_+(I)$ generate a dense subgroup of $Homeo_+(I)$. By contrast, if $M$ is any compact connected manifold with boundary other than the interval, we observe that an open dense set of pairs from the associated boundary-fixing homeomorphism group $Homeo(M,\partial M)$ will generate a discrete subgroup. We make similar observations for homeomorphism groups of manifolds without boundary including $\mathbb{S}^1$.

math.GR↗
arXiv2014

$\mathop{\rm PL}_+(I)$ is not a Polish group

The group $\mathop{\rm PL}_+(I)$ of increasing piecewise linear self-homeomorphisms of the interval $I=[0,1]$ may not be assigned a topology in such a way that it becomes a Polish group. The same statement holds for the groups $\mathop{\rm Homeo}_+^{Lip}(I)$ of bi-Lipschitz homeomorphisms of $I$, and $\mathop{\rm Diff}_+^{1+ε}(I)$ of diffeomorphisms of $I$ whose derivatives are Hölder continuous with exponent $ε$.

math.GR↗