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Andrzej Szczepański

Publications and source records attributed to Andrzej Szczepański.

13 recordsLinked to original sources
arXiv2024

Automorphism groups of combinatorial Hantzsche-Wendt groups

Combinatorial Hantzsche-Wendt groups were introduced by W. Craig and P.A. Linnell. Every such a group $G_n$, where $n$ is a natural number, encodes the holonomy action of any $n+1$-dimensional Hantzsche-Wendt manifold. $G_2$ is the fundamental group of the classical Hantzsche-Wendt manifold -- the only one $3$-dimensional oriented flat manifold with non-cyclic holonomy group. In this article, we describe the structure of the automorphism and of the outer automorphism groups of combinatorial Hantzsche-Wendt groups.

math.GR↗
arXiv2023

Minimal non-solvable Bieberbach groups

It has been shown by several authors that there exists a non-solvable Bieberbach group of dimension $15$. In this note we show that this is in fact a minimal dimension for such kind of groups.

math.GR↗
arXiv2021

Spin$^c$ structures on Hantzsche-Wendt manifolds

Using a combinatorial description of Stiefel-Whitney classes of closed flat manifolds with diagonal holonomy representation, we show that no Hantzsche-Wendt manifold of dimension greater than three does not admit a spin$^c$ structure.

math.AT↗
arXiv2020

Flat manifolds with holonomy representation of quaternionic type

We are interested in the question of the existence of flat manifolds for which all $\mathbb R$-irreducible components of the holonomy representation are either absolutely irreducible, of complex or of quaternionic type. In the first two cases such examples are well known. But the existence of the third type of flat manifolds was unknown to the authors. In this article we construct such an example. Moreover, we present a list of finite groups for which a construction of manifolds of quaternionic type is impossible.

math.GR↗
arXiv2017

A short note about diffuse Bieberbach groups

We consider low dimensional diffuse Bieberbach groups. In particular we classify diffuse Bieberbach groups up to dimension 6. We also answer a question from [S. Kionke, J. Raimbault, On geometric aspects of diffuse groups, Doc. Math. 21 (2016), page 887] about minimal dimension of a non-diffuse Bieberbach group which does not contain three-dimensional Hantzsche-Wendt group.

math.GR↗
arXiv2017

Classification of spin structures on 4-dimensional almost-flat manifolds

Almost-flat manifolds were defined by Gromov as a natural generalisation of flat manifolds and as such share many of their properties. Similarly to flat manifolds, it turns out that the existence of a spin structure on an almost-flat manifold is determined by the canonical orthogonal representation of its fundamental group. Utilising this, we classify the spin structures on all four-dimensional almost-flat manifolds that are not flat. Out of 127 orientable families, there are exactly 15 that are non-spin, the rest are in fact parallelizable.

math.AT↗
arXiv2016

Spin structures of flat manifolds of diagonal type

For each integer $d$ at least two, we construct non-spin closed oriented flat manifolds with holonomy group $\mathbb Z_2^d$ and with the property that all of their finite proper covers have a spin structure. Moreover, all such covers have trivial Stiefel-Whitney classes.

math.AT↗
arXiv2013

Spin structures on almost-flat manifolds

We give a necessary and suffcient condition for almost-flat manifolds with cyclic holonomy to admit a Spin structure. Using this condition we find all 4-dimensional orientable almost- flat manifolds with cyclic holonomy that do not admit a Spin structure.

math.GT↗
arXiv2013

Holonomy groups of flat manifolds with $R_\infty$ property

Let $M$ be a flat manifold. We say that $M$ has $R_\infty$ property if the Reidemeister number $R(f) = \infty$ for every homeomorphism $f \colon M \to M.$ In this paper, we investigate a relation between the holonomy representation $ρ$ of a flat manifold $M$ and the $R_\infty$ property. In case when the holonomy group of $M$ is solvable we show that, if $ρ$ has a unique $\mathbb{R}$-irreducible subrepresentation of odd degree, then $M$ has $R_\infty$ property. The result is related to conjecture 4.8 from [1]. [1] K. Dekimpe, B. De Rock, P. Penninckx, \emph{The $R_{\infty}$ property for infra-nilmanifolds}, Topol. Methods Nonlinear Anal. 34 (2009), no.2, 353 - 373

math.RT↗
arXiv2013

Tangent bundles of Hantzsche-Wendt manifolds

We formulate a condition for an existence of a $Spin^C$ - structure on an oriented at manifold $M^n$ with $H^2(Mn;R) = 0$. As an application we shall prove that all cyclic Hantzsche - Wendt manifolds have not the $Spin^C$-structure.

math.AT↗
arXiv2012

Flat manifolds with holonomy group Z_2^k of diagonal type

We consider relations between two families of flat manifolds with holonomy group (Z_2)^k of diagonal type. The family ${\cal RBM}$ of real Bott manifolds and the family ${\cal GHW}$ of generalized Hantzsche-Wendt manifolds. In particular, we prove that the intersection ${\cal GHW}\cap {\cal RBM}$ is not empty. We also consider some class of real Bott manifolds without $\operatorname{Spin}$ and $\operatorname{Spin}^{\C}$ structure. There are given conditions for the (non)existence of such structures.

math.AT↗