Recent zbMATH articles in MSC 55R10https://zbmath.org/atom/cc/55R102024-03-13T18:33:02.981707ZWerkzeugOn the coverings of Hantzsche-Wendt manifoldhttps://zbmath.org/1528.200922024-03-13T18:33:02.981707Z"Chelnokov, Grigory"https://zbmath.org/authors/?q=ai:chelnokov.grigory-r"Mednykh, Alexander"https://zbmath.org/authors/?q=ai:mednykh.alexander-dThe authors' paper is a part of a series of papers devoted to the enumeration of finite-sheeted coverings over closed Euclidean 3-manifolds, which are also known as flat 3-dimensional manifolds or Euclidean 3-forms.
In their previous papers, the authors also described isomorphism types of finite index subgroups \(H\) in the fundamental group \(G\) of manifolds \(\mathcal{B}_1\) and \(\mathcal{B}_2\) and calculated the respective numbers \(s_{H,G}(n)\) and \(c_{H,G}(n)\) for each isomorphism type \(H\). Similar questions also were solved by the authors for manifolds \(\mathcal{G}_2\), \(\mathcal{G}_3\), \(\mathcal{G}_4\), \(\mathcal{G}_5\), \(\mathcal{B}_3\) and \(\mathcal{B}_4\) too.
In the reviewed paper, the authors solve the same questions for the Hantzsche-Wendt manifold \(\mathcal{G}_6,\) undoubtedly the most weird among Euclidean \(3\)-manifolds. This is the unique Euclidean \(3\)-form with finite first homology group \(H_1(\mathcal{G}_6) = \mathbb{Z}^2_4.\) The authors also classify finite index subgroups in the fundamental group \(\pi_1(\mathcal{G}_{6})\) up to isomorphism. Given index \(n\), the authors calculate the numbers of subgroups and the numbers of conjugacy classes of subgroups for each isomorphism type and provide the Dirichlet generating series for the above sequences too.
Reviewer: Stepan Moskaliuk (Wien)Signature cocycles on the mapping class group and symplectic groupshttps://zbmath.org/1528.200972024-03-13T18:33:02.981707Z"Benson, Dave"https://zbmath.org/authors/?q=ai:benson.david-john"Campagnolo, Caterina"https://zbmath.org/authors/?q=ai:campagnolo.caterina"Ranicki, Andrew"https://zbmath.org/authors/?q=ai:ranicki.andrew-a"Rovi, Carmen"https://zbmath.org/authors/?q=ai:rovi.carmenSummary: \textit{W. Meyer} [Math. Ann. 201, 239--264 (1973; Zbl 0241.55019)] constructed a cocycle in \(H^2(\mathsf{Sp}(2g, \mathbb{Z}); \mathbb{Z})\) which computes the signature of a closed oriented surface bundle over a surface. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer's decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall-Maslov index. The main theorem of the paper, Theorem 6.6, provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer \(N\). Using these results, we are able to give a complete answer for \(N = 2, 4, \text{ and } 8\), and based on a theorem of \textit{P. Deligne} [C. R. Acad. Sci., Paris, Sér. A 287, 203--208 (1978; Zbl 0416.20042)], we show that this is the best we can hope for using this method.