Recent zbMATH articles in MSC 55Rhttps://zbmath.org/atom/cc/55R2024-03-13T18:33:02.981707ZWerkzeugOn 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.Iterated \(S^3\) Sasaki joins and Bott orbifoldshttps://zbmath.org/1528.530452024-03-13T18:33:02.981707Z"Boyer, Charles P."https://zbmath.org/authors/?q=ai:boyer.charles-p"Tønnesen-Friedman, Christina W."https://zbmath.org/authors/?q=ai:tonnesen-friedman.christina-wiisSummary: We present a categorical relationship between iterated \(S^3\) Sasaki-joins and Bott orbifolds. Then we show how to construct smooth Sasaki-Einstein (SE) structures on the iterated joins. These become increasingly complicated as dimension grows. We give an explicit nontrivial construction of (infinitely many) smooth SE structures up through dimension eleven, and conjecture the existence of smooth SE structures in all odd dimensions.Abelian cycles in the homology of the Torelli grouphttps://zbmath.org/1528.550112024-03-13T18:33:02.981707Z"Lindell, Erik"https://zbmath.org/authors/?q=ai:lindell.erikLet \(S_{g,r}^{s}\) be a connected, orientable surface, of genus \(g\), \(r\) marked points and \(s\) boundary components. Let \(\Gamma_{g,r}^{s}\) be its mapping class group and let \(H=H_{1}(S_{g},\mathbb{Q})\) and \(H_{\mathbb{Z}}=H_{1}(S_{g},\mathbb{Z})\). The natural action of \(\Gamma_{g,r}^{s}\) on \(S_{g}\) preserves the symplectic form and gives a homomorphism \(\Gamma_{g,r}^{s}\to Sp(H)\). The Torelli group of \(S_{g,r}^{s}\), denoted by \(\mathcal{I}_{g,r}^{s}\), is the kernel of this homomorphism. Since the image of the above homomorphism is \(Sp(H_{\mathbb{Z}})\), the natural action of this group on \(H_{n}(\mathcal{I}_{g,r}^{s})\) transforms this into an \(Sp(H_{\mathbb{Z}})\)-representation for all \(n\geq 0\). On the other hand D. Johnson constructed a homomorphism \(\tau: H_{1}(\mathcal{I}_{g,r}^{s})\to \bigwedge^{3}H_{\mathbb{Z}}\) that naturally extends to homomorphisms, for \(n\geq1\),
\[
\psi_{n}:=(\tau_{J})_{*}: H_{n}(\mathcal{I}_{g,r}^{s})\to \bigwedge^{n}(\bigwedge^{3}H).
\]
The author uses these \(\psi_{n}\) to explore properties of \(H_{n}(\mathcal{I}_{g,r}^{s})\) such as finite dimensionality, or when they are algebraic \(S_{p}(H_{\mathbb{Z}})\)-representations, and considers the injectivity of \(\psi_{n}\).
The first result is that the image \(Im(\psi_{n})\), of \(\psi_{n}\), contains all irreducible subrepresentations of weight \(3n\) for \(n\geq 1\) and \(g\geq 3n\). The second result is that \(Im(\psi_{n})\) contains many more subrepresentation defined by certain partitions in a stable range, that is for \(g\geq n+2k+l\) for certain \(k\) and \(l\). The author also proves that for \(n\geq 2\) and \(g>>0\) one has that \(im(\psi_{n})\) is a proper subgroup of \(H_{n}(\mathcal{I}_{g,r}^{s})\). The main tool is to construct certain \textit{abelian cycles} in \(H_{n}(\mathcal{I}_{g,r}^{s})\) to obtain elements in these groups and detect elements in \(Im(\psi_{n})\).
Reviewer: Daniel Juan Pineda (Michoacán)Extremal stability for configuration spaceshttps://zbmath.org/1528.550122024-03-13T18:33:02.981707Z"Knudsen, Ben"https://zbmath.org/authors/?q=ai:knudsen.ben"Miller, Jeremy"https://zbmath.org/authors/?q=ai:miller.jeremy-a"Tosteson, Philip"https://zbmath.org/authors/?q=ai:tosteson.philipLet \(M\) be a manifold of dimension \(d \geq 2\). Classical work of \textit{D. McDuff} [Topology 14, 91--107 (1975; Zbl 0296.57001)], \textit{G. Segal} [Acta Math. 143, 39--72 (1979; Zbl 0427.55006)], and \textit{T. Church} [Invent. Math. 188, No. 2, 465--504 (2012; Zbl 1244.55012)] establishes rational homological stability for the unordered configuration spaces \(B_n(M)\). If \(d\) is even, then their results can be interpreted as saying that there exists a polynomial in \(n\) of degree at most \(\dim(H_0(M;\mathbb{Q}))-1\), which coincides with the \(i\)-th rational Betti number of \(B_n(M)\) for \(n\) large enough. In particular, this is a pattern in (constant) low homological dimension \(i\). If \(d\) is odd, then work of \textit{C. F. Bödigheimer} et al. [Topology 28, No. 1, 111--123 (1989; Zbl 0689.55012)] gives a closed form expression for the rational homology.
Focussing therefore on the case \(d\) even, the authors prove \textit{extremal stability}, a new phenomenon occurring in (constant) low homological \textit{codimension}. Precisely, letting \(v_n = n(d-1)+|\pi_0(M)|\), they show that for each \(i\) there are two polynomials in \(n\), each of degree at most \(H_{d-1}(M;\mathbb{Q}^{w})-1\), such that for \(n\) large, the dimension of \(H_{v_n-i}(B_n(M;\mathbb{Q}))\) coincides with the first polynomial if \(n\) is even and the second if \(n\) is odd. The example of the surface of genus 2 shows that their degree bound is sharp. The case \(M = \mathbb{C}P^3\) was already known, and is due to \textit{M. Maguire} [``Computing cohomology of configuration spaces'', Preprint, \url{arXiv:1612.06314}]. The only other known example of extremal stability, due to \textit{J. Miller} et al. [Compos. Math. 156, No. 4, 822--861 (2020; Zbl 1458.11085)], is for congruence subgroups of \(SL_n(\mathbb{Z})\).
Reviewer: Guy Boyde (Utrecht)