Recent zbMATH articles in MSC 55Qhttps://zbmath.org/atom/cc/55Q2024-03-13T18:33:02.981707ZWerkzeugDenominators of special values of \(\zeta\)-functions count \(KU\)-local homotopy groups of mod \(p\) Moore spectrahttps://zbmath.org/1528.110772024-03-13T18:33:02.981707Z"Salch, Andrew"https://zbmath.org/authors/?q=ai:salch.andrewSummary: For each odd prime \(p\), we show that the orders of the \(KU\)-local homotopy groups of the \(\operatorname{mod}\,p\) Moore spectrum are equal to denominators of special values of certain quotients of Dedekind zeta-functions of totally real number fields. With this observation in hand, we give a cute topological proof of the Leopoldt conjecture for those number fields, by showing that it is a consequence of periodicity properties of \(KU\)-local stable homotopy groups.Bad representations and homotopy of character varietieshttps://zbmath.org/1528.140062024-03-13T18:33:02.981707Z"Guérin, Clément"https://zbmath.org/authors/?q=ai:guerin.clement"Lawton, Sean"https://zbmath.org/authors/?q=ai:lawton.sean"Ramras, Daniel"https://zbmath.org/authors/?q=ai:ramras.daniel-aGiven a connected reductive complex affine algebraic group \(G\) and a finitely generated group \(\Gamma\), the \(G\)-character variety of \(\Gamma\) is defined as the GIT quotient
\[
\mathcal{X}_{\Gamma}(G)=\Hom(\Gamma,G) /\!\!/ G,
\]
where \(G\) acts by conjugation. Character varieties play an important role in several areas, such as Representation Theory, Nonabelian Hodge Theory or Geometric Topology. In this paper, the authors focus on the case where \(\Gamma=F_r\), the free group of rank r, and compute the higher homotopy groups \(\pi_k(\mathcal{X}_{F_r}(G))\), for \(0\leq k \leq 4\), extending previous results [\textit{C. Florentino} et al., Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 1, 143--185 (2017; Zbl 1403.14011)]. They also prove that
\[
\pi_k(\mathcal{X}_{F_r}(G))\cong \pi_k(G)^r \times \pi_{k-1}(PG),
\]
in a certain range, where \(PG\) is the quotient of \(G\) by its center \(Z(G)\), for both classical \(\left( \text{type } A_n,B_n,C_n,D_n \right)\) and exceptional groups \(G\) \(\left(G_2,F_4, E_6,E_7,E_8 \right)\).
Results are based on a detailed analysis of the singular locus of the character variety, which leads to the study of bad and ugly representations, which are irreducible representations whose \(G\)-stabilizer is larger than \(Z(G)\) and representations that produce topological singularities of \(\mathcal{X}_{F_r}(G)\), respectively. Proofs rely on previous results of Richardson on the algebraic singularities of \(G^r\) [\textit{R. W. Richardson}, Duke Math. J. 57, No. 1, 1--35 (1988; Zbl 0685.20035)] and codimension bounds of the singular locus, built upon results of \textit{C. Guérin} [J. Group Theory 21, No. 5, 789--816 (2018; Zbl 1437.20027); Geom. Dedicata 195, 23--55 (2018; Zbl 1418.20005)]. Their analysis leads to the study of Borel-de Siebenthal subalgebras of the Lie algebra \(\mathfrak{g}\) of \(G\), which are also classified in the article and used to describe bad representations.
Reviewer: Javier Martínez-Martínez (Madrid)Dominant energy condition and spinors on Lorentzian manifoldshttps://zbmath.org/1528.530632024-03-13T18:33:02.981707Z"Ammann, Bernd"https://zbmath.org/authors/?q=ai:ammann.bernd-eberhard"Glöckle, Jonathan"https://zbmath.org/authors/?q=ai:glockle.jonathanThis article is devoted to the study of the topology of initial data sets \(\mathcal{I}^{\ge}(M)\), in particular their homotopy groups, satisfying the dominant energy condition. The authors use index theory and a Lorentzian version of the index difference in order to detect homotopy groups of \(\mathcal{I}^{\ge}(M)\), and construct non-trivial elements in \(\pi_{k}(\mathcal{I}^{\ge}(M))\) combining known methods for constructing non-trivial homotopy groups in the space of positive scalar curvature metrics together with a given suspension mechanism. The index theory used by the authors revolves around the Dirac-Witten operator, introduced by \textit{E. Witten} [Commun. Math. Phys. 80, 381--402 (1981; Zbl 1051.83532)] and formalized by \textit{T. Parker} and \textit{C. H. Taubes} [Commun. Math. Phys. 84, 223--238 (1982; Zbl 0528.58040)] in their proof of the positive mass theorem in general relativity, which is obtained from the standard Dirac operator via a modification depending on a symmetric two-form. For initial data satisfying the strict dominant energy condition, the Dirac-Witten operator is an invertible, self-adjoint, Fredholm operator. This allowed the second author to study the topology of initial data sets satisfying the strict dominant energy condition in a prior publication via a Lorentzian analog of the \(\alpha\)-invariant introduced by \textit{N. J. Hitchin} [Adv. Math. 14, 1--55 (1974; Zbl 0284.58016)]. In this regard, the present article focuses on the differences and difficulties that arise when the strictness in the dominant energy condition is relaxed. In this case, the Diract-Witten operator may be non-invertible and the authors study its kernel, proving that spinors in this kernel define solutions to the initial data problem determined by a parallel spinor. Given this connection, the authors characterize compact manifolds admitting such initial data, proving that their fundamental group is virtually solvable of derived length at most two.
For the entire collection see [Zbl 1517.53004].
Reviewer: Carlos Shabazi (Hamburg)Stable homotopy groups of spheres: from dimension 0 to 90https://zbmath.org/1528.550102024-03-13T18:33:02.981707Z"Isaksen, Daniel C."https://zbmath.org/authors/?q=ai:isaksen.daniel-c"Wang, Guozhen"https://zbmath.org/authors/?q=ai:wang.guozhen"Xu, Zhouli"https://zbmath.org/authors/?q=ai:xu.zhouliThis paper provides \(2\)-primary information on the stable homotopy groups of spheres up to dimension \(90\), with only a few undetermined cases. Whilst this continues on from the first author's monograph [\textit{D. C. Isaksen}, Stable stems. Providence, RI: American Mathematical Society (AMS) (2019; Zbl 1454.55001)], the authors exploit crucial new ingredients here. This is an impressive body of work, representing a snapshot of substantial progress achieved using modern methods.
A key ingredient is the interplay between classical and \(\mathbb{C}\)-motivic (cellular) homotopy theory. Recall that motivic homotopy theory provides a second grading (the weight) and one has the motivic homotopy class \(\tau\) (working \(2\)-completely); inverting \(\tau\) allows passage to classical homotopy theory. (As the authors note, the usage of \(\mathbb{C}\)-motivic cellular stable homotopy theory can now be circumvented by using synthetic homotopy theory [\textit{P. Pstragowski}, Invent. Math. 232, No. 2, 553--681 (2023; Zbl 07676257)], but essentially applying the same approach to the calculations.)
Here, these methods are made much more powerful and precise by using the results of [\textit{B. Gheorghe} et al., Acta Math. 226, No. 2, 319--407 (2021; Zbl 1478.55006)] which show that \(\mathbb{C}\)-motivic cellular stable homotopy theory is a `deformation' of classical stable homotopy theory: the `generic fibre' is classical stable homotopy theory whereas the `special fibre' is algebraic, identifying with the stable derived category of \(BP_*BP\)-comodules [\textit{M. Hovey}, Contemp. Math. 346, 261--304 (2004; Zbl 1067.18012)]. This is intimately related to the role of the cofibre \(C \tau\) of \(\tau\) in the \(\mathbb{C}\)-motivic stable homotopy category. In particular, the \(\mathbb{C}\)-motivic Adams spectral sequence for \(C \tau\) is isomorphic to the algebraic Novikov spectral sequence that computes the \(E_2\)-page of the \(BP\) Adams-Novikov spectral sequence. The work of the second author [\textit{G. Wang}, Chin. Ann. Math., Ser. B 42, No. 4, 551--560 (2021; Zbl 1471.55001)] has made the latter accessible to machine computation.
A further new ingredient is the usage of the motivic modular forms spectrum \(mmf\). The \(\mathbb{C}\)-motivic Adams spectral sequence of \(mmf\) is understood [\textit{D. C. Isaksen}, Homology Homotopy Appl. 11, No. 2, 251--274 (2009; Zbl 1193.55009); \textit{B. Gheorghe} et al., J. Eur. Math. Soc. (JEMS) 24, No. 10, 3597--3628 (2022; Zbl 1498.14050)]; via the unit map this provides information for the motivic sphere spectrum.
With these tools in hand, the authors proceed as follows. They first compute by machine the cohomology of the \(\mathbb{C}\)-motivic Steenrod algebra and the algebraic Novikov spectral sequence. Using the cell structure of \(C\tau\), Adams differentials for \(C\tau\) can be pushed forward or pulled back to yield Adams differentials for the motivic sphere spectrum.
Further information on Adams differentials is given by standard arguments (such as Toda bracket shuffles) and comparison with the case of \(mmf\). Thereafter, hidden \(\tau\)-extensions are treated.
Finally, inverting \(\tau\) yields the required classical information.
For some background material, the reader is referred to the first author's monograph; some details are provided on the finer points of Adams spectral sequence calculations, such as hidden extensions, Massey products, Toda brackets and their relations.
Of necessity, the paper is not self-contained: the charts and data sets (available on Zenobo, as per references) are essential companions. Indeed, one of the main results is stated as follows: \textit{The classical Adams spectral sequence for the sphere spectrum is displayed in the charts in [\textit{D. C. Isaksen} et al., ``Classical and \(\mathbf{C} \)-motivic Adams charts'', Zenobo (2022; \url{doi:10.5281/zenodo.6987156})], up to the \(90\)-stem}.
Most of the results covered in the text are provided in tabular form in the final section of the paper (containing 25 tables over 31 pages). These are mostly for the motivic setting, using the trigrading \((s,f,w)\), where \(s\) is the stem, \(f\) is the Adams filtration, and \(w\) is the motivic weight. The body of the paper serves to explain how these computations are obtained and to offer a guide to the techniques used. The text is not necessarily intended for linear reading; the user will truly appreciate its value when delving deep to understand particular calculations and phenomena.
The authors document carefully the remaining unresolved uncertainties that start in dimension \(84\). There are some differentials that are undetermined and also some unresolved hidden \(2\)-extensions. This means that the additive structure of a few stable homotopy groups in this range is not determined here.
Reviewer: Geoffrey Powell (Angers)