Papers

We explicitly determine the first Galois obstruction in the cokernel of the Johnson homomorphism of the mapping class group of a surface of genus $g$, for every genus $g \ge 2$. It is described as a sum of two terms which are considerably different in character. One lies in the kernel of the Enomoto-Satoh trace map, whereas the other belongs to a certain ideal which vanishes upon passage to the closed surface case.

We prove that, under certain homological conditions, the attaching map of the top cell of a Poincaré duality complex is inert when localised away from a finite set of primes. This improves on a result of Félix and Tanré in these cases. As an additional application of the methods, we give a loop space decomposition of simply-connected $6$-dimensional Poincaré duality complexes satisfying certain hypotheses. We also show that, under the hypotheses of the inertness theorem, the $(n-1)$-skeleton of an $n$-dimensional Poincaré duality complex satisfies the hyperbolic form of Moore's Conjecture after localising away from an explicit finite set of primes, and use this to obtain new examples of \(p\)-local maps between spheres that are not inert.

We give a systematic account of the shape theory of $\infty$-topoi, viewing the shape of an $\infty$-topos as its generalized homotopy type. We establish the basic functorial properties of the shape, including preservation of colimits, descent, and homotopy invariance. We then prove that shape preserves cofiltered limits under compactness and perfectness hypotheses and establish Künneth-type formulas for products. Finally, we give conditions under which the shape of an $\infty$-topos determines its cohomology and homology.

Polygon equations form a family of equations generalizing the pentagon equation. In this paper, we construct semi-simplicial sets of permitted colorings associated with set-theoretic solutions of polygon equations and use them to define the corresponding (co)homology groups. We investigate several properties of these groups and establish an equivalence of categories between set-theoretic solutions of polygon equations and higher Segal semi-simplicial sets satisfying certain conditions. As a special case, our result recovers the correspondence between bijective set-theoretic solutions of the pentagon equation and $2$-Segal semi-simplicial sets proved by Dyckerhoff–Kapranov.

We prove that any two smooth toric Fano fourfolds whose integral cohomology rings are isomorphic as graded rings are diffeomorphic.

The first singular homology of a Peano continuum $X$ with torsion-free first Cech homology, $\check H_1(x)$, splits as $H_1(X) = \check H_1(X) \oplus K$ where $K$ is the homology shape kernel of $X$. Consequently if a Peano continuum $X$ is a subspace of $\mathbb R^3$, then $H_1(X) = \mathbb Z^\lambda \oplus K$ where $K$ is the homology shape kernel of $X$ and $\lambda$ is a countable cardinal. In the process we construct cotorsion quotients of subgroups of the first homology which correspond to path-connected fibrations of $X$.

We construct a six-functor formalism for syntomic cohomology of p-adic formal schemes. This, in particular, generalizes Poincaré duality to general smooth morphisms.

We study reductions of the tangent bundle of $S^{\dim G}$ through the adjoint representation of a compact connected Lie group $G$. If $d=\dim G$ and $r=\operatorname{rank} G$, we show that the adjoint map $\operatorname{Ad}\colon G\longrightarrow \mathrm{SO}(d)$ is homotopic, as an ordinary map, to one with values in $\mathrm{SO}(d-r+1)$. It follows that every vector bundle over a sphere associated to a principal $G$-bundle via the adjoint representation admits $r-1$ linearly independent sections. Combined with Adams's theorem on vector fields on spheres, this gives the necessary condition $r\le \rho(d+1)$ for an adjoint reduction of $TS^d$. In particular, no such reduction exists for a non-trivial compact connected group, simple or not, of dimension $d>1$ with $d\not\equiv3\pmod4$. As an application, this excludes the adjoint $E_7$-reduction of $TS^{133}$ that is the exceptional branch in the structure-group argument of Bor, Hernández-Lamoneda, Jiménez-Desantiago and Montejano for Banach's isometric subspace problem.

We show that for a finite group $G$, the $G$-category $\underline{\mathrm{Sp}}_{G,\mathbb{Q}}$ of rational genuine $G$-spectra is parametrized semiadditive, in the sense of Cnossen–Lenz–Linskens, for morphisms of genuine $G$-spaces whose fixed point maps have $\pi$-finite fibers. In particular, this means that the canonical norm map between a colimit and a limit of rational genuine $G$-spectra indexed by such $G$-spaces is an equivalence. This extends the Wirthmüller isomorphism from $G$-orbits to a broader class of $G$-spaces and combines the theory of ambidexterity of Hopkins and Lurie with parametrized category theory.

In the late 1980s, Baulieu and Grossman demonstrated that the supersymmetric formulation of Langevin stochastic differential equations (SDEs), proposed earlier by Parisi and Sourlas, belongs to the family of Witten-type topological field theories (TFTs). From a certain angle, this finding may appear puzzling: TFTs have no local degrees of freedom, whereas SDEs do exhibit local fluctuations. In this paper, we address this apparent contradiction in the context of the supersymmetric theory of stochastic dynamics (STS), a generalization of the Parisi-Sourlas-Baulieu-Grossman approach to SDEs of arbitrary form. We further discuss how, within STS, dynamical chaos can be identified with the spontaneous breakdown of the corresponding topological supersymmetry, while 1/f noise can be understood as a consequence of the Goldstone theorem. The butterfly effect, in turn, calls for an effective field-theoretic description that may itself possess a hidden topological structure, as we speculate on the basis of the Ginzburg-Landau approach and insights from AdS/CFT duality.

An admissible configuration $\Lambda\subset\Hh^m\simeq\Rr^{4m}$ defines a compact intersection of quadrics $Z_{\Hh}(\Lambda)$ and a free diagonal $\Sp(1)$ quotient $N_{\Hh}(\Lambda)$. We realize this quotient as the leaf space of a real dilation action on an open subset of quaternionic projective space. The total space is a quaternionic moment-angle polyhedral product. Every quotient is $2$-connected; when there are no indispensable coordinates, it is $3$-connected and its fourth integral cohomology is generated by the Euler class $e$ of the associated oriented rank-four bundle. We compute $p_1(TN_{\Hh})=2(n-2)e$ and prove that no LVMQ manifold admits a quaternionic Kähler metric. An explicit infinite family consists of products of spheres admitting both homogeneous positive Einstein metrics and quaternionic toric structures in the sense of Gentili–Gori–Sarfatti. We also describe the trace foliations of the real action in the Poincaré domain. For the separate rank-one distribution $X\Hh$, an invertible quaternionic-linear field is integrable only in the real scalar case. A nonlinear field with positive real scalar derivative at the origin and involutive $X\Hh$ has exactly the standard quaternionic Hopf foliation on every sufficiently small centered sphere.

CBCT (Cone Beam Computed Tomography) scans provide detailed three-dimensional images, widely used in dentistry for diagnostic and treatment planning tasks. While invaluable, analyzing and documenting these scans is labor-intensive, prompting efforts to automate key steps like the classification and segmentation of anatomical structures to identify tooth types and associated pathologies. In this article, we propose an approach to automation that leverages persistent homology, a framework from topological data analysis that studies the shape of data by identifying features like connected components, holes, and voids across multiple scales. Persistent homology, together with a support vector machine, allows us to classify teeth in a CBCT scan and to perform diagnostics. Our method advances the state of the art, reaching average accuracy scores of 97.67% for tooth-labeling and 96.77% for diagnostic tasks, outperforming a CNN trained on the same data with accuracy of 70.27% and 86.67%, respectively.

We develop a general framework for constructing pseudometrics on objects of Abelian categories using valuations, which assign non-negative costs to morphisms. Specializing to finitely presented vector space representations of the poset $[0,\infty)^\ell$, a natural setting for multiparameter persistence modules, we show that a large class of these distances is governed by contours, right-continuous lax actions of $[0,\infty)$ on the parameter poset. Properties of contours translate into geometric properties of the induced distances, including conditions under which the resulting pseudometrics are metrics and allowing us to identify a rich family of compact subsets of finitely presented representations.

Since the 1980s, the Bogomolov multiplier of a finite group has been known to obstruct rationality in complex algebraic geometry, and more recently it is understood to be responsible for any torsion in the oriented and stable unitary 2-dimensional $G$-equivariant bordism groups $\Omega_2^{SO,G}$ and $\Omega_2^{U,G}$. In this note, as a small step toward building a bridge between these two far-flung roles, we discuss the vanishing of Bogomolov multipliers of two specific families of finite groups. First, we revisit Kunyavskiĭ's result that the Bogomolov multipliers of all finite simple groups vanish, taking inspiration from the low-dimensional topological interpretation of the Ore conjecture. Second, in lieu of arguments in complex birational geometry (such as the hard direction of the Chevalley-Shephard-Todd theorem), we combine cut-and-paste combinatorial-topological techniques with elementary calculations of Ihara-Yokonuma to show that all finite Coxeter groups have vanishing Bogomolov multiplier.

In this article, the cyclotomic Galois action on topological K-theory adjoined with a primitive $n$th root of unity and the famous $GL_1(\mathbf{Z}/n)$- and $GL_2(\mathbf{Z}/n)$-Galois actions on topological modular forms with $\Gamma_1(n)$- and $\Gamma(n)$-level structures are unified and generalised. This is done by defining a quotient stack in derived algebraic geometry parametrising constant finite abelian subgroups of $\mathbf{P}$-divisible groups, and studying how these quotient stacks and various notions of torsors interact with Galois extensions formed by taking algebraic and categorical invariants. Taking global sections then yields the titular Galois extensions on $K$-geometric fixed points, recovering and refining the well-known examples above and providing new ones. As an application, the $\infty$-category of perfect modules over a variety of $H$-equivariant ring spectra $R$ are decomposed into simple pullbacks of nonequivariant categories, leading to Mayer–Vietoris sequences for localising invariants of $R$. For example, this occurs for equivariant topological K-theory for all $p$-groups as well as any finite nonabelian simple group of order less than 500.

For a finite connected simplicial complex $X$, the strand map $\iota_\ast$, from $\mathbb{P}_n(X)$ to $\prod_{i=1}^n\pi_1(X,x_i^0)$, sends a pure braid to the homotopy classes of its strands. A theorem of Goldberg (1973) computes its kernel when $X$ is a closed surface other than $S^2$ and $\mathbb{RP}^2$: the kernel is the normal closure of the pure braids supported in an embedded disc. We extend this picture to arbitrary finite connected simplicial complexes. Call $X$ $\textit{weakly Goldberg}$ if some contractible subcomplex $X_0\subseteq X$ realises Goldberg's description, $\ker\iota_\ast=\left\langle \operatorname{im}(\mathbb{P}_n(X_0)\to\mathbb{P}_n(X)) \right\rangle$, and $\textit{Goldberg}$ if $X_0$ can moreover be chosen so that $\mathbb{P}_n(X_0)\to\mathbb{P}_n(X)$ is injective. We prove that the strand map is surjective if and only if $X\not\cong S^1$; that $X$ is weakly Goldberg if and only if its free part is a forest; and that $X$ is Goldberg if and only if it admits an $\textit{admissible tree}$ – a maximal tree of a scaffold, compatible with the boundary and interior types of the attachments of the free part to the thick components. We also classify the complexes for which the kernel is trivial, settle the exceptional surfaces $S^2$ and $\mathbb{RP}^2$, and obtain complete answers for manifolds and for graphs. The main tools are a graph-of-spaces decomposition of the configuration space at a point of $X$ and a resolution procedure reducing an arbitrary complex to a simple model.

Computations in simplicial homology are crucial for algebraic topology and computational geometry in general. Although several mature software systems provide facilities for working with simplicial complexes, there is currently limited support for such computations within the Wolfram language ecosystem. This paper presents SimplicialHomology, an open-source Wolfram language paclet for constructing abstract simplicial complexes and computing their homology groups over the integers, rational numbers, and finite fields. Furthermore, the package supports many other functionalities such as calculating Betti numbers, automorphism groups, and other constructions such as joins, stars, links, etc. Several examples are presented to illustrate the package and its performance.

The relative sectional category, introduced by González, Grant, and Vandembroucq for fibrations and later extended by García-Calcines to arbitrary maps, provides a common framework encompassing several numerical homotopy invariants, including the Lusternik–Schnirelmann category, the topological complexity of a map, and homotopic distance. In this paper, we introduce and study the equivariant analogue of the relative sectional category for $G$-maps. We establish its fundamental homotopy-theoretic properties, including comparison, product, and composition inequalities, as well as its behavior under changes of domain and codomain. As applications, we introduce and investigate equivariant analogues of the topological complexity of a map, in the sense of Scott and Murillo–Wu, and the equivariant Lusternik–Schnirelmann category of a map. Several examples are provided to illustrate the theory and demonstrate that these invariants extend the corresponding classical equivariant notions.

We introduce synthetic buildings for each finite group $G$ and prime $p$. These are $G$-simplicial complexes, among which are the $p$-subgroups complex of K.S. Brown, and also the buildings of finite groups of Lie type in characteristic $p$. Synthetic buildings are useful because of special properties, including providing a formula for the cohomology of $G$. Our goal is to describe the kinds of synthetic buildings that can arise, and ideally to classify them. The minimal synthetic buildings play a special role and for most groups there are not very many of these. However, we show that it is possible to find sequences of finite groups with increasingly many minimal synthetic buildings as we move along the sequence, and also groups with minimal synthetic buildings of different dimensions. It is also possible to find groups with infinitely many non-minimal synthetic buildings of a given dimension. In other situations, we show that there are no minimal synthetic buildings of equivariant homotopy type distinct from those of Brown's complex and the complex consisting of a single point. We make a number of conjectures.

We introduce reparametrization categories with cuts. For every such category $\mathcal{P}$, the tensor product of two objectwise weak homotopy equivalences of $\mathcal{P}$-spaces is an objectwise weak homotopy equivalence, and the q-model structure on $\mathcal{P}$-flows is left proper. The reparametrization categories $\mathcal{G}$ and $\mathcal{M}$, as well as the terminal category $\mathbf{1}$, have cuts. The terminal category recovers the left properness of the q-model structure on ordinary flows. We also give a symmetric reparametrization category with cuts and a strict unit. This category is not equivalent to $\mathbf{1}$.

We prove that the q-model category of $\mathcal P$-multipointed d-spaces for $\mathcal P\in\{\mathcal G,\mathcal M\}$ is not left proper by constructing a weak equivalence whose pushout along a q-cofibration obtained by attaching a single 1-dimensional globular cell is not a weak equivalence. The counterexample is constructed in the category of $\Delta$-Hausdorff $\Delta$-generated spaces. For $\mathcal P=\mathcal G$, all objects are automatically saturated, and for $\mathcal P=\mathcal M$, the original weak equivalence is between saturated objects. The failure is caused by a family of nonconstant execution paths that converges in the ambient mapping space to a constant map which is not an execution path; after the globular cell is attached, this degeneration causes two previously distinct path components to merge.

In this note, we show the category of $\Gamma$-sets with the sphere spectrum, the smash product, and the substitution product gives rise to an isomix linearly distributive category, which is a categorical semantics of the multiplicative fragment of linear logic. We also show Lydakis' assembly map between these two monoidal products corresponds to the mixor.

We establish lower bounds for the Smith–Thom deficiency of the Hilbert square of a maximal nonsingular real projective variety. These bounds recover the known surface case and provide new obstructions to Smith-Thom maximality in every dimension at least two. As applications, we prove that the Hilbert square of any real abelian variety of dimension at least two is not Smith-Thom maximal. We further show that the deficiencies of the Hilbert squares of maximal real abelian varieties grow exponentially with the dimension. We also obtain nonmaximality results for the Hilbert squares of certain Cartesian products.

The distance matrix of a finite point cloud can be visualized as a heatmap. When the data arise from a time series, the sublevel sets of this image are known as recurrence plots and are widely used in time series analysis. Motivated by this perspective, we establish a relationship between the distance-matrix filtration of the time series and the Čech (or Vietoris–Rips) filtration of its state-space embedding in the form of a degree-one chain map. We study the induced maps in homology, showing that the map from $H_0$ into $H_1$ is essentially surjective and providing an example where the map from $H_1$ into $H_2$ is nontrivial. These chain maps can be applied to simplify image persistence computations arising in the computation of cycling signatures, a topological tool for time series analysis. Moreover, these computations yield finer information that allows the analysis of transitions between different types of cycling motion.

Tornehave and Yalçin proved that the tom Dieck homomorphism, which sends a virtual real representation to a unit of the Burnside ring, is surjective for any $p$-group $S$. We prove that this homomorphism, and the sign homomorphism it factors through, remain surjective when restricted to fusion-stable subgroups associated to a saturated fusion system on $S$. As a corollary, we close the main question posed by Mazza–Miller in arXiv:2508.07404 by showing that given a field $k$ of positive characteristic, the Lefschetz homomorphism from the Picard group of the bounded homotopy category of $p$-permutation modules to the unit group of its Grothendieck ring is surjective for all finite groups if and only if $k = \mathbb{F}_2$.

We study the extended persistent homology of the radial filtration of a shape $M\subseteq \mathbb{R}^n$. A radial filtration is formed by choosing a center point $c$ and taking points of $M$ within distance $r$ of $c$. We show that, under mild assumptions, we can recover the radial extended persistence of a manifold with boundary from the radial persistence of its boundary. We also establish algorithms to compute radial extended persistence when $M$ is a set of pixels in a 2D digital grid. The methods are similar to those used to compute the extended persistent homology of a height filtration of a shape embedded in Euclidean space. We envisage these results will be useful in biomedical image analysis settings where it is natural to consider a radial filtration with respect to a fixed center, for example in the study of neuronal structures.

We introduce $C$-lifting functors, which axiomatize lifting properties of non-abelian tensor and exterior products with respect to prescribed classes of group extensions. For a homomorphism $f\colon\Gamma\to G$, we associate to every $C$-lifting functor $F$ a relative quotient $F_f(G)$. This quotient maps epimorphically onto the subgroup determined by $F$ in every $f$-extension belonging to $C$. We show that the construction is functorial in $f$ and that, for an $f$-extension $p\colon\widetilde G\to G$, it gives a morphism of natural exact sequences. In degree two this yields an epimorphism $ \frac{H_2(G;\mathbb Z)}{f_*H_2(\Gamma;\mathbb Z)} \longrightarrow \ker p\cap[\widetilde G,\widetilde G]. $ Applying the construction to iterated tensor and exterior powers, we obtain relative Schur-Baer theorems for the lower central and derived series. We also compare the relative tensor and exterior squares, relate the exterior construction to the relative Schur multiplier $H_2(G,\Gamma;\mathbb Z)$, and derive applications to orderability. As a consequence, we prove that a virtual knot group is left-orderable if and only if it is circularly orderable.

The dual Stiefel-Whitney classes wbar_j(M) of a manifold M are elements of H^j(M;Z/2) which give information about embedding and immersing M in Euclidean space. We consider the question of finding, for each n, the largest j such that there is an n-dimensional Spin manifold with wbar_j nonzero. We obtain upper and lower bounds. Since one theorem proves existence of manifolds without giving explicit examples, we consider a parallel question of finding explicit manifolds with wbar_j nonzero for j as large as possible.

We introduce Probabilistic Ball Mapper, a formulation of Ball Mapper in which each data point is assigned a probability distribution supported only on the metric balls that contain it. This assignment defines both a partition subordinate to the Ball Mapper cover and a Markov kernel from the finite data space to the cover. We study two assignment schemes: a uniform-on-support rule and a localized radial-basis rule that incorporates distance to landmarks while preserving the underlying cover. Pushing the empirical data distribution through the kernel produces a probability distribution over vertices. Drawing twice, conditionally and independently, from each pointwise distribution produces a soft-overlap matrix. This matrix is symmetric, nonnegative, positive semidefinite, and has the vertex distribution as both marginals. It therefore provides a mass-normalized refinement of classical Ball Mapper overlap rather than another unnormalized edge count. For graphs constructed on a common cover, the vertex and overlap distributions can be compared directly. For independently fitted covers, we formulate Wasserstein and fused Gromov–Wasserstein-type discrepancies that account for vertex mass, landmark geometry when a common ambient metric is available, and intrinsic graph relations. For a fixed cover, we derive explicit perturbation bounds controlled by the sensitivity of the assignment rule, the magnitude of the data perturbation, and the data mass near cover boundaries. When the cover is recomputed, landmark motion creates an additional source of variation, for which we state a transport-based stability principle rather than an unconditional theorem. The resulting framework turns Ball Mapper into a probability-valued representation suitable for quantitative comparison while retaining its geometric interpretability and computational simplicity.

One of the most effective approaches to computations in algebraic $K$-theory is trace methods, which compare algebraic $K$-theory with topological Hochschild homology and topological cyclic homology. In recent work, two of the authors, together with Gerhardt, construct an equivariant refinement of topological Hochschild homology ($\mathrm{ETHH}$) which receives a trace map from Merling's genuine equivariant algebraic $K$-theory. In this paper, we perform foundational computations of $\mathrm{ETHH}$ that can serve as input for future computations of $\mathrm{ETHH}$ and equivariant topological cyclic homology. Namely, we compute $\mathrm{ETHH}(H\underline{\mathbb{F}}_p)$ for odd primes, showcasing the complexity of Bökstedt periodicity in this setting. Furthermore, we give computations of $\mathrm{ETHH}$ for the equivariant complex cobordism spectra $MU_G$ and $MU_{\mathbb{R}}$.