Igor Vlasenko, Sergiy Maksymenko
This paper studies one-dimensional non-Hausdorff manifolds that are similar to "graphs with split vertices". It is shown that if $M$ is a connected one-dimensional non-Hausdorff manifold such that the set of its "non-Hausdorff" points is locally finite, and each component of its complement has a countable base, then there exists a quotient map $π\colon M \to Γ$ onto an open one-dimensional CW complex, which maps the non-Hausdorff points of $M$ to the vertices of $Γ$. Moreover, $Γ$ is the minimal Hausdorff quotient of $M$, that is, for every continuous map $f\colon M \to N$ into a Hausdorff space $N$, there exists a unique continuous map $\hat{f}\colon Γ\to N$ such that $f = \hat{f} \circ π$.
Marius Huber, David R. Reich, Lena A. Jäger
Persistent homology, a method from topological data analysis, extracts robust, multi-scale features from data. It produces stable representations of time series by applying varying thresholds to their values (a process known as a \textit{filtration}). We develop novel filtrations for time series and introduce topological methods for the analysis of eye-tracking data, by interpreting fixation sequences as time series, and constructing ``hybrid models'' that combine topological features with traditional statistical features. We empirically evaluate our method by applying it to the task of dyslexia detection from eye-tracking-while-reading data using the Copenhagen Corpus, which contains scanpaths from dyslexic and non-dyslexic L1 and L2 readers. Our hybrid models outperform existing approaches that rely solely on traditional features, showing that persistent homology captures complementary information encoded in fixation sequences. The strength of these topological features is further underscored by their achieving performance comparable to established baseline methods. Importantly, our proposed filtrations outperform existing ones.
B. Shapiro
Let Y be a compact Riemann surface, phi:Y -> CP^1 a meromorphic function, and Gamma in Y a ribbon graph avoiding the critical points of phi. Then phi(Gamma) is an immersed graph in CP^1. Conversely, given an immersion im:Theta to bCP^1 of an abstract multigraph Theta without vertices of valence 1 or 2, we describe a construction of a compact Riemann surface Y and a meromorphic function phi_{im}:Y in CP^1 such that phi_{im}(Gamma)=im(Theta). We investigate the relation between the topology of Y and the combinatorics of Gamma. In particular, for a surface of genus g we construct spanning ribbon graphs whose underlying abstract graphs have arbitrary prescribed graph genus g' smaller or equal g, including the planar case. As a consequence, the number of self-intersections of φ(Gamma) cannot, in general, be controlled solely by the genus of Y. We establish general lower bounds for the number of self-intersections and formulate several open problems, with emphasis on planar ribbon graphs.
Marco Praderio Bova
We develop tools which use common fusion systems building techniques in order to compute higher limits over the centric orbit category. We apply these tools in order to study both the Diaz-Park sharpness conjecture as well as the weaker cohomological sharpness conjecture which predicts vanishing of higher limits only for the cohomology Mackey functors . Our approach leads to proving cohomological sharpness (but not sharpness) for all saturated fusion systems over p-groups of either maximal nihlpotency or of rank 2 and all polynomial, Henke-Shpectorov and van Beek fusion systems. This list includes all but 2 of the cases for which cohomological sharpness was previously known as well as most currently known families of exotic fusion systems. For the polynomial, Henke-Shpectorov and 6 of the van Beek fusion systems, sharpness is also approximated by proving vanishing of all but the first higher limits of any Mackey functor. The distinction our approach makes between sharpness and cohomological sharpness is somewhat surprising and interesting by itself. Our approach draws a new connection between cohomological sharpness and fusion system building techniques. We believe that this connection will lead to a better understanding of both fusion systems and Mackey functors over them.
Steven Amelotte, Vladimir Gorchakov
We construct models for the classifying spaces of coabelian subgroups of right-angled Coxeter groups as homotopy orbit spaces of real moment-angle complexes, generalizing well-known models for the classifying space of a right-angled Coxeter group and its commutator subgroup. This identifies the cohomology of these groups with the Borel equivariant cohomology of elementary abelian $2$-group actions on cubical subcomplexes of a cube $[-1,1]^m$. We then characterize equivariant formality for these actions, leading to a simple graph-theoretic criterion for when the cohomology of a coabelian subgroup is free as a module over the cohomology of the quotient by the commutator subgroup of the right-angled Coxeter group.
Nikola Milićević
We develop new aspects of the homological algebra theory for persistence modules, in both the one-parameter and multi-parameter settings. For a poset $P$ and an order preserving map $\varphi:P\times P\to P$, we introduce a novel tensor product of persistence modules indexed by $P$, $\otimes_{\varphi}$. We prove that each $\otimes_{\varphi}$ has a right adjoint, $\mathbf{Hom}^{\varphi}$, the internal hom of persistence modules that also depends on $\varphi$. We prove that every $\otimes_{\varphi}$ yields a Künneth short exact sequence of chain complexes of persistence modules. Dually, the $\mathbf{Hom}^{\varphi}$ also has an associated Künneth short exact sequence in cohomology. As special cases both of these short exact sequences yield Universal Coefficient Theorems. We show how to apply these to chain complexes of persistence modules arising from filtered CW complexes. For the special case of $P=\mathbb{R}_+$, the $p$-quasinorms for each $p\in (0,\infty]$ yield a distinct $\otimes_{\ell^p_c}$ and its adjoint $\mathbf{Hom}^{\ell^p_c}$. We compute their derived functors, $\mathbf{Tor}^{\ell^p_c}$ and $\mathbf{Ext}_{\ell^p_c}$ explicitly for interval modules. We show that the Universal Coefficient Theorem developed can be used to compute persistent Borel-Moore homology of a filtration of non-compact spaces. Finally, we show that for every $p\in [1,\infty]$ the associated Künneth short exact sequence can be used to significantly speed up and approximate persistent homology computations in a product metric space $(X\times Y,d^p)$ with the distance $d^p((x,y),(x',y'))=||d_X(x,x'),d_Y(y,y')||_p$.
Drew Heard
For $G$ a finite group and $T$ a $G$-Tambara functor, we construct the frame $\mathop{RadId}_G(T)$ of radical Tambara ideals and show that its points are the Nakaoka primes. We show that this frame is spatial and coherent, and deduce that the Nakaoka spectrum is a spectral space, recovering a recent result of Chan and Spitz.
Oscar Harr
For a topological space that is homeomorphic to a finite simplicial complex, we prove that the Bartels--Nikolaus assembly functor has a fully faithful right adjoint. Using this, we define for each such topological space $X$ a {\em Whitehead category}, whose K-theory is canonically identified with the Whitehead spectrum of $X$; and for a homotopy equivalence between two such spaces, we define an object in the Whitehead category of $X$ called the {\em torsion cosheaf} of the map, whose K-theory class recovers the classical Whitehead torsion.
Ioannis Gkeneralis
We investigate the equivariant topological rigidity of complex and quaternionic moment--angle manifolds. By reducing the classification to the equivariant rigidity of their quasitoric (or quoric) quotients and the classification of the associated principal bundles, we establish new rigidity results within the category of locally linear actions. We prove that complex moment-angle manifolds are equivariantly rigid: any locally linear manifold equivariantly homotopy equivalent to a complex moment--angle manifold is equivariantly homeomorphic to it. In the quaternionic setting, we establish full equivariant rigidity for manifolds with four-dimensional quoric quotients and provide a primary rigidity statement for higher dimensions based on degree-4 characteristic classes. These results characterize moment--angle manifolds as equivariant strong Borel manifolds, demonstrating that their equivariant homotopy type completely determines their equivariant homeomorphism type.
Nicholas A. Scoville
The complex of discrete Morse matchings $\M(K)$, introduced by Chari and Joswig, is a simplicial complex whose simplices are the acyclic matchings on the Hasse diagram of $K$. Its homotopy type is known in only a handful of cases. In this paper, we compute the homotopy types of $\M(Δ^3)$ and $\M(\partialΔ^3)$, the corresponding pure complexes $\M_{P}(Δ^3) \simeq \M_{P}(\partialΔ^3)$, and the generalized complex of discrete Morse matchings $\GM(Δ^3) \simeq \GM(\partialΔ^3)$. For general $n$ we prove the identity $f(n) = (n+1) \cdot |\text{top-dimensional facets of } \M(Δ^n_{(n-2)})|$, reducing the enumeration of optimal matchings on $Δ^n$ to an enumeration on its $(n-2)$-skeleton, and we show that the inclusion $\M(K) \hookrightarrow \M(CK)$ is null-homotopic for any cone. We also compute the $f$-vector of $\M(Δ^4)$, whose top entry $f(4) = 380{,}125$ is the number of optimal discrete Morse matchings on $Δ^4$. We conclude with two conjectures extending the $\M_{P}$ and $\GM$ equivalences to all $n$.
Makoto Ozawa
We introduce a persistent geometric framework for knot types based on normalized spaces of representatives. For a knot type $K$ and a scale parameter $Λ>0$, we consider the space \[ Y_Λ(K)= R_{1,Λ}(K) \] of representatives of $K$ with thickness at least $1$ and length at most $Λ$, modulo orientation-preserving reparametrization and rigid motions. This space may be viewed as a normalized moduli-type space of unparametrized representatives of the knot type $K$, equipped with a ropelength sublevel filtration. We equip $Y_Λ(K)$ with an extended pseudometric defined by the infimum of swept areas of admissible deformations. This leads to a deformation-theoretic notion of admissible components and hence to a natural $0$-dimensional persistence module as $Λ$ increases. We show that the first birth time of this persistence is exactly the ropelength of the knot type. Accordingly, the layer \[ I(K)=Y_{Rop(K)}(K) \] appearing at the first birth is distinguished; we call it the ideal stratum of $K$. From the moduli-theoretic viewpoint, the ideal stratum is the minimizer locus of the ropelength functional on the normalized representative space. In this way, ideal knots are interpreted not merely as minimizers of ropelength, but as the initial stratum of a persistent shape profile associated with the knot type. We also introduce ideal admissible components and ideal merge scales, which suggest further geometric invariants beyond ropelength.
Heng Xie
We calculate the Witt ring of the real sphere.
Mattie Ji, Indradyumna Roy, Vikas Garg
Persistent homology (PH) encodes global information, such as cycles, and is thus increasingly integrated into graph neural networks (GNNs). PH methods in GNNs typically traverse an increasing sequence of subgraphs. In this work, we first expose limitations of this inclusion procedure. To remedy these shortcomings, we analyze contractions as a principled topological operation, in particular, for graph representation learning. We study the persistence of contraction sequences, which we call Contraction Homology (CH). We establish that forward PH and CH differ in expressivity. We then introduce Hourglass Persistence, a class of topological descriptors that interleave a sequence of inclusions and contractions to boost expressivity, learnability, and stability. We also study related families parametrized by two paradigms. We also discuss how our framework extends to simplicial and cellular networks. We further design efficient algorithms that are pluggable into end-to-end differentiable GNN pipelines, enabling consistent empirical improvements over many PH methods across standard real-world graph datasets. Code is available at \href{https://github.com/Aalto-QuML/Hourglass}{this https URL}.
Mohit Dubey
Multi-agent systems (MAS) powered by large language models suffer from severe token inefficiency arising from two compounding sources: (i) unstructured parallel execution, where all agents activate simultaneously irrespective of input readiness; and (ii) unrestricted context sharing, where every agent receives the full accumulated context regardless of relevance. Existing mitigation strategies - static pruning, hierarchical decomposition, and learned routing - treat coordination as a structural allocation problem and fundamentally ignore its temporal dimension. We propose Phase-Scheduled Multi-Agent Systems (PSMAS), a framework that reconceptualizes agent activation as continuous control over a shared attention space modeled on a circular manifold. Each agent i is assigned a fixed angular phase theta_i in the range [0, 2*pi], derived from the task dependency topology; a global sweep signal phi(t) rotates at velocity omega, activating only agents within an angular window epsilon. Idle agents receive compressed context summaries, reducing per-step token consumption. We implement PSMAS on LangGraph, evaluate on four structured benchmarks (HotPotQA-MAS, HumanEval-MAS, ALFWorld-Multi, WebArena-Coord) and two unstructured conversational settings, and prove stability, convergence, and optimality results for the sweep dynamics. PSMAS achieves a mean token reduction of 27.3 percent (range 21.4-34.8 percent) while maintaining task performance within 2.1 percentage points of a fully activated baseline (p < 0.01, n = 500 per configuration), and outperforms the strongest learned routing baseline by 5.6 percentage points in token reduction with 2.0 percentage points less performance drop. Crucially, we show that scheduling and compression are independent sources of gain: scheduling alone accounts for 18-20 percentage points of reduction, robust to compression degradation up to alpha = 0.40.
Chad M. Topaz
The Mapper algorithm from topological data analysis constructs a graph summarizing the shape of a high-dimensional dataset, and groups of data points identified within this graph are widely interpreted as evidence of distinct subtypes. However, the covariance structure of the data alone can make such groups appear differentiated, even when no subtypes are present. Existing validation approaches do not account for this effect and thus cannot distinguish covariance artifacts from genuine subtypes. We propose a Gaussian null model that generates reference data matching the sample covariance matrix. We pair it with a test statistic that measures mean-level differentiation between communities. In an idealized setting, we prove that covariance geometry alone causes Mapper communities to differ in their average feature profiles, and we show that a simpler label-permutation baseline cannot detect this effect. Simulations confirm well-controlled Type I error under Gaussian data. We apply the framework to four published Mapper analyses spanning breast cancer gene expression, Congressional voting, NBA player performance, and lower-grade glioma genomics. In every case, once outlier singleton communities are accounted for, the observed differentiation does not exceed what the null produces at the α = 0.05 level. This result does not rule out subtypes in these datasets, but it does indicate that the observed structure is consistent with what covariance geometry alone can produce. Stronger evidence would be needed to support a subtype claim.
Daisuke Kishimoto, Donald Stanley, Carlos Gabriel Valenzuela Ruiz
For a field $\mathbb{F}$ and a triangulated compact $\mathbb{F}$-orientable manifold, consider the homology of the associated Moment-Angle ccomplex $H_*(\mathcal{Z}_{\mathcal{K}})$. We show the total homology rank $β(\mathcal{Z}_{\mathcal{K}})$ satisfies the inequality $β(\mathcal{Z}_{\mathcal{K}};\mathbb{F})\geq 2^{m-1}(β(\mathcal{K};\mathbb{F})-2)+2$, with equality occurring exactly when the triangulation is $\mathbb{F}$-tight. Using Lefschetz duality, we introduce a short exact sequence of functors that, in turn, introduces a new duality theorem in Double Homology for tight manifold triangulations.
Shih-Yu Chang
We introduce an operadic notion of spectrum for algebras over colored operads in a symmetric monoidal category. The construction is defined via a canonical Hochschild-type object together with an operadic residue, which together encode spectral information in a manner compatible with operadic composition. A central result of this work is that classical spectral invariants do not, in general, admit a natural base change in the operadic setting. More precisely, we show that there is no functorial procedure that transports spectra along strong monoidal functors while preserving their expected structural properties. This establishes a fundamental obstruction to spectral base change. To address this issue, we construct a universal operadic residue object and show that it induces a well-defined and functorial notion of operadic spectrum. We further prove that this construction is canonical and reduces to the classical spectrum in the case of the trivial operad. These results provide a conceptual foundation for spectral theory in operadic and higher algebraic contexts, and clarify the limitations of extending classical spectral invariants beyond the linear setting.
Gian Maria Dall'Ara, Roberto Frigerio, Ervin Hadziosmanovic
Let $M$ be a complete hyperbolic $n$-manifold, $n\geq 2$. Via integration over geodesic simplices, any closed bounded differential 2-form on $M$ defines a bounded cohomology class in $H^2_b(M)$. It was proved by Barge and Ghys (for $n=2$) and by Battista et al. (for $n>2$) that, if $M$ is closed, then this procedure defines an injective embedding of the (infinite-dimensional) space of closed differential $2$-forms on $M$ into $H^2_b(M)$. We extend this result to the case when the fundamental group of $M$ is of the first kind, i.e. its limit set is equal to the whole boundary at infinity of hyperbolic space (this holds, for example, when $M$ has finite volume). Our argument is different from Barge and Ghys' original one, and relies on the following fact of independent interest: an $L^\infty$ function on the hyperbolic plane is determined by its integrals over all ideal triangles. We prove this fact by way of Fourier analysis on the hyperbolic plane.
Sagnik Biswas
We study two problems concerning the Stiefel manifolds $V_{n,2}$ and their products with spheres. First, we address a rigidity problem: we determine, for most values of~$n$, all self-maps of $V_{n,2}$ that are homotopic to an almost diffeomorphism. Second, we classify smooth closed manifolds tangentially homotopy equivalent to $V_{n,2} \times S^k$ up to almost diffeomorphism, for $k = 3, 5$ or $7 \leq k \leq n-3$, $k \neq 2^i - 2$. Our method is to find explicit inverses in the structure set via normal invariants of specific tangential homotopy equivalences. In favourable cases -- notably $V_{12,2} \times S^3$, $V_{16,2} \times S^3$, $V_{12,2} \times S^5$, $V_{10,2} \times S^5$ -- the classification is complete: every such manifold is almost diffeomorphic to $V_{n,2} \mathbin{\#} Σ\times S^k$ for some exotic sphere $Σ$. In the general case, we identify inverses for a large subgroup of $\operatorname{Im}(η)$ and provide a possible way forward to the remainder.
Paul Shick
We study $v_n$-periodic phenomena in $C_2$-equivariant stable homotopy through the lens of the $C_2$-equivariant Adams spectral sequence at the prime 2. In particular, we construct/detect certain classes related to powers of the $v_n$ generators of $π_*(BP)$ in the cohomology of certain finitely generated subalgebras $A^{C_2}(m)$ of the $C_2$-equivariant Steenrod algebra. We define the notion of classes in $\text{Ext}_{A^{C_2}}(\underline{H}^\star, \underline{H}^\star)$ being $v_n$-periodic or $v_n$-torsion and exhibit a chromatic filtration by showing that $v_n$-torsion classes are also $v_k$-torsion for $0\le k < n.$ We also promote the Lin-Davis-Mahowald-Adams splitting of Ext of the suitable version of ``$R P_{-\infty}^\infty$" to the $C_2$-equivariant setting and use this to define appropriate algebraic versions of Mahowald's root invariant. We establish that whenever a class corresponding to a power of $v_{n}$ is nonzero in $ \text{Ext}_{A^{C_2}(m)}(\underline{H}^\star, \underline{H}^\star),$ then the same power of $v_{n-1}$ is also nonzero in $ \text{Ext}_{A^{C_2}(m-1)}(\underline{H}^\star, \underline{H}^\star),$ and its algebraic Mahowald invariant $M_m^{C_2-alg}(v_{n-1}^{2^f}) \subset \text{Ext}_{A^{C_2}(m)}(\underline{H}^\star, \underline{H}^\star)$ contains class(es) corresponding to $v_n^{2^f}.$ Real motivic versions of these results hold as well.