Tootfinder

The condensed homotopy type of a scheme
Peter J. Haine, Tim Holzschuh, Marcin Lara, Catrin Mair, Louis Martini, Sebastian Wolf with an appendix by Bogdan Zavyalov
https://arxiv.org/abs/2510.07443

The condensed homotopy type of a scheme
We study a condensed version of the étale homotopy type of a scheme, which refines both the usual étale homotopy type of Friedlander-Artin-Mazur and the proétale fundamental group of Bhatt-Scholze. In the first part of this paper, we prove that this condensed homotopy type satisfies descent along integral morphisms and that the expected fiber sequences hold. We also provide explicit computations, for example, for rings of continuous functions. A key ingredient in many of our arguments is a d…

The homotopy type of the Poincar\'e cobordism category for surfaces
Az\'elie Picot
https://arxiv.org/abs/2510.07659 https://arxiv.org/pdf/2510.0765…

The homotopy type of the Poincaré cobordism category for surfaces
We define a version of the surface cobordism category $\mathrm{Cob}_2^{\mathrm{SG}}(X)$ over a base space $X$ where surfaces are considered up to self homotopy equivalences instead of diffeomorphisms. We prove the induced functor $B\mathrm{Cob}_2^{\mathrm{SG}}(-): \mathcal{S} \rightarrow \mathcal{S}$ is not $1$-excisive. We show its first derivative $\partial_1 B\mathrm{Cob}_2^{\mathrm{SG}}(-)$ in the Goodwillie sense is equivalent to a Thom spectrum over $B\mathrm{haut}_*^+(S^2)$.

Simple homotopy theory for Fukaya categories
Yonghwan Kim
https://arxiv.org/abs/2509.05856 https://arxiv.org/pdf/2509.05856

Simple homotopy theory for Fukaya categories
We develop a categorical framework for simple homotopy theory in Fukaya categories, based on the fundamental group of the ambient symplectic manifold. When the first Chern class vanishes, we show that any isomorphism in the Fukaya category of a Weinstein manifold has trivial Whitehead torsion. As an application, we prove that any pair of closed connected Lagrangians that are isomorphic in the Fukaya category of such Weinstein manifolds are simple homotopy equivalent, provided one of the Lagrang…

Replaced article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Homotopy type theory as a language for diagrams of $\infty$-logoses
Taichi Uemura

Homotopy theory of simplicial parametrized operads
Gregoire Marc
https://arxiv.org/abs/2510.06932 https://arxiv.org/pdf/2510.06932

Homotopy theory of simplicial parametrized operads
We define a generalization of (coloured) operads based on double lax functors and we construct a model structure on the associated category of generalized simplicial (coloured) operads. In particular, we obtain a model structure on the category of simplicial (coloured) O-operads of Nardin and Shah.

Formalizing the zigzag construction of path spaces of pushouts
Vojt\v{e}ch \v{S}t\v{e}pan\v{c}\'ik
https://arxiv.org/abs/2510.08452 https://arxiv.org/p…

Formalizing the zigzag construction of path spaces of pushouts
A recent pre-print of Wärn gives a novel pen-and-paper construction of a type family characterizing the path spaces of an arbitrary pushout, and a natural language argument for its correctness. We present the first formalization of the construction and a proof that it is fiberwise equivalent to the path spaces. The formalization is carried out in axiomatic homotopy type theory, using the Agda proof assistant and the agda-unimath library.

Universal constructions in homotopical algebra
Ricardo Campos, Bruno Vallette
https://arxiv.org/abs/2510.06941 https://arxiv.org/pdf/2510.06941

Universal constructions in homotopical algebra
We apply the effective integration theory of Lie-graph algebras, developed recently by the authors, to the deformation and homotopy theories of types of bialgebras, that is structures controlled by a properad, like associative bialgebras, (involutive) Lie bialgebras, Frobenius bialgebras, double Poisson bialgebras, pre-Calabi--Yau algebras, quantum Airy structures, etc. In these cases, we provide their associated Deligne groupoid with an explicit homotopical description. We settle the Koszul hi…

Monoidal model structures over infinite groups
Ioannis Emmanouil, Olympia Talelli
https://arxiv.org/abs/2510.08195 https://arxiv.org/pdf/2510.08195

Monoidal model structures over infinite groups
The stable category of modules over the algebra of a finite group with coefficients in a field is a compactly generated tensor triangulated category, that has been studied extensively in representation theory. In this paper, we provide a plethora of infinite groups G, for which the category of kG-modules (where k is a commutative coherent ring of finite global dimension) admits a monoidal model structure, in the sense of Hovey, whose associated homotopy category is a compactly generated tensor …

Replaced article(s) found for cs.LO. https://arxiv.org/list/cs.LO/new
[1/1]:
- Ordinal Exponentiation in Homotopy Type Theory
Tom de Jong, Nicolai Kraus, Fredrik Nordvall Forsberg, Chuangjie Xu

Replaced article(s) found for math.CT. https://arxiv.org/list/math.CT/new
[1/1]:
- Strict Rezk completions of models of HoTT and homotopy canonicity
Rafa\"el Bocquet

Isotopy invariance and stratified $\mathbb{E}_2$-structure of the Ran Grassmannian
Guglielmo Nocera, Morena Porzio
https://arxiv.org/abs/2509.06222 https://

Isotopy invariance and stratified $\mathbb{E}_2$-structure of the Ran Grassmannian
Let $G$ be a complex reductive group. A folklore result asserts the existence of an $\mathbb{E}_2$-algebra structure on the Ran Grassmannian of $G$ over $\mathbb{A}^1_{\mathbb{C}}$, seen as a topological space with the complex-analytic topology. The aim of this paper is to prove this theorem, by establishing a homotopy invariance result: namely, an inclusion of open balls $D' \subset D$ in $\mathbb{C}$ induces a homotopy equivalence between the respective Beilinson--Drinfeld Grassmannians $\mat…

The $\mathbb{A}^1$-Euler characteristic of symmetric powers
Louisa F. Br\"oring, Jesse Pajwani, Anna M. Viergever
https://arxiv.org/abs/2510.06922 https://

The $\mathbb{A}^1$-Euler characteristic of symmetric powers
The $\mathbb{A}^1$-Euler characteristic is a refinement in algebraic geometry of the classical topological Euler characteristic, which can be constructed using motivic homotopy theory. This invariant is a quadratic form rather than an integer, which carries a lot of information, but is difficult to compute in practice. In this survey, we discuss a conjectural way for computing the $\mathbb{A}^1$-Euler characteristic of the symmetric powers of a variety in terms of the $\mathbb{A}^1$-Euler chara…

Non-characterizing slopes for more 3-manifolds
Matthew Elpers
https://arxiv.org/abs/2510.05412 https://arxiv.org/pdf/2510.05412

Non-characterizing slopes for more 3-manifolds
For any homotopy class h in any compact orientable 3-manifold M which is closed or has exclusively torus boundary components, we produce infinitely many pairs of distinct knots representing h with orientation-preserving homeomorphic 0-surgeries.

Yang-Mills kinematic algebra via homotopy transfer from a worldline operator algebra
Roberto Bonezzi, Christoph Chiaffrino, Olaf Hohm, Maria Foteini Kallimani
https://arxiv.org/abs/2508.17933

Yang-Mills kinematic algebra via homotopy transfer from a worldline operator algebra
The homotopy Lie or $L_{\infty}$ algebra encoding Yang-Mills theory is the tensor product of a color Lie algebra with the kinematic $C_{\infty}$ algebra. We derive this $C_{\infty}$ algebra, via homotopy transfer, from a strict operator algebra of a worldline theory, realized as an associative star product algebra. This gives a homotopy transfer interpretation to worldline vertex operators introduced in previous work.

Homotopy Languages
C\'esar Bardomiano Mart\'inez, Simon Henry
https://arxiv.org/abs/2510.02607 https://arxiv.org/pdf/2510.02607

Homotopy Languages
We attach to each weak model category $\mathcal{M}$ a class of first order formulas about the fibrant objects of $\mathcal{M}$ whose validity is invariant under homotopies and weak equivalences. This is a generalization of the classical Blanc-Freyd Language of categories -- which involves formula avoiding equality on objects and which are invariant under isomorphism and equivalences of categories. In particular, we obtain similar homotopy invariant languages for $2$-categories, bicategories, ch…

from my link log —
Homotopy type theory for dummies.
http://www.chriswarbo.net/blog/2015-09-11-hott_for_dummies.html
saved 2025-11-03 …

A strong finiteness condition for smashing localisations
Isabel Longbottom
https://arxiv.org/abs/2509.07344 https://arxiv.org/pdf/2509.07344

A strong finiteness condition for smashing localisations
We define a class of smashing localisations which we call compactly central, and classify compactly central localisations of $Sp_{(p)}$ and of $Sp$. Our main result is that $L_n^f$ is a compactly central localisation. A map $α: 1 \to A$ in a presentably symmetric monoidal $\infty$-category $\mathscr{C}$ is central if there exists a homotopy $α\otimes id_A \simeq id_A \otimes α: A \to A \otimes A$. A central map $α$ can be used to produce a smashing localisation $L_α$ of $\mathscr{C}$, be…

HANN: Homotopy auxiliary neural network for solving nonlinear algebraic equations
Ling-Zhe Zai, Lei-Lei Guo, Zhi-Yong Zhang
https://arxiv.org/abs/2509.26358 https://

HANN: Homotopy auxiliary neural network for solving nonlinear algebraic equations
Solving nonlinear algebraic equations is a fundamental but challenging problem in scientific computations and also has many applications in system engineering. Though traditional iterative methods and modern optimization algorithms have exerted effective roles in addressing certain specific problems, there still exist certain weaknesses such as the initial value sensitivity, limited accuracy and slow convergence rate, particulary without flexible input for the neural network methods. In this pa…

Motivic homotopy theory for perfect schemes
Christian Dahlhausen, Jeroen Hekking, Storm Wolters
https://arxiv.org/abs/2510.01390 https://arxiv.org/pdf/2510…

Motivic homotopy theory for perfect schemes
We construct a perfect version of Morel--Voevodsky's motivic homotopy category over a perfect base scheme in positive characteristic. By checking the axioms of a coefficient system, we establish a six-functor formalism. We show that multiplication by $p$ is already invertible in the perfect motivic homotopy catgory. By work of Elmanto--Khan the functor sending an $\mathbb{F}_p$-scheme $S$ to the category $\mathrm{S}\mathcal{H}(S)[1/p]$ is invariant under universal homeomorphisms, hence under pe…

Planar ternary graphs, flag spheres, and Delannoy polynomials
Margaret Bayer, Richard Danner, Thiago Holleben, Marie Kramer, Yirong Yang
https://arxiv.org/abs/2509.21705 https:/…

Planar ternary graphs, flag spheres, and Delannoy polynomials
In 2022 Kim showed when a graph $G$ is ternary (without induced cycles of length divisible by three), its independence complex $\text{Ind}(G)$ is either contractible or homotopy equivalent to a sphere. In this paper, we show that when $\text{Ind}(G)$ is homotopy equivalent to a sphere of dimension $\dim \text{Ind}(G)$, the complex is Gorenstein. Equivalently, $G$ is a $1$-well-covered graph. This answers a question by Faridi and Holleben. We then focus on the independence complexes of Gorenst…

Homotopy equivalence of digital pictures in $\mathbb{Z}^2$
Dae-Woong Lee, P. Christopher Staecker
https://arxiv.org/abs/2509.03023 https://arxiv.org/pdf/25…

Homotopy equivalence of digital pictures in $\mathbb{Z}^2$
We investigate the properties of digital homotopy in the context of digital pictures $(X,κ,\bar κ)$, where $X\subsetneq \mathbb{Z}^n$ is a finite set, $κ$ is an adjacency relation on $X$, and $\bar κ$ is an adjacency relation on the complement of $X$. In particular we focus on homotopy equivalence between digital pictures in $\mathbb{Z}^2$. We define a numerical homotopy-type invariant for digital pictures in $\mathbb{Z}^2$ called the outer perimeter, which is a basic tool for distinguishin…

$K$-Theory of Adelic and Rational Group $C^*$-algebras via Generalized Winding Numbers
Wenqing Wu, Hang Wang
https://arxiv.org/abs/2509.00390 https://arxiv…

$K$-Theory of Adelic and Rational Group $C^*$-algebras via Generalized Winding Numbers
We take the following approach to analyze homotopy equivalence in periodic adelic functions. First, we introduce the concept of pre-periodic functions and define their homotopy invariant through the construction of a generalized winding number. Subsequently, we establish a fundamental correspondence between periodic adelic functions and pre-periodic functions. By extending the generalized winding number to periodic adelic functions, we demonstrate that this invariant completely characterizes ho…

Homotopy classification of closed polygonal lines: results and problems
E. Alkin, O. Nikitenko, A. Skopenkov
https://arxiv.org/abs/2508.16287 https://arxiv…

Homotopy classification of closed polygonal lines: results and problems
In this text we expose (as a sequence of problems) basic cases of some fundamental ideas and methods of mathematics. Namely, of homotopy, degree, fundamental group, covering, Whitehead invariant, etc. This is done by considering the elementary example: closed polygonal lines in a subset of the plane. Although these ideas and methods are parts of topology, they are used in many other areas including computer science. This text is expository and is accessible to mathematicians not specialized in …

Newton-Flow Particle Filters based on Generalized Cram\'er Distance
Uwe D. Hanebeck
https://arxiv.org/abs/2509.00182 https://arxiv.org/pdf/2509.00182…

Newton-Flow Particle Filters based on Generalized Cramér Distance
We propose a recursive particle filter for high-dimensional problems that inherently never degenerates. The state estimate is represented by deterministic low-discrepancy particle sets. We focus on the measurement update step, where a likelihood function is used for representing the measurement and its uncertainty. This likelihood is progressively introduced into the filtering procedure by homotopy continuation over an artificial time. A generalized Cramér distance between particle sets is der…

Artin--Mazur formal groups and Milne duality via unipotent spectra
Shubhodip Mondal, Tasos Moulinos, Lucy Yang
https://arxiv.org/abs/2510.06152 https://arx…

Artin--Mazur formal groups and Milne duality via unipotent spectra
We introduce and develop the notion of "unipotent spectra." This is defined to be the stabilization of Toën's category of affine stacks, and is related to recent work of Mondal--Reinecke. Unipotent spectra give rise to unipotent stable homotopy groups and unipotent homology, which are new invariants for schemes valued in unipotent group schemes. As applications, we recover the Artin--Mazur formal groups associated to schemes without any vanishing assumptions. Further, we show that syntomic coh…

Replaced article(s) found for math.DS. https://arxiv.org/list/math.DS/new
[1/1]:
- Symmetric periodic solutions in the generalized Sitnikov Problem with homotopy methods
Carlos Barrera-Anzaldo, Carlos Garc\'ia-Azpeitia

Morse sequences on stacks and flooding sequences
Gilles Bertrand
https://arxiv.org/abs/2509.01384 https://arxiv.org/pdf/2509.01384

Morse sequences on stacks and flooding sequences
This paper builds upon the framework of \emph{Morse sequences}, a simple and effective approach to discrete Morse theory. A Morse sequence on a simplicial complex consists of a sequence of nested subcomplexes generated by expansions and fillings-two operations originally introduced by Whitehead. Expansions preserve homotopy, while fillings introduce critical simplexes that capture essential topological features. We extend the notion of Morse sequences to \emph{stacks}, which are monotonic funct…

UTOPY: Unrolling Algorithm Learning via Fidelity Homotopy for Inverse Problems
Roman Jacome, Romario Gualdr\'on-Hurtado, Leon Suarez-Rodriguez, Henry Arguello
https://arxiv.org/abs/2509.14394

UTOPY: Unrolling Algorithm Learning via Fidelity Homotopy for Inverse Problems
Imaging Inverse problems aim to reconstruct an underlying image from undersampled, coded, and noisy observations. Within the wide range of reconstruction frameworks, the unrolling algorithm is one of the most popular due to the synergistic integration of traditional model-based reconstruction methods and modern neural networks, providing an interpretable and highly accurate reconstruction. However, when the sensing operator is highly ill-posed, gradient steps on the data-fidelity term can hinde…

Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy
Jean-Michel Coron, Joachim Krieger, Shengquan Xiang
https://arxiv.org/abs/2509.12779

Wave maps from circle to Riemannian manifold: global controllability is equivalent to homotopy
We study wave maps from the circle to a general compact Riemannian manifold. We prove that the global controllability of this geometric equation is characterized precisely by the homotopy class of the data. As a remarkable intermediate result, we establish uniform-time global controllability between steady states, providing a partial answer to an open problem raised by Dehman, Lebeau and Zuazua (2003). Finally, we obtain quantitative exponential stability around closed geodesics with negative s…

Lorentz Covariance of the $4d$ Nonlinear Higher-Spin Equations via BRST
O. A. Gelfond, M. A. Vasiliev
https://arxiv.org/abs/2509.22852 https://arxiv.org/pd…

Lorentz Covariance of the $4d$ Nonlinear Higher-Spin Equations via BRST
We propose a BRST extension of the higher-spin gauge theory in $AdS_4$ with the BRST operator associated with the local Lorentz symmetry. Our construction supports manifest local Lorentz covariance and is applicable both to any homotopy scheme of the perturbative analysis including the recently proposed differential homotopy and to the variety of further extended higher-spin models in $AdS_3$ and $AdS_4$ with higher differential forms and Coxeter higher-spin algebras.

Wronskians as $N$-ary brackets in finite-dimensional analogues of $sl(2)$
Arthemy V. Kiselev
https://arxiv.org/abs/2510.02145 https://arxiv.org/pdf/2510.02…

Wronskians as $N$-ary brackets in finite-dimensional analogues of $sl(2)$
The Wronskian determinants (for coefficients of higher-order differential operators on the affine real line or circle) satisfy the table of Jacobi-type quadratic identities for strong homotopy Lie algebras -- i.e. for a particular case of $L_\infty$-deformations -- for the Lie algebra of vector fields on that one-dimensional affine manifold. We show that the standard realisation of $\mathfrak{sl}(2)$ by quadratic-coefficient vector fields is the bottom structure in a sequence of finite-dimensio…

On generalized morphisms associated to endofunctors of C*-algebras
Georgii S. Makeev
https://arxiv.org/abs/2509.02001 https://arxiv.org/pdf/2509.02001

On generalized morphisms associated to endofunctors of C*-algebras
We introduce a class of good endofunctors of $C^{*}$-algebras, endow it with a structure of a bimonoidal category, and define homotopies of natural transformations between such endofunctors. For every pair of $C^{*}$-algebras and a good endofunctor, we construct a commutative monoid of generalized morphisms, and endow these monoids with a bilinear composition. This construction generalizes the homotopy category of asymptotic homomorphisms used in the definition of the Connes-Higson $E$-theory. …

Kapranov $L_{\infty}[1]$ algebras
Ruggero Bandiera, Seokbong Seol, Mathieu Sti\'enon, Ping Xu
https://arxiv.org/abs/2509.26341 https://arxiv.org/pdf/25…

Kapranov $L_{\infty}[1]$ algebras
Given any Kähler manifold $X$, Kapranov discovered an $L_\infty[1]$ algebra structure on $Ω^{0,\bullet}_X(T^{1,0}_X)$. Motivated by this result, we introduce, as a generalization of $L_\infty[1]$ algebras, a notion of $L_\infty[1]$ $\mathfrak{R}$-algebra, where $\mathfrak{R}$ is a differential graded commutative algebra with unit. We show that standard notions (such as quasi-isomorphism and linearization) and results (including homotopy transfer theorems) can be extended to this context. For …

On the number of components of folds of image simple fold maps
O. Saeki, R. Sadykov
https://arxiv.org/abs/2509.03878 https://arxiv.org/pdf/2509.03878

On the number of components of folds of image simple fold maps
A smooth map between manifolds is said to be \emph{image simple} if its restriction to its singular point set is a topological embedding. We study the parity of the number of connected components of the singular point set for image simple fold maps from a closed manifold of dimension $\ge 2$ to a surface. It is known that this parity is a homotopy invariant when the source manifold is of even dimension and the target surface is orientable. In this paper we show that for an arbitrary image simpl…

Decomposing multipersistence modules using functor calculus
Bj{\o}rnar Gullikstad Hem
https://arxiv.org/abs/2510.06178 https://arxiv.org/pdf/2510.06178

Decomposing multipersistence modules using functor calculus
We apply poset cocalculus, a functor calculus framework for functors out of a poset, to study the problem of decomposing multipersistence modules into simpler components. We both prove new results in this topic and offer a new perspective on already established results. In particular, we show that a pointwise finite-dimensional bipersistence module is middle-exact if and only if it is isomorphic to the homology of a homotopy degree 1 functor, from which we deduce a novel, more synthetic proof o…

A High-Dimensional Extension of Wagner's Theorem and the Geometrization of Hypergraphs
Qiming Fang, Sihong Shao
https://arxiv.org/abs/2510.01926 https://

A High-Dimensional Extension of Wagner's Theorem and the Geometrization of Hypergraphs
This paper introduces a geometric representation of hypergraphs by representing hyperedges as simplices. Building on this framework, we employ homotopy groups to analyze the topological structure of hypergraphs embedded in high-dimensional Euclidean spaces. Leveraging this foundation, we extend Wagner's theorem to $\mathbb{R}^d$. Specifically, we establish that a triangulated $d$-uniform topological hypergraph embeds into $\mathbb{R}^d$ if and only if it contains neither $K_{d+3}^d$ nor $K_{3,d…

Differential Contracting Homotopy in the Linearized 3d Higher-Spin Theory
M. A. Vasiliev, V. A. Vereitin
https://arxiv.org/abs/2508.11500 https://arxiv.org…

Differential Contracting Homotopy in the Linearized 3d Higher-Spin Theory
In this paper, the recently developed differential homotopy approach is applied to the problem of disentangling dynamical and topological fields of the $3d$ higher-spin gauge theory at the linear level. This formalism allows us to reproduce all known disentangling solutions in a unified form, including both the solutions obtained previously within the shifted homotopy approach in \cite{Korybut:2022kdx} and that derived by hand in \cite{Vasiliev:1992ix}, as well as other solutions including thos…

On the equivalence invariance of formal category theory
Paula Verdugo
https://arxiv.org/abs/2509.04255 https://arxiv.org/pdf/2509.04255

On the equivalence invariance of formal category theory
We prove a result of equivalence invariance of formal category theory for statements that can be expressed within an equipment. To do this, we exploit Henry and Bardomiano Martínez's link between Makkai's FOLDS (first order logic with dependent sorts) and abstract homotopy theory. In the process, we construct a model structure on the category DblCat of double categories and double functors, whose trivial fibrations are the double functors that are surjective on objects, full on horizontal and …

From Stein manifolds to Oka manifolds: the h-principle in complex analysis
Franc Forstneric
https://arxiv.org/abs/2509.21197 https://arxiv.org/pdf/2509.211…

From Stein manifolds to Oka manifolds: the h-principle in complex analysis
This introduction to the homotopy principle in complex analysis and geometry, better known as the Oka theory, is aimed at wide mathematical audience. After a brief historical survey of the h-principle in smooth analysis and geometry, I present the key notions of Oka manifolds and Oka maps, which developed from the Oka-Grauert principle and Gromov's theory of elliptic complex manifolds and elliptic holomorphic submersions. I discuss recent and ongoing developments, open problems, and mention som…

$G\operatorname{-CGWH}$ spaces and their (homotopy) colimits
Aleksandar Miladinovi\'c
https://arxiv.org/abs/2508.19037 https://arxiv.org/pdf/2508.19037…

$G\operatorname{-CGWH}$ spaces and their (homotopy) colimits
In this paper we take a look at compactly generated weak Hausdorff spaces equipped with an action of a compact Lie group $G$ together with their colimits and homotopy colimits. In particular, we investigate relations between (homotopy) colimits and mapping spaces, and consequently, homotopy groups.

On Steenrod squares for even and odd Khovanov homology
Advika Rajapakse
https://arxiv.org/abs/2509.03396 https://arxiv.org/pdf/2509.03396

On Steenrod squares for even and odd Khovanov homology
For an arbitrary link $L \subset S^3$ , Sarkar-Scaduto-Stoffregen construct a family of spatial refinements of even and odd Khovanov homology. We give a computation of $\text{Sq}^2$ on these spaces, determining their stable homotopy types for all knots K up to 11 crossings. We also prove that the Steenrod squares $\text{Sq}_0^2$ , $\text{Sq}_1^2$ defined by Schütz do arise as Steenrod squares on these spaces.

Moduli Stacks of $G$-Curves in Homotopy Theory at $h=p-1$
Rin Ray
https://arxiv.org/abs/2509.23428 https://arxiv.org/pdf/2509.23428

Moduli Stacks of $G$-Curves in Homotopy Theory at $h=p-1$
We study the action on the deformation space of a formal group by the maximal finite subgroup $G$ of its automorphisms, at the first height where the group has nontrivial $p$-torsion for odd $p$. We show given this group $G$ there is a universal construction of a geometric model of the $G$-action via inverse Galois theory which generalizes the use of level structure to ramification data. We use configuration spaces to understand the model, and conclude that the Lubin-Tate action at $h=p-1$ is a…

Homotopy Theory for Ordered Simplicial Complexes
Melissa Wei
https://arxiv.org/abs/2509.22968 https://arxiv.org/pdf/2509.22968

Homotopy Theory for Ordered Simplicial Complexes
We construct a model structure on the category of ordered simplicial complexes, Quillen equivalent to the standard model structure on simplicial sets. This shows that simplicial complexes, which are fully combinatorial in nature, provide a new model for the homotopy theory of spaces.

Morse-based Modular Homology for Evolving Simplicial Complexes
Anqiao Ouyang
https://arxiv.org/abs/2508.14429 https://arxiv.org/pdf/2508.14429

Morse-based Modular Homology for Evolving Simplicial Complexes
The computation of homology groups for evolving simplicial complexes often requires repeated reconstruction of boundary operators, resulting in prohibitive costs for large-scale or frequently updated data. This work introduces MMHM, a Morse-based Modular Homology Maintenance framework that preserves homological invariants under local complex modifications. An initial discrete Morse reduction produces a critical cell complex chain-homotopy equivalent to the input; subsequent edits trigger locali…

Non-Standard Models of Homotopy Type Theory
Nima Rasekh
https://arxiv.org/abs/2508.07736 https://arxiv.org/pdf/2508.07736

Non-Standard Models of Homotopy Type Theory
Homotopy type theory is a modern foundation for mathematics that introduces the univalence axiom and is particularly suitable for the study of homotopical mathematics and its formalization via proof assistants. In order to better comprehend the mathematical implications of homotopy type theory, a variety of models have been constructed and studied. Here a model is understood as a model category with suitable properties implementing the various type theoretical constructors and axioms. A first e…

Homotopy Iterators
Paul Breiding, Taylor Brysiewicz, Hannah Friedman
https://arxiv.org/abs/2509.08084 https://arxiv.org/pdf/2509.08084

Homotopy Iterators
We introduce the concept of homotopy iterators for performing polynomial homotopy continuation tasks in a memory efficient manner. The main idea is to push forward an iterator for the start solutions of a homotopy via the function which tracks them along the homotopy. Doing so produces a representation of the target solutions, bypassing the need to hold all solutions in memory. We discuss several applications of this datatype ranging from solution counting to data compression.

Mutations of quivers with 2-cycles
Fang Li, Siyang Liu, Lang Mou, Jie Pan
https://arxiv.org/abs/2508.15022 https://arxiv.org/pdf/2508.15022

Mutations of quivers with 2-cycles
We develop a mutation theory for quivers with oriented 2-cycles using a structure called a homotopy, defined as a normal subgroupoid of the quiver's fundamental groupoid. This framework extends Fomin-Zelevinsky mutations of 2-acyclic quivers and yields involutive mutations that preserve the fundamental groupoid quotient by the homotopy. It generalizes orbit mutations arising from quiver coverings and allows for infinite mutation sequences even when orbit mutations are obstructed. We further con…

Serre Duality and the Whitehead Link
Cailan Li
https://arxiv.org/abs/2509.22133 https://arxiv.org/pdf/2509.22133 …

Serre Duality and the Whitehead Link
We prove the action of the full twist is a Serre functor in the homotopy category of dihedral Soergel Bimodules. Our proof relies on the representability of a certain partial trace functor, which we prove using the diagrammatics for Hochschild cohomology of Soergel Bimodules. Leveraging these representability results, we then compute the triply-graded link homology of the Whitehead link.

Replaced article(s) found for math.LO. https://arxiv.org/list/math.LO/new
[1/1]:
- Relating homotopy equivalences to conservativity in dependent type theories with computation axioms
Matteo Spadetto

On torus equivariant $S^4$-bundles over $S^4$ and Petrie-type questions for GKM manifolds
Oliver Goertsches, Panagiotis Konstantis, Leopold Zoller
https://arxiv.org/abs/2509.03100

On torus equivariant $S^4$-bundles over $S^4$ and Petrie-type questions for GKM manifolds
We classify $T^2$-GKM fibrations in which both fiber and base are the GKM graph of $S^4$, with standard weights in the base. For each case in which the total space is orientable, we construct, by explicit clutching, a realization as a $T^2$-equivariant linear $S^4$-bundle over $S^4$. We determine which of the total spaces of these examples are non-equivariantly homotopy equivalent, homeomorphic or diffeomorphic, thereby finding many examples of a) pairs of homotopy equivalent, non-homeomorphic …

Lipshitz--Sarkar stable homotopy type for certain planar trivalent graphs with perfect matchings
Nilangshu Bhattacharyya
https://arxiv.org/abs/2508.12272 https://

Lipshitz--Sarkar stable homotopy type for certain planar trivalent graphs with perfect matchings
We develop a space-level refinement of the $2$-factor homology by constructing a stable homotopy type associated to a certain family $\mathscr{G}$ of planar trivalent graphs equipped with perfect matchings. Specifically, we define a cover functor from the $2$-factor flow category $\mathscr{C}(Γ_{M})$ to the cube flow category $\mathscr{C}_{C}(n)$, where the perfect matching graph $Γ_{M}$ represents a planar trivalent graph $G$ together with a perfect matching $M$, such that $(G,M) \in \mathsc…

The Groupoid-syntax of Type Theory is a Set
Thorsten Altenkirch, Ambrus Kaposi, Szumi Xie
https://arxiv.org/abs/2509.14988 https://arxiv.org/pdf/2509.14988…

The Groupoid-syntax of Type Theory is a Set
Categories with families (CwFs) have been used to define the semantics of type theory in type theory. In the setting of Homotopy Type Theory (HoTT), one of the limitations of the traditional notion of CwFs is the requirement to set-truncate types, which excludes models based on univalent categories, such as the standard set model. To address this limitation, we introduce the concept of a Groupoid Category with Families (GCwF). This framework truncates types at the groupoid level and incorporate…

On the attaching map of the 3-cell skeleton of the homotopy fibre
Juxin Yang, Fengchun Lei, Jingyan Li, Jie Wu
https://arxiv.org/abs/2509.21444 https://arx…

On the attaching map of the 3-cell skeleton of the homotopy fibre
Localize spaces at 2. Let $f\inπ_{n+k}(S^{n})$, ($n\geq2$ and $k\geq0$) and let $F$ be the homotopy fibre of the pinch map $ΣC_{f}=S^{n+1}\cup e^{n+k+2} \rightarrow S^{n+k+2}$, (in the case $k=0$, further assume $ΣC_{f}\not\Simeq*$). As a type of Gray's relative James constructions, it is known that $ F$ has a CW decomposition $F=(S^{n+1}\cup_{\af} e^{2n+k+2})\cup_β e^{3n+2k+3}\cup \cdots$, where $\af=[1_{S^{n+1}},\s f]\inπ_{n+k+1}(S^{n})$ is the Whitehead product and $β$ is a higher Whit…

Simplicial Homotopy Type Theory is not just Simplicial: What are $\infty$-Categories?
Nima Rasekh
https://arxiv.org/abs/2508.07737 https://arxiv.org/pdf/25…

Simplicial Homotopy Type Theory is not just Simplicial: What are $\infty$-Categories?
$\infty$-category theory was originally developed in the context of classical homotopy theory using standard set theoretical assumptions, but has since been extended to a variety of mathematical foundations. One such successful effort, primarily due to Martini and Wolf, introduced a theory of $\infty$-categories internal to the foundation of an arbitrary Grothendieck $\infty$-topos, meaning they used categorical foundations. Another approach, due to Riehl and Shulman, developed a theory of $\in…

The Double Copy of Maximal Supersymmetry in $D=10$
Roberto Bonezzi, Giuseppe Casale, Olaf Hohm
https://arxiv.org/abs/2508.11753 https://arxiv.org/pdf/2508.…

The Double Copy of Maximal Supersymmetry in $D=10$
We continue the program of using homotopy algebras to obtain off-shell, local and gauge redundant derivations of the double copy relations between gauge theory and gravity. We apply it to $N=1$ super-Yang-Mills theory in $D=10$ in order to obtain type IIA or type IIB supergravity, at least to cubic order in fields. Furthermore, we show how the super-Lie algebra of global supersymmetries, acting on the homotopy algebra of $N=1$ super-Yang-Mills theory, double copies to the maximal supersymmetry …

Non-negative curvature on certain product manifolds
Wen Shen
https://arxiv.org/abs/2508.15194 https://arxiv.org/pdf/2508.15194

Non-negative curvature on certain product manifolds
Let $G/H$ be a closed, simply connected homogeneous manifold. Suppose every stable class of real vector bundles over $G/H$ contains a homogeneous bundle. Then, for any closed, simply connected smooth manifold $M$ homotopy equivalent to $G/H$, there exists $n>\mathrm{dim}(M)$ such that the product manifold $M\times S^{n}$ admits a metric with non-negative sectional curvature. Many homogeneous manifolds satisfy this assumption, including simply connected compact rank-one symmetric spaces, and amo…

Obstruction sequences to homotopy equivalences
Coline Emprin
https://arxiv.org/abs/2509.17895 https://arxiv.org/pdf/2509.17895

Obstruction sequences to homotopy equivalences
We develop an obstruction theory for the existence of gauge equivalences in complete differential graded Lie algebras. Specifically, this theory provides a characterization of homotopy equivalences between differential graded algebras governed by operads or properads, potentially colored in a groupoid. We apply this framework to establish new homotopy equivalence results in both algebraic topology and algebraic geometry, with a particular focus on the study of minimal models for highly connecte…

Homotopy classification of $4$-manifolds with $3$-manifold fundamental group
Jonathan Hillman, Daniel Kasprowski, Mark Powell, Arunima Ray
https://arxiv.org/abs/2508.07504 https…

Homotopy classification of $4$-manifolds with $3$-manifold fundamental group
We give a criterion on a group $π$ and a homomorphism $w \colon π\to C_2$ under which closed $4$-manifolds with fundamental group $π$ and orientation character $w$ are classified up to homotopy equivalence by their quadratic $2$-types. We verify the criterion for a large class of $3$-manifold groups and orientation characters, in particular for the fundamental group $π$ of any closed, orientable $3$-manifold whose finite subgroups are cyclic, provided $w$ vanishes on every element of $π$ o…

Crosslisted article(s) found for math.AT. https://arxiv.org/list/math.AT/new
[1/1]:
- Homotopy Languages
C\'esar Bardomiano Mart\'inez, Simon Henry
http…

$\mathbb{A}^1$-invariant motivic cohomology of schemes
Tom Bachmann, Elden Elmanto, Matthew Morrow
https://arxiv.org/abs/2508.09915 https://arxiv.org/pdf/2…

$\mathbb{A}^1$-invariant motivic cohomology of schemes
Voevodsky outlined a conjectural programme that his slice filtration in motivic homotopy theory should give rise to a good theory of $\mathbb{A}^1$-invariant motivic cohomology. This paper achieves his vision in the generality of arbitrary quasicompact, quasiseparated schemes, by introducing a theory of $\mathbb{A}^1$-invariant motivic cohomology which is related to Weibel's homotopy $K$-theory via an Atiyah--Hirzebruch spectral sequence, and which we compare to étale and syntomic cohomology i…

Rational K3 Homotopy and the Largest Mathieu Group
Federico Carta, John F. R. Duncan, Yang-Hui He
https://arxiv.org/abs/2509.18969 https://arxiv.org/pdf/25…

Homotopy classification of $S^{2k-1}$-bundles over $S^{2k}$
Zhongjian Zhu, Jianzhong Pan
https://arxiv.org/abs/2508.14341 https://arxiv.org/pdf/2508.14341

Homotopy classification of $S^{2k-1}$-bundles over $S^{2k}$
In this paper, we classify the homotopy types of the total spaces of $S^{2k-1}$-bundles (or fibrations) over $S^{2k}$ for $2\leq k\leq 6$ by listing the attaching map of the top cell of these as CW-complexes. The method is using the new necessary and sufficient conditions we gave in our previous work to determine whether a CW complex has the homotopy type of total space of these sphere bundles.

Transversality Methods for Homotopy Groups of Stable Loci in Affine GIT Quotients
Yizhi Wang
https://arxiv.org/abs/2508.06213 https://arxiv.org/pdf/2508.06…

Transversality Methods for Homotopy Groups of Stable Loci in Affine GIT Quotients
We investigate the homotopy groups of stable loci in affine Geometric Invariant Theory (GIT), arising from linear actions of complex reductive algebraic groups on complex affine spaces. Our approach extends the infinite-dimensional transversality framework of Daskalopoulos-Uhlenbeck and Wilkin to this general GIT setting. Central to our method is the construction of a G-equivariant holomorphic vector bundle over the conjugation orbit of a one-parameter subgroup (1-PS), whose fibres are precisel…

A new approach to rational stable parametrized homotopy theory
Yves F\'elix, Aniceto Murillo, Alejandro Saiz
https://arxiv.org/abs/2509.11393 https://a…

A new approach to rational stable parametrized homotopy theory
This work develops a comprehensive algebraic model for rational stable parametrized homotopy theory over arbitrary base spaces. Building on the simplicial analogue of the foundational framework of May-Sigurdsson for parametrized spectra, and the homotopy theory of complete differential graded Lie algebras, we construct an explicit sequence of Quillen equivalences that translate the homotopy theory of rational spectra of retractive simplicial sets into the purely algebraic framework of complete …

Linking numbers for periodic tangles
Yuka Kotorii, Ken'ichi Yoshida
https://arxiv.org/abs/2509.24568 https://arxiv.org/pdf/2509.24568

Linking numbers for periodic tangles
Periodic tangles are 1-dimensional submanifolds in the 3-space with translational symmetry. In this paper, we define the linking numbers for singly, doubly, and triply periodic tangles using appropriate motifs and show that they are well-defined and invariant under link-homotopy. Furthermore, we extend the notion to the higher order linking numbers for singly periodic tangles.

B-RNS-GSS formalism and $L_{\infty}$-actions
Andrei Mikhailov
https://arxiv.org/abs/2510.10400 https://arxiv.org/pdf/2510.10400

B-RNS-GSS formalism and $L_{\infty}$-actions
Pure spinor formalism and RNS formalism are related by a chain of equivalences constructed by introducing and integrating-out BRST quartets. This is known as B-RNS-GSS formalism. One of the steps can be understood as adding auxiliary fields to lift a strong homotopy action of the SUSY Lie superalgebra in the large Hilbert space to a strict action. We develop a general prescription for this ``strictification'' procedure, which can be applied for any strong homotopy action of a Lie superalgebra. …

Homotopy types of S^{2k-1}-bundles over S^{2k}
Zhongjian Zhu, Jianzhong Pan
https://arxiv.org/abs/2508.13800 https://arxiv.org/pdf/2508.13800

Homotopy types of S^{2k-1}-bundles over S^{2k}
In this paper, we give the new necessary and sufficient conditions on the attaching map $f$ of the top cell of a CW-complex having the homotopy type of the total space of $S^{2k-1}$-fibration over $S^{2k}$ for any $k\geq 2$. As an application, we give the homotopy classification of the $S^{2k-1}$-bundles over $S^{2k}$ for some specific numbers $k\geq 2$ in our subsequent paper.

Branching space of precubical set
Philippe Gaucher
https://arxiv.org/abs/2508.14839 https://arxiv.org/pdf/2508.14839

Branching space of precubical set
Using the notion of short natural directed path, we introduce the homotopy branching space of a precubical set. It is unique only up to homotopy equivalence. We prove that, for any precubical set, it is homotopy equivalent to the branching space of any q-realization, any m-realization and any h-realization of the precubical set as a flow. As an application, we deduce the invariance of the homotopy branching space and of the branching homology up to cubical subdivision. By reversing the time dir…

Motivic Homotopy Groups of Spheres and Free Summands of Stably Free Modules
Sebastian Gant, Ben Williams
https://arxiv.org/abs/2510.10033 https://arxiv.org…

Motivic Homotopy Groups of Spheres and Free Summands of Stably Free Modules
Working over an algebraically closed field of characteristic $0$, we show that the motivic stable homotopy groups of the sphere spectrum can be determined entirely from the motivic homotopy groups of the $p$-completed sphere spectra and the motivic cohomology of the ground field, except possibly for the $0$ and $-1$-stems. Using this, we show that the complex realization map from the motivic cohomology group to the classical stable homotopy group is an isomorphism in a range of bidegrees. We ap…

Modular forms for chromatic homotopy: Supersingular congruences
Ken Ono
https://arxiv.org/abs/2509.16175 https://arxiv.org/pdf/2509.16175

Modular forms for chromatic homotopy: Supersingular congruences
In this note, we confirm a conjecture of Larson that arises in the Adams--Novikov spectral sequence (ANSS) for the stable homotopy groups of spheres and, specifically, in Behrens' program on explicit modular forms detecting $v_2$--periodic classes in the divided $β$-family. The conjecture predicts the supersingular order of the weight $12t$ form $L_2(Δ^t)$, when $(p-1)\mid 12t,$ attached to the $Γ_0(2)$ Hecke correspondence. We prove the prediction for all primes $p\ge5$, thereby providing t…

Exotic diffeomorphisms on a contractible 4-manifold surviving two stabilizations
Sungkyung Kang, JungHwan Park, Masaki Taniguchi
https://arxiv.org/abs/2510.12394 https://…

Exotic diffeomorphisms on a contractible 4-manifold surviving two stabilizations
We develop a $\mathrm{Pin}(2) \times \mathbb{Z}_2$-equivariant refinement of the lattice homotopy type for computing equivariant Seiberg--Witten Floer homotopy types. As an application, we construct a relatively exotic diffeomorphism on a compact contractible 4-manifold that survives two stabilizations.

A motivic approach to rational $p$-adic cohomologies
Federico Binda, Alberto Vezzani
https://arxiv.org/abs/2508.16196 https://arxiv.org/pdf/2508.16196

A motivic approach to rational $p$-adic cohomologies
We survey over some recent applications of motivic homotopy theory in the definition and the study of $p$-adic cohomology theories. In particular, we revisit the proof of the $p$-adic weight-monodromy conjecture for smooth projective hypersurfaces in light of the motivic definition of nearby cycles and monodromy operators.

The Rational Homotopy of Stable $C_p$-Smoothings
Oliver H. Wang
https://arxiv.org/abs/2510.11622 https://arxiv.org/pdf/2510.11622

The Rational Homotopy of Stable $C_p$-Smoothings
Smooth structures on high dimensional manifolds are classified by maps to the infinite loop space $TOP/O$. The homotopy groups of this space are known to be finite. Given a compact Lie group $G$, this space can be regarded as an equivariant infinite loop space and equivariant maps from a locally linear, high dimensional $G$-manifold to $TOP/O$ classify stable $G$-smoothings. We compute the equivariant homotopy groups $π_V^{C_p}TOP/O\otimes\Q$ where $C_p$ denotes the cyclic group of order $p$.

On the Coarse Lusternik-Schnirelmann Category of Groups
Aditya De Saha
https://arxiv.org/abs/2510.10367 https://arxiv.org/pdf/2510.10367

On the Coarse Lusternik-Schnirelmann Category of Groups
We introduce a coarse analog of the classical Lusternik-Schnirelmann category which we denote by $\text{c-cat}$, defined for metric spaces in the coarse homotopy category. This provides a new tool for studying large-scale topological properties of groups and spaces. We establish that $\text{c-cat}$ is a coarse homotopy invariant and prove a lower-bound $\text{p-cat}(Γ)\leq \text{c-cat}(Γ)$ for geometrically finite groups $Γ$, where $\text{p-cat}$ denotes the proper LS-category introduced in …

Operadic twisting as an adjunction
Guillaume Laplante-Anfossi, Adrian Petr, Vivek Shende
https://arxiv.org/abs/2510.02229 https://arxiv.org/pdf/2510.02229

Operadic twisting as an adjunction
For operads with a map from the curved homotopy Lie operad, we introduce a corresponding curved variant `cTw' of Willwacher's operadic twisting comonad `Tw'. We show that cTw-coalgebra structures on such an operad are in bijection with certain splittings (not respecting the differential) of the projection to its quotient by the curvature operation. We derive a similar classification of Tw-coalgebras. For the class of operads whose Koszul dual admits a unital extension, we give explicit formul…

A motivic proof of the finiteness of relative de Rham cohomology
Alberto Vezzani
https://arxiv.org/abs/2508.17773 https://arxiv.org/pdf/2508.17773

A motivic proof of the finiteness of relative de Rham cohomology
We give a quick proof of the fact that the relative de Rham cohomogy groups of a smooth and proper map X/S between schemes over Q are vector bundles on the base, replacing Hodge-theoretic and transcendental methods with A1-homotopy theory

Spectral moduli problems for level structures and an integral Jacquet-Langlands dual of Morava E-theory
Xuecai Ma, Yifei Zhu
https://arxiv.org/abs/2509.00690 https://

Spectral moduli problems for level structures and an integral Jacquet-Langlands dual of Morava E-theory
Given an E-infinity ring spectrum R, with motivation from chromatic homotopy theory, we define relative effective Cartier divisors for a spectral Deligne-Mumford stack over Spet(R) and prove that, as a functor from connective R-algebras to topological spaces, it is representable. This enables us to solve various moduli problems of level structures on spectral abelian varieties, overcoming difficulty at primes dividing the level. In particular, we obtain higher-homotopical refinement for finite …

The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra
Takahito Kuriya
https://arxiv.org/abs/2508.18985 https://arxiv…

The LMO Spectrum: Factorization Homology and the E_3-Structure of the Jacobi Diagram Algebra
This paper introduces a framework for the categorification of the Le-Murakami-Ohtsuki (LMO) invariant of 3-manifolds, defining a new invariant, the LMO Spectrum, via factorization homology. The theoretical foundation of this framework is the main algebraic result of this paper (Theorem A): a proof that the algebra of Jacobi diagrams, A_Jac, possesses the structure of a homotopy E3-algebra. This algebraic structure is shown to be a consequence of 3-dimensional geometry, originating from a "Tetra…

On reductions of Selmer sections
Wojciech Porowski
https://arxiv.org/abs/2509.15660 https://arxiv.org/pdf/2509.15660

On reductions of Selmer sections
We consider reductions of Selmer sections of the étale homotopy sequence of a hyperbolic curve over a number field. We show that the conjugacy class of a noncuspidal Selmer section is uniquely determined by its reduction on a set of density one. Moreover, we show that a noncuspidal Selmer section reduces to cusps only on a set of density zero, at least after a finite field extension.

Operads of moduli spaces of points in $\mathbb{C}^d$ revisited
Vladimir Dotsenko, Eduardo Hoefel, Sergey Shadrin, Grigory Solomadin
https://arxiv.org/abs/2509.20331 https://

Operads of moduli spaces of points in $\mathbb{C}^d$ revisited
We study the operad structure on the homology of moduli spaces of pointed rooted trees of $d$-dimensional projective spaces, introduced by Chen, Gibney and Krashen a couple of decades ago. We describe this operad by generators and relations, show that it is homotopy Koszul, exhibit a Givental-type action on representations of that operad, and prove that this operad represents the homotopy quotient of the operad of chains of $S^1$-framed little $2d$-disks by its natural circle action. Our approa…

Hermitian $K$-theory and Milnor-Witt motivic cohomology over $\mathbb Z$
H{\aa}kon Kolderup, Oliver R\"ondigs, Paul Arne {\O}stv{\ae}r
https://arxiv.org/abs/2509.16404 http…

Hermitian $K$-theory and Milnor-Witt motivic cohomology over $\mathbb Z$
The theme of this paper is to compute hermitian $K$-groups in terms of the recently developed theory of Milnor-Witt motivic cohomology. Our approach makes use of the very effective slice spectral sequence within the motivic stable homotopy category, which we analyze in detail for base schemes of arithmetic interest. We show a Grothendieck-Riemann-Roch theorem, determine the map between Milnor-Witt and hermitian $K$-theory up to degree five for all fields, and compute the hermitian $K$-groups an…

Classification for smooth manifolds looking like $\mathbb{CP}^3\times S^7$
Wen Shen
https://arxiv.org/abs/2510.00613 https://arxiv.org/pdf/2510.00613

Classification for smooth manifolds looking like $\mathbb{CP}^3\times S^7$
In this paper, we classify simply connected closed smooth $13$-dimensional manifolds whose cohomology ring is isomorphic to that of $\mb{CP}^3\times S^7$, up to diffeomorphism, homeomorphism, and homotopy equivalence. Furthermore, if such a manifold satisfies certain conditions, either itself or its connected sum with an exotic $13$-sphere $Σ^{13}$ admits a Riemannian metric of non-negative sectional curvature. As an additional application of our classification, we classify the diagonal $S^1$-…

Almost-concordance of knots in aspherical 3-manifolds
Ryan Stees
https://arxiv.org/abs/2508.14638 https://arxiv.org/pdf/2508.14638

Almost-concordance of knots in aspherical 3-manifolds
In this paper, we study topological concordance modulo local knotting, or almost-concordance, of knots in 3-manifolds $M\neq S^3$. A. Levine, Celoria (arXiv:1602.05476v4), and Friedl-Nagel-Orson-Powell (arXiv:1611.09114v2) conjecture that, absent the presence of an embedded dual 2-sphere, any free homotopy class $x$ of knots in $M$ contains infinitely many concordance classes modulo the action of the concordance group of knots in $S^3$ by local knotting. We develop a method for confirming this …

Replaced article(s) found for math.AG. https://arxiv.org/list/math.AG/new
[1/1]:
- Unipotent homotopy theory of schemes
Shubhodip Mondal, Emanuel Reinecke
https…

Models for rational $(\infty, 1)$-categories
Eleftherios Chatzitheodoridis
https://arxiv.org/abs/2509.22413 https://arxiv.org/pdf/2509.22413

Models for rational $(\infty, 1)$-categories
We introduce rational $(\infty, 1)$-categories, which are $(\infty, 1)$-categories enriched in spaces whose higher homotopy groups are rational vector spaces. We provide two models for rational $(\infty, 1)$-categories, rational complete Segal spaces and rational Segal categories, and we show that they are equivalent.

On Topology of the Infinite-Dimensional Space of Fibrations
Ziqi Fang
https://arxiv.org/abs/2508.14038 https://arxiv.org/pdf/2508.14038

On Topology of the Infinite-Dimensional Space of Fibrations
This work serves as an opening and basis of an ongoing program investigating topological and geometric aspects of the moduli space of smooth fiberings on a manifold. The present paper focuses on the algebraic and differential topology of this space, and particularly addresses the following three quests in a top-down manner: the classification, for each class the path components, and for each component the homotopy type (as loosely analogous to the those three for studying the moduli of smooth s…

Equivariant homotopic distance
Navnath Daundkar, J. M. Garc\'ia-Calcines
https://arxiv.org/abs/2508.20485 https://arxiv.org/pdf/2508.20485

Equivariant homotopic distance
In this paper, we introduce and investigate the equivariant homotopic distance between G-maps, an equivariant homotopy invariant that extends the notion of homotopic distance to the setting of spaces equipped with a continuous action of a topological group. This new invariant provides a unifying framework that generalizes both the equivariant Lusternik-Schnirelmann category and the equivariant topological complexity. We establish fundamental properties of the equivariant homotopic distance, inc…

Replaced article(s) found for math.AG. https://arxiv.org/list/math.AG/new
[1/1]:
- Punctured tubular neighborhoods and stable homotopy at infinity
Fr\'ed\'eric D\'eglise, Adrien Dubouloz, Paul Arne {\O}stv{\ae}r

A chromatic approach to homological stability
Oscar Randal-Williams
https://arxiv.org/abs/2508.20629 https://arxiv.org/pdf/2508.20629

A chromatic approach to homological stability
We propose a way to organise the subject of ``higher-order homological stability'', in the context of a graded $E_2$-algebra $\mathbf{R}$, along the same lines that the chromatic perspective organises stable homotopy theory. From this point of view proving a (higher-order) homological stability theorem corresponds to producing Smith--Toda complexes in the category of $\mathbf{R}$-modules: using this perspective we prove that whenever $\mathbf{R}$ is defined over a field of positive characteri…

Rigidifying simplicial complexes and realizing group actions
Cristina Costoya, Rafael Gomes, Antonio Viruel
https://arxiv.org/abs/2509.09646 https://arxiv.…

Rigidifying simplicial complexes and realizing group actions
We show that any action of a finite group on a finitely presentable group arises as the action of the group of self-homotopy equivalences of a space on its fundamental group. In doing so, we prove that any finite connected (abstract) simplicial complex $\mathbf{K}$ can be rigidified -- meaning it can be perturbed in a way that reduces the full automorphism group to any subgroup -- while preserving the homotopy type of the geometric realization $| \mathbf{K} |$.

Framed configuration spaces and exotic spheres
Manuel Krannich, Alexander Kupers, Fadi Mezher
https://arxiv.org/abs/2509.19074 https://arxiv.org/pdf/2509.1…

Cohomology of non-finite CL-shellable posets
Guille Carri\'on Santiago, Antonio D\'iaz Ramos
https://arxiv.org/abs/2509.17574 https://arxiv.org/pdf…

Cohomology of non-finite CL-shellable posets
Shellable complexes are homotopy equivalent to a wedge of spheres of possibly different dimensions, so that the (co)homology of the constant functor over the complex is concentrated in those degrees. In this work, we introduce the concept of a stable functor -a local weakening of fibrancy- over a shellable poset, which ensures the vanishing of the (co)homology of such a functor in specific degrees. In addition, we extend the notion of shellability to non-finite and non-pure posets. The methods …

On the complexity of parametrized motion planning algorithms
Navnath Daundkar, Ekansh Jauhari
https://arxiv.org/abs/2508.17629 https://arxiv.org/pdf/2508.1…

On the complexity of parametrized motion planning algorithms
We study a probabilistic variant of the r-th sequential parametrized topological complexity, which bounds this classical invariant from below and measures the difficulty in constructing permissive parametrized motion planning algorithms. On one hand, we use cohomology to show that this new invariant behaves similarly to the classical invariant on Fadell-Neuwirth fibrations and oriented sphere bundles; on the other hand, we use equivariant homotopy theory to prove that its behavior is wildly dif…

Crosslisted article(s) found for math.AT. https://arxiv.org/list/math.AT/new
[1/1]:
- Homotopy classification of closed polygonal lines: results and problems
E. Alkin, O. Nikitenko, A. Skopenkov

Enriched model categories and the Dold-Kan correspondence
Martin Frankland, Arnaud Ngopnang Ngomp\'e
https://arxiv.org/abs/2508.14291 https://arxiv.org…

Enriched model categories and the Dold-Kan correspondence
The monoidal properties of the Dold-Kan correspondence have been studied in homotopy theory, notably by Schwede and Shipley. Changing the enrichment of an enriched, tensored, and cotensored category along the Dold-Kan correspondence does not preserve the tensoring nor the cotensoring. More generally, what happens to an enriched model category if we change the enrichment along a weak monoidal Quillen pair? We prove a change of base theorem that describes which properties are preserved and which …

A new approach to (3 1)-dimensional TQFTs via topological modular forms
Sergei Gukov, Vyacheslav Krushkal, Lennart Meier, Du Pei
https://arxiv.org/abs/2509.12402 https://…

A new approach to (3+1)-dimensional TQFTs via topological modular forms
In this paper, we present a construction toward a new type of TQFTs at the crossroads of low-dimensional topology, algebraic geometry, physics, and homotopy theory. It assigns TMF-modules to closed 3-manifolds and maps of TMF-modules to 4-dimensional cobordisms. This is a mathematical proposal for one of the simplest examples in a family of $π_*({\rm TMF})$-valued invariants of 4-manifolds which are expected to arise from 6-dimensional superconformal field theories. As part of the construction…

Replaced article(s) found for math.AT. https://arxiv.org/list/math.AT/new
[1/1]:
- The \v{C}ech homotopy groups of a shrinking wedge of spheres
Jeremy Brazas
ht…

Replaced article(s) found for math.AT. https://arxiv.org/list/math.AT/new
[1/1]:
- Massey products for homotopy inner products
Kate Poirier, Thomas Tradler, Scott O. Wilson

The cohomology of $\mathbb{R}$-motivic $\mathcal{A}(2)$
Konstantin Emming
https://arxiv.org/abs/2509.11266 https://arxiv.org/pdf/2509.11266

The cohomology of $\mathbb{R}$-motivic $\mathcal{A}(2)$
We compute the cohomology of the quotient algebra $\mathcal{A}(2)$ of the $\mathbb{R}$-motivic dual Steenrod algebra. We do so by running a $ρ$-Bockstein spectral sequence whose input is the cohomology of $\mathbb{C}$-motivic $\mathcal{A}(2)$. The purpose of our computation is that the cohomology of $\mathcal{A}(2)$ is the input to an Adams spectral sequence of a hypothetical $\mathbb{R}$-motivic modular forms spectrum. This Adams spectral sequence computes the homotopy groups of such an $\mat…

Derived mapping spaces of $\infty$-categories
Kensuke Arakawa, Daniel Carranza, Chris Kapulkin
https://arxiv.org/abs/2509.10288 https://arxiv.org/pdf/2509.…

Derived mapping spaces of $\infty$-categories
We prove the Derived Mapping Space Lemma, which generalizes the central theorem of Cisinski's work on calculus of fractions for $\infty$-categories, and allows us to provide a unified framework for analyzing mapping spaces in localizations of ($\infty$-)categories. As an application, we give a sufficient condition for when a cubical or simplicial category is the localization of its underlying category at homotopy equivalences.