Tootfinder

A Computational Approach to the Homotopy Theory of DG categories
Dogancan Karabas, Sangjin Lee
https://arxiv.org/abs/2405.03258 https://

A Computational Approach to the Homotopy Theory of DG categories
We give a specific cylinder functor for semifree dg categories. This allows us to construct a homotopy colimit functor explicitly. These two functors are "computable", specifically, the constructed cylinder functor sends a dg category of finite type, i.e., a semifree dg category having finitely many generating morphisms, to a dg category of finite type. The homotopy colimit functor has a similar property. Moreover, using the cylinder functor, we give a cofibration category of semifree dg catego…

This https://arxiv.org/abs/2403.00373 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Frobenius rigidity in $\mathbb A^1$-homotopy theory
We study the homotopy fixed points under the Frobenius endomorphism on the stable $\mathbb A^1$-homotopy category of schemes in characteristic $p>0$ and prove a rigidity result for cellular objects in these categories after inverting $p$. As a consequence we determine the analogous fixed points on the $K$-theory of algebraically closed fields in positive characteristic. We also prove a rigidity result for the homotopy fixed points of the partial Frobenius pullback on motivic cohomology groups i…

This https://arxiv.org/abs/2309.15089 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Equivariant Floer homotopy via Morse-Bott theory
We generalize the Cohen-Jones-Segal construction to the Morse-Bott setting. In other words, we define framings for Morse-Bott analogues of flow categories and associate a stable homotopy type to this data. We use this to recover the stable homotopy type of a closed manifold from Morse-Bott theory, and the stable equivariant homotopy type of a closed manifold with the action of a compact Lie group from Morse theory. We use this machinery in Floer theory to construct a genuine circle equivarian…

Homology and homotopy for arbitrary categories
Suddhasattwa Das
https://arxiv.org/abs/2404.03735 https://arxiv.org/pdf/2404.03735

Homology and homotopy for arbitrary categories
One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was discovered with the development of Homology theory. Some of the deepest results in topology are about the connections between Homotopy and Homology. These results are proved using intricate constructions. This paper takes an axiomatic approach that provides a comm…

Homotopy methods for higher order shape optimization: A globalized shape-Newton method and Pareto-front tracing
A. Cesarano, B. Endtmayer, P. Gangl
https://arxiv.org/abs/2405.03421

Homotopy methods for higher order shape optimization: A globalized shape-Newton method and Pareto-front tracing
First order shape optimization methods, in general, require a large number of iterations until they reach a locally optimal design. While higher order methods can significantly reduce the number of iterations, they exhibit only local convergence properties, necessitating a sufficiently close initial guess. In this work, we present an unregularized shape-Newton method and combine shape optimization with homotopy (or continuation) methods in order to allow for the use of higher order methods even…

This https://arxiv.org/abs/2102.07894 has been replaced.
link: https://scholar.google.com/scholar?q=a

The path-missing and path-free complexes of a directed graph
We study two simplicial complexes arising from a directed graph $G = (V, E)$ with two chosen vertices $s$ and $t$: the *path-free complex*, consisting of all subsets $F \subseteq E$ that contain no path from $s$ to $t$, and the *path-missing complex*, its Alexander dual. Using discrete Morse theory, we prove that both complexes have well-behaved homotopy types -- either contractible or homotopy-equivalent to spheres.

Immersions of 3-manifolds into the 5-space, almost contact structures, and complex tangents
Masato Tanabe
https://arxiv.org/abs/2405.02513 https://<…

Immersions of 3-manifolds into the 5-space, almost contact structures, and complex tangents
The Wu invariant is a regular homotopy invariant for immersions of oriented 3-manifolds into the 5-space. In this paper, we present new expressions and vanishing theorems for the invariant, from the viewpoint of almost contact structures and complex tangents. As an application, we determine the regular homotopy classes of inclusion maps of the surface singularity links of type ADE into the 5-sphere.

This https://arxiv.org/abs/2301.05932 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On continuation and convex Lyapunov functions
Suppose that the origin is globally asymptotically stable under a set of continuous vector fields on Euclidean space and suppose that all those vector fields come equipped with -- possibly different -- convex Lyapunov functions. We show that this implies there is a homotopy between any two of those vector fields such that the origin remains globally asymptotically stable along the homotopy. Relaxing the assumption on the origin to any compact convex set or relaxing convexity to geodesic convexi…

Regularity of semi-valuation rings and homotopy invariance of algebraic K-theory
Christian Dahlhausen
https://arxiv.org/abs/2403.02413 https://

Regularity of semi-valuation rings and homotopy invariance of algebraic K-theory
We show that the algebraic K-theory of semi-valuation rings with stably coherent regular semi-fraction ring satisfies homotopy invariance. Moreover, we show that these rings are regular if their valuation is non-trivial. Thus they yield examples of regular rings which are not homotopy invariant for algebraic K-theory. On the other hand, they are not necessarily coherent, so that they provide a class of possibly non-coherent examples for homotopy invariance of algebraic K-theory. As an applicati…

This https://arxiv.org/abs/2102.07894 has been replaced.
link: https://scholar.google.com/scholar?q=a

The path-missing and path-free complexes of a directed graph
We study two simplicial complexes arising from a directed graph $G = (V, E)$ with two chosen vertices $s$ and $t$: the *path-free complex*, consisting of all subsets $F \subseteq E$ that contain no path from $s$ to $t$, and the *path-missing complex*, its Alexander dual. Using discrete Morse theory, we prove that both complexes have well-behaved homotopy types -- either contractible or homotopy-equivalent to spheres.

This https://arxiv.org/abs/2311.16642 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The homotopy decomposition of the suspension of a non-simply-connected $5$-manifold
In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth $5$-manifolds. As applications, we compute the reduced $K$-groups of $M$ and show that the suspension map between the third cohomotopy set $π^3(M)$ and the fourth cohomotopy set $π^4(ΣM)$ is a bijection.

Foundation of Floer homotopy theory I: Flow categories
Mohammed Abouzaid, Andrew J. Blumberg
https://arxiv.org/abs/2404.03193 https://

Foundation of Floer homotopy theory I: Flow categories
We construct a stable infinity category with objects flow categories and morphisms flow bimodules; our construction has many flavors, related to a choice of bordism theory, and we discuss in particular framed bordism and the bordism theory of complex oriented derived orbifolds. In this setup, the construction of homotopy types associated to Floer-theoretic data is immediate: the moduli spaces of solutions to Floer's equation assemble into a flow category with respect to the appropriate bordism …

Homotopical Minimal Measures for Geodesic flows on Surfaces of Higher Genus
Fang Wang, Zhihong Xia
https://arxiv.org/abs/2403.04452 https://

Homotopical Minimal Measures for Geodesic flows on Surfaces of Higher Genus
We study the homotopical minimal measures for positive definite autonomous Lagrangian systems. Homotopical minimal measures are action-minimizers in their homotopy classes, while the classical minimal measures (Mather measures) are action-minimizers in homology classes. Homotopical minimal measures are much more general, they are not necessarily homological action-minimizers. However, some of them can be obtained from the classical ones by lifting them to finite-fold covering spaces. We apply t…

Frobenius rigidity in $\mathbb A^1$-homotopy theory
Timo Richarz, Jakob Scholbach
https://arxiv.org/abs/2403.00373 https://arxiv.org/…

Frobenius rigidity in $\mathbb A^1$-homotopy theory
We study the homotopy fixed points under the Frobenius endomorphism on the stable $\mathbb A^1$-homotopy category of schemes in characteristic p>0 and prove a rigidity result for cellular objects in these categories after inverting p. As a consequence we determine the analogous fixed points on the K-theory of algebraically closed fields in positive characteristic. We also prove a rigidity result for the homotopy fixed points of the partial Frobenius pullback on motivic cohomology groups in weig…

Cofinality Theorems of Infinity Categories and Algebraic K-Theory
Hisato Matsukawa
https://arxiv.org/abs/2405.03498 https://arxiv.org…

Cofinality Theorems of Infinity Categories and Algebraic K-Theory
In this paper, we establish a theorem that proves a condition when an inclusion morphism between simplicial sets becomes a weak homotopy equivalence. Additionally, we present two applications of this result. The first application demonstrates that cofinal full inclusion functors of (\infty)-categories are weak homotopy equivalences. For our second application, we provide an alternative proof of Barwick's cofinality theorem of algebraic (K)-theory.

This https://arxiv.org/abs/2404.19747 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Obstruction Complexes in Grid Homology
Recently, Manolescu-Sarkar constructed a stable homotopy type for link Floer homology, which uses grid homology and accounts for all domains that do not pass through a specific square. In doing so, they produced an obstruction chain complex of the grid diagram with that square removed. We define the obstruction chain complex of the full grid, without the square removed, and compute its homology. Though this homology is too complicated to immediately extend the Manolescu-Sarkar construction, we …

This https://arxiv.org/abs/2312.00382 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_hepp…

The effective potential of composite operator in the first order region of QCD phase transition
We propose a method to determine the effective potential of QCD from the gap equation, by introducing the homotopy method between the solutions of the equation of motion. Via this method, the effective potential beyond the bare vertex approximation is obtained, which then generalizes the Cornwall, Jackiw and Tomboulis (CJT) effective potential for the bilocal composite operators. Moreover, the extended effective potential is set to be a function of self energy instead of the propagator, which i…

This https://arxiv.org/abs/2312.11022 has been replaced.
link: https://scholar.google.com/scholar?q=a

Solving mathematical programs with complementarity constraints arising in nonsmooth optimal control
This paper examines solution methods for mathematical programs with complementarity constraints (MPCC) obtained from the time-discretization of optimal control problems (OCPs) subject to nonsmooth dynamical systems. The MPCC theory and stationarity concepts are reviewed and summarized. The focus is on relaxation-based methods for MPCCs, which solve a (finite) sequence of more regular nonlinear programs (NLP), where a regularization/homotopy parameter is driven to zero. Such methods perform reas…

This https://arxiv.org/abs/2302.08585 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Numerical Nonlinear Algebra
Numerical nonlinear algebra is a computational paradigm that uses numerical analysis to study polynomial equations. Its origins were methods to solve systems of polynomial equations based on the classical theorem of Bézout. This was decisively linked to modern developments in algebraic geometry by the polyhedral homotopy algorithm of Huber and Sturmfels, which exploits the combinatorial structure of the equations and led to efficient software for solving polynomial equations. Subsequent grow…

A support preserving homotopy for the de Rham complex with boundary decay estimates
Andrea N\"utzi
https://arxiv.org/abs/2404.18005 https://

A support preserving homotopy for the de Rham complex with boundary decay estimates
We study the de Rham complex of relative differential forms on compact manifolds with boundary. Chain homotopies for this complex are highly non-unique, and different homotopies can have different analytic properties, particularly near the boundary. We construct a chain homotopy that has desirable support propagation properties, and that satisfies estimates relative to weighted Sobolev norms, where the weights measure decay at the boundary. The estimates are optimal given the homogeneity proper…

This https://arxiv.org/abs/2307.14028 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Knotted families from graspers
For any smooth manifold $M$ of dimension $d\geq4$ we construct explicit classes in homotopy groups of spaces of embeddings of either an arc or a circle into $M$, in every degree that is a multiple of $d-3$, and show that they are detected in the Taylor tower of Goodwillie and Weiss. The classes are obtained from families of string links constructed in the $d$-ball.

homotopy.io: a proof assistant for finitely-presented globular $n$-categories
Nathan Corbyn, Lukas Heidemann, Nick Hu, Chiara Sarti, Calin Tataru, Jamie Vicary
https://arxiv.org/abs/2402.13179

homotopy.io: a proof assistant for finitely-presented globular $n$-categories
We present the proof assistant homotopy.io for working with finitely-presented semistrict higher categories. The tool runs in the browser with a point-and-click interface, allowing direct manipulation of proof objects via a graphical representation. We describe the user interface and explain how the tool can be used in practice. We also describe the essential subsystems of the tool, including collapse, contraction, expansion, typechecking, and layout, as well as key implementation details inclu…

This https://arxiv.org/abs/2212.06937 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the $\infty$-topos semantics of homotopy type theory
Many introductions to homotopy type theory and the univalence axiom gloss over the semantics of this new formal system in traditional set-based foundations. This expository article, written as lecture notes to accompany a 3-part mini course delivered at the Logic and Higher Structures workshop at CIRM-Luminy, attempts to survey the state of the art, first presenting Voevodsky's simplicial model of univalent foundations and then touring Shulman's vast generalization, which provides an interpreta…

Mapping fiber, loop and suspension graphs in naive discrete homotopy theory
So Yamagata
https://arxiv.org/abs/2402.15714 https://arxi…

Mapping fiber, loop and suspension graphs in naive discrete homotopy theory
Discrete homotopy theory or A-homotopy theory is a combinatorial homotopy theory defined on graphs, simplicial complexes and metric spaces, reflecting information about their connectivity. The purpose of the present paper is to further understanding of the similarities between the A-homotopy and ordinary homotopy theories. More precisely, we define the notion of a mapping fiber in the category of graphs and study its basic properties, providing non-compatibility with projection.

Homotopy types of diagrams of chain complexes
David Blanc, Surojit Ghosh, Aziz Kharoof
https://arxiv.org/abs/2404.03589 https://arxiv…

Homotopy types of diagrams of chain complexes
We study the homotopy theory of diagrams of chain complexes over a field indexed by a finite poset, and show that it can be completely described in terms of appropriate diagrams of graded vector spaces.

This https://arxiv.org/abs/2206.10485 has been replaced.
link: https://scholar.google.com/scholar?q=a

Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds
In this article we extend and strengthen the seminal work by Niyogi, Smale, and Weinberger on the learning of the homotopy type from a sample of an underlying space. In their work, Niyogi, Smale, and Weinberger studied samples of $C^2$ manifolds with positive reach embedded in $\mathbb{R}^d$. We extend their results in the following ways: In the first part of our paper we consider both manifolds of positive reach -- a more general setting than $C^2$ manifolds -- and sets of positive reach embed…

This https://arxiv.org/abs/2210.15792 has been replaced.
link: https://scholar.google.com/scholar?q=a

Lattice homology, formality, and plumbed L-space links
We define a link lattice complex for plumbed links, generalizing constructions of Ozsváth, Stipsicz and Szabó, and of Gorsky and Némethi. We prove that for all plumbed links in rational homology 3-spheres, the link lattice complex is homotopy equivalent to the link Floer complex as an $A_\infty$-module. Additionally, we prove that the link Floer complex of a plumbed L-space link is a free resolution of its homology. As a consequence, we give an algorithm to compute the link Floer complexes o…

Homotopy types of diagrams of chain complexes
David Blanc, Surojit Ghosh, Aziz Kharoof
https://arxiv.org/abs/2404.03589 https://arxiv…

Homotopy types of diagrams of chain complexes
We study the homotopy theory of diagrams of chain complexes over a field indexed by a finite poset, and show that it can be completely described in terms of appropriate diagrams of graded vector spaces.

Extension Theory and Fermionic Strongly Fusion 2-Categories
Thibault D. D\'ecoppet
https://arxiv.org/abs/2403.03211 https://arxiv…

Extension Theory and Fermionic Strongly Fusion 2-Categories
We study group graded extensions of fusion 2-categories. As an application, we obtain a homotopy theoretic classification of fermionic strongly fusion 2-categories. We examine various examples in detail.

This https://arxiv.org/abs/2210.15792 has been replaced.
link: https://scholar.google.com/scholar?q=a

Lattice homology, formality, and plumbed L-space links
We define a link lattice complex for plumbed links, generalizing constructions of Ozsváth, Stipsicz and Szabó, and of Gorsky and Némethi. We prove that for all plumbed links in rational homology 3-spheres, the link lattice complex is homotopy equivalent to the link Floer complex as an $A_\infty$-module. Additionally, we prove that the link Floer complex of a plumbed L-space link is a free resolution of its homology. As a consequence, we give an algorithm to compute the link Floer complexes o…

This https://arxiv.org/abs/2403.14844 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csMS_…

Extrapolating Solution Paths of Polynomial Homotopies towards Singularities with PHCpack and phcpy
PHCpack is a software package for polynomial homotopy continuation, which provides a robust path tracker [Telen, Van Barel, Verschelde, SISC 2020]. This tracker computes the radius of convergence of Newton's method, estimates the distance to the nearest path, and then applies Padé approximants to predict the next point on the path. A priori step size control is less sensitive to finely tuned tolerances than a posteriori step size control, and is therefore robust. The Python interface phcpy is …

Fray functors and equivalence of colored HOMFLYPT homologies
Luke Conners
https://arxiv.org/abs/2405.00875 https://arxiv.org/pdf/2405…

Fray functors and equivalence of colored HOMFLYPT homologies
We construct several families of functors on the homotopy category of singular Soergel bimodules that mimic cabling and insertion of column-colored projectors. We use these functors to identify the intrinsically-colored homology of Webster--Williamson and the projector-colored homology of Elias--Hogancamp for an arbitrary link, up to multiplication by a polynomial in the quantum degree $q$. Combined with the results of arXiv:2303.16271, this establishes parity results for the intrinsic column-c…

This https://arxiv.org/abs/2311.06677 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An asymptotic homotopy lifting property
A $C^*$-algebra $A$ is said to have the homotopy lifting property if for all $C^*$-algebras $B$ and $E$, for every surjective $^*$-homomorphism $π\colon E \rightarrow B$ and for every $^*$-homomorphism $ϕ\colon A \rightarrow E$, any path of $^*$-homomorphisms $A \rightarrow B$ starting at $πϕ$ lifts to a path of $^*$-homomorphisms $A \rightarrow E$ starting at $ϕ$. Blackadar has shown that this property holds for all semiprojective $C^*$-algebras. We show that a version of the homotopy l…

Modularity of higher theta series II: Chow group of the generic fiber
Tony Feng, Adeel Khan
https://arxiv.org/abs/2403.19711 https://…

Modularity of higher theta series II: Chow group of the generic fiber
Higher theta series on moduli spaces of Hermitian shtukas were constructed by Feng--Yun--Zhang and conjectured to be modular, parallel to classical conjectures in the Kudla program. In this paper we prove the modularity of higher theta series after restriction to the generic locus. The proof is an upgrade, using motivic homotopy theory, of earlier work of Feng--Yun--Zhang which established generic modularity of $\ell$-adic realizations. In the process, we develop some general tools of broader u…

A Gradually Reinforced Sample-Average-Approximation Differentiable Homotopy Method for a System of Stochastic Equations
Peixuan Li, Chuangyin Dang, Yang Zhan
https://arxiv.org/abs/2403.00294

A Gradually Reinforced Sample-Average-Approximation Differentiable Homotopy Method for a System of Stochastic Equations
This paper intends to apply the sample-average-approximation (SAA) scheme to solve a system of stochastic equations (SSE), which has many applications in a variety of fields. The SAA is an effective paradigm to address risks and uncertainty in stochastic models from the perspective of Monte Carlo principle. Nonetheless, a numerical conflict arises from the sample size of SAA when one has to make a tradeoff between the accuracy of solutions and the computational cost. To alleviate this issue, we…

Homotopy commutativity in quasitoric manifolds
Sho Hasui, Daisuke Kishimoto, Yichen Tong, Mitsunobu Tsutaya
https://arxiv.org/abs/2404.01510 https://

Homotopy commutativity in quasitoric manifolds
We prove that the loop space of a quasitoric manifold is homotopy commutative if and only if the underlying polytope is $(Δ^3)^n$ and the characteristic matrix is equivalent to a matrix of certain type. We also construct for each $n\ge 2$ and a positive integer $k$, a quasitoric manifold $M(k,n)$ over $(Δ^3)^n$ such that its loop space is homotopy commutative if and only if $k$ is even, where every quasitoric manifold over $Δ^3$ is equivalent to $\mathbb{C} P^3$ whose loop space is homotopy …

This https://arxiv.org/abs/2210.11098 has been replaced.
link: https://scholar.google.com/scholar?q=a

The definable content of homological invariants II: Čech cohomology and homotopy classification
This is the second installment in a series of papers applying descriptive set theoretic techniques to both analyze and enrich classical functors from homological algebra and algebraic topology. In it, we show that the Čech cohomology functors $\check{\mathrm{H}}^n$ on the category of locally compact separable metric spaces each factor into (i) what we term their definable version, a functor $\check{\mathrm{H}}^n_{\mathrm{def}}$ taking values in the category $\mathsf{GPC}$ of groups with a Poli…

This https://arxiv.org/abs/2209.02667 has been replaced.
link: https://scholar.google.com/scholar?q=a

Directed degeneracy maps for precubical sets
Symmetric transverse sets were introduced to make the construction of the parallel product with synchronization for process algebras functorial. It is proved that one can do directed homotopy on symmetric transverse sets in the following sense. A q-realization functor from symmetric transverse sets to flows is introduced using a q-cofibrant replacement functor of flows. By topologizing the cotransverse maps, the cotransverse topological cube is constructed. It can be regarded both as a cotransv…

On the logarithmic slice filtration
Federico Binda, Doosung Park, Paul Arne {\O}stv{\ae}r
https://arxiv.org/abs/2403.03056 https://ar…

On the logarithmic slice filtration
We consider slice filtrations in logarithmic motivic homotopy theory. Our main results establish conjectured compatibilities with the Beilinson, BMS, and HKR filtrations on (topological, log) Hochschild homology and related invariants. In the case of perfect fields admitting resolution of singularities, the motivic trace map is compatible with the slice and BMS filtrations, yielding a natural morphism from the motivic Atiyah-Hirzerbruch spectral sequence to the BMS spectral sequence. Finally, w…

Homotopy classification of knotted defects in ordered media
Yuta Nozaki, Tam\'as K\'alm\'an, Masakazu Teragaito, Yuya Koda
https://arxiv.org/abs/2402.16079

Homotopy classification of knotted defects in ordered media
We give a homotopy classification of the global defects in ordered media, and explain it via the example of biaxial nematic liquid crystals, i.e., systems where the order parameter space is the quotient of the $3$-sphere $S^3$ by the quaternion group $Q$. As our mathematical model we consider continuous maps from complements of spatial graphs to the space $S^3/Q$ modulo a certain equivalence relation, and find that the equivalence classes are enumerated by the six subgroups of $Q$. Through mono…

This https://arxiv.org/abs/2304.10529 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The first step towards symplectic homotopy theory
We consider two categories related to symplectic manifolds: 1. Objects are symplectic manifolds and morphisms are symplectic embeddings. 2. Objects are symplectic manifolds endowed with compatible almost complex structure and morphisms are pseudoholomorphic maps. We define new homotopy theories for these categories. In particular, we give definitions of homotopy equivalent symplectic manifolds and define new cohomology theories. Theses cohomology theories are functorial, homotopy invarian…

This https://arxiv.org/abs/2304.10763 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Finiteness of CAT(0) group actions
We prove some finiteness results for discrete isometry groups $Γ$ of uniformly packed CAT$(0)$-spaces $X$ with uniformly bounded codiameter (up to group isomorphism), and for CAT$(0)$-orbispaces $M = Γ\backslash X$ (up to equivariant homotopy equivalence or equivariant diffeomorphism); these results generalize, in nonpositive curvature, classical finiteness theorems of Riemannian geometry. As a corollary, the order of every torsion subgroup of $Γ$ is bounded above by a universal constant onl…

New look at Milnor Spheres
Leonardo Cavenaghi, Lino Grama, Ludmil Katzarkov
https://arxiv.org/abs/2404.19088 https://arxiv.org/pdf/24…

New look at Milnor Spheres
S. Donaldson disproved the smooth $\mathrm{h}$-cobordism conjecture in dimension $4$ by studying invariants coming from the moduli space of connections on $\mathrm{SO}(3)$-bundles over smooth $4$-dimension manifolds $X$. In this paper, we reverse his point of view in dimension $8$. Namely, we consider fibrations coming from diffeomorphisms classes of homotopy Hopf manifolds $Σ^7\times\mathrm{S}^1$, i.e., $Σ^7$ is a homotopy sphere, seeing the bases as moduli spaces for additional structures. …

Higher structures for Lie $H$-pseudoalgebras
Apurba Das
https://arxiv.org/abs/2403.10777 https://arxiv.org/pdf/2403.10777

Higher structures for Lie $H$-pseudoalgebras
Let $H$ be a cocommutative Hopf algebra. The notion of Lie $H$-pseudoalgebra is a multivariable generalization of Lie conformal algebras. In this paper, we study some higher structures related to Lie $H$-pseudoalgebras where we increase the flexibility of the Jacobi identity. Namely, we first introduce $L_\infty$ $H$-pseudoalgebras (also called strongly homotopy Lie $H$-pseudoalgebras) as the homotopy analogue of Lie $H$-pseudoalgebras. We give several equivalent descriptions of such homotopy a…

This https://arxiv.org/abs/2404.03735 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Homology and homotopy for arbitrary categories
One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was discovered with the development of Homology theory. Some of the deepest results in topology are about the connections between Homotopy and Homology. These results are proved using intricate constructions. This paper takes an axiomatic approach that provides a comm…

Flux Quantization
Hisham Sati, Urs Schreiber
https://arxiv.org/abs/2402.18473 https://arxiv.org/pdf/2402.18473

Flux Quantization
Flux- and charge-quantization laws for higher gauge fields of Maxwell type -- e.g. the common electromagnetic field (the "A-field") but also the B-, RR-, and C-fields considered in string/M-theory -- specify non-perturbative completions of these fields by encoding their solitonic behaviour and hence by specifying the discrete charges carried by the individual branes (higher-dimensional monopoles or solitons) that source the field fluxes. This article surveys the general (rational-)homotopy th…

This https://arxiv.org/abs/2310.14425 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Reformulation of the stable Adams conjecture
We revisit methods of proof of the Adams Conjecture in order to correct and supplement earlier efforts to prove analogous conjectures in the stable homotopy category. We utilize simplicial schemes over an algebraically closed field of positive characteristic and a rigid version of Artin-Mazur étale homotopy theory. Consideration of special $\mathcal F$-spaces and together with Bousfield-Kan $\mathbb Z/\ell$-completion enables us to employ an "étale functor" which commutes up to homotopy with …

Discrete harmonic maps between hyperbolic surfaces
Wai Yeung Lam
https://arxiv.org/abs/2405.02205 https://arxiv.org/pdf/2405.02205

Discrete harmonic maps between hyperbolic surfaces
Given a topological cell decomposition of a closed surface equipped with edge weights, we consider the Dirichlet energy of any geodesic realization of the 1-skeleton graph to a hyperbolic surface. By minimizing the energy over all possible hyperbolic structures and over all realizations within a fixed homotopy class, one obtains a discrete harmonic map into an optimal hyperbolic surface. We characterize the extremum by showing that at the optimal hyperbolic structure, the discrete harmonic map …

This https://arxiv.org/abs/2302.00151 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Formalising and Computing the Fourth Homotopy Group of the $3$-Sphere in Cubical Agda
Brunerie's 2016 PhD thesis contains the first synthetic proof in Homotopy Type Theory (HoTT) of the classical result that the fourth homotopy group of the 3-sphere is $\mathbb{Z}/2\mathbb{Z}$. The proof is one of the most impressive pieces of synthetic homotopy theory to date and uses a lot of advanced classical algebraic topology rephrased synthetically. Furthermore, the proof is fully constructive and the main result can be reduced to the question of whether a particular "Brunerie number" $β…

This https://arxiv.org/abs/2304.10763 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Finiteness of CAT(0) group actions
We prove some finiteness results for discrete isometry groups $Γ$ of uniformly packed CAT$(0)$-spaces $X$ with uniformly bounded codiameter (up to group isomorphism), and for CAT$(0)$-orbispaces $M = Γ\backslash X$ (up to equivariant homotopy equivalence or equivariant diffeomorphism); these results generalize, in nonpositive curvature, classical finiteness theorems of Riemannian geometry. As a corollary, the order of every torsion subgroup of $Γ$ is bounded above by a universal constant onl…

Recovering generalized homology from Floer homology: the complex oriented case
Laurent C\^ot\'e, Yusuf Bar{\i}\c{s} Kartal
https://arxiv.org/abs/2404.02776

Recovering generalized homology from Floer homology: the complex oriented case
We associate an invariant called the completed Tate cohomology to a filtered circle-equivariant spectrum and a complex oriented cohomology theory. We show that when the filtered spectrum is the spectral symplectic cohomology of a Liouville manifold, this invariant depends only on the stable homotopy type of the underlying manifold. We make explicit computations for several complex oriented cohomology theories, including Eilenberg-Maclane spectra, Morava K-theories, their integral counterparts, …

This https://arxiv.org/abs/2302.00151 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Formalising and Computing the Fourth Homotopy Group of the $3$-Sphere in Cubical Agda
Brunerie's 2016 PhD thesis contains the first synthetic proof in Homotopy Type Theory (HoTT) of the classical result that the fourth homotopy group of the 3-sphere is $\mathbb{Z}/2\mathbb{Z}$. The proof is one of the most impressive pieces of synthetic homotopy theory to date and uses a lot of advanced classical algebraic topology rephrased synthetically. Furthermore, the proof is fully constructive and the main result can be reduced to the question of whether a particular "Brunerie number" $β…

This https://arxiv.org/abs/2210.05933 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the Whitehead theorem for nilpotent motivic spaces
We improve some foundational connectivity results and the relative Hurewicz theorem in motivic homotopy theory, study functorial central series in motivic local group theory, establish the existence of functorial Moore--Postnikov factorizations for nilpotent morphisms of motivic spaces under a mild technical hypothesis and establish an analog of the Whitehead theorem for nilpotent motivic spaces. As an application, we deduce a surprising unstable motivic periodicity result.

This https://arxiv.org/abs/2401.11752 has been replaced.
link: https://scholar.google.com/scholar?q=a

Univalent Enriched Categories and the Enriched Rezk Completion
Enriched categories are categories whose sets of morphisms are enriched with extra structure. Such categories play a prominent role in the study of higher categories, homotopy theory, and the semantics of programming languages. In this paper, we study univalent enriched categories. We prove that all essentially surjective and fully faithful functors between univalent enriched categories are equivalences, and we show that every enriched category admits a Rezk completion. Finally, we use the Rezk…

On the Drinfeld formal group
Deven Manam
https://arxiv.org/abs/2403.02555 https://arxiv.org/pdf/2403.02555

On the Drinfeld formal group
We identify Drinfeld's formal group on the prismatization of $\mathrm{Spf}\,\mathbb{Z}_p$ with a formal group arising from homotopy theory, given locally by the Quillen formal group of a decompleted variant of topological periodic cyclic homology. We also prove that this formal group extends to the syntomization of $\mathrm{Spf}\,\mathbb{Z}_p$, and that it admits a certain algebraization conjectured by Drinfeld.

This https://arxiv.org/abs/2404.03735 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Homology and homotopy for arbitrary categories
One of the prime motivation for topology was Homotopy theory, which captures the general idea of a continuous transformation between two entities, which may be spaces or maps. In later decades, an algebraic formulation of topology was discovered with the development of Homology theory. Some of the deepest results in topology are about the connections between Homotopy and Homology. These results are proved using intricate constructions. This paper takes an axiomatic approach that provides a comm…

This https://arxiv.org/abs/2002.12083 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the group of homotopy classes of relative homotopy automorphisms
We prove that the group of homotopy classes of relative homotopy automorphisms of a simply connected finite CW-complex is finitely presented and that the rationalization map from this group to its rational analogue has a finite kernel.

Homotopy type of shellable $q$-complexes and their homology groups
Sudhir R. Ghorpade, Rakhi Pratihar, Tovohery H. Randrianarisoa, Hugues Verdure, Glen Wilson
https://arxiv.org/abs/2403.07102

Homotopy type of shellable $q$-complexes and their homology groups
The theory of shellable simplicial complexes brings together combinatorics, algebra, and topology in a remarkable way. Initially introduced by Alder for $q$-simplicial complexes, recent work of Ghorpade, Pratihar, and Randrianarisoa extends the study of shellability to $q$-matroid complexes and determines singular homology groups for a subclass of these $q$-simplicial complexes. In this paper, we determine the homotopy type of shellable $q$-simplicial complexes. Moreover, we establish the shell…

A note on homotopy and pseudoisotopy of diffeomorphisms of $4$-manifolds
Manuel Krannich, Alexander Kupers
https://arxiv.org/abs/2402.16167 https://…

A note on homotopy and pseudoisotopy of diffeomorphisms of $4$-manifolds
This note serves to record examples of diffeomorphisms of closed smooth $4$-manifolds $X$ that are homotopic but not pseudoisotopic to the identity, and to explain why there are no such examples when $X$ is orientable and its fundamental group is a free group.

This https://arxiv.org/abs/2307.01726 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Homotopy cofinality for Non-Abelian homology of group diagrams
We prove that a homotopy cofinal functor between small categories induces a weak equivalence between homotopy colimits of pointed simplicial sets. This is used to prove that the non-Abelian homology of a group diagram is isomorphic to the homology of its inverse image under a homotopy cofinal functor. This also made it possible to establish that the non-Abelian homology of group diagrams are invariant under the left Kan extension along virtual discrete cofibrations. With the help of these resul…

This https://arxiv.org/abs/2212.02617 has been replaced.
link: https://scholar.google.com/scholar?q=a

Ribbonness of Kervaire's sphere-link in homotopy 4-sphere and its consequences to 2-complexes
M. A. Kervaire showed that every group of deficiency $d$ and weight $d$ is the fundamental group of a smooth sphere-link of $d$ components in a smooth homotopy 4-sphere. In the use of the smooth unknotting conjecture and the smooth 4D Poincar{é} conjecture, any such sphere-link is shown to be a sublink of a free ribbon sphere-link in the 4-sphere. Since every ribbon sphere-link in the 4-sphere is also shown to be a sublink of a free ribbon sphere-link in the 4-sphere, Kervaire's sphere-link an…

This https://arxiv.org/abs/2401.16081 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On some topological equivalences for moduli spaces of $G$-bundles
Let $X$ be a smooth projective curve of genus $g \geq 3$, and let $G$ be a nontrivial connected reductive affine algebraic group over $\mathbb{C}$. Examining the moduli spaces of regularly stable $G$-Higgs bundles and holomorphic $G$-connections with a fixed topological type $d\in π_1(G)$ over $X$, we establish that the $k$-th homotopy groups of these two moduli spaces are isomorphic for $k \leq 2g-4$. We also prove that the mixed Hodge structures on the rational cohomology groups of these two…

Intersection Multiplicity in Loop Spaces and the String Topology Coproduct
Philippe Kupper, Maximilian Stegemeyer
https://arxiv.org/abs/2404.03460 https://…

Intersection Multiplicity in Loop Spaces and the String Topology Coproduct
The string topology coproduct is often perceived as a counterpart in string topology to the Chas-Sullivan product. However, in certain aspects the string topology coproduct is much harder to understand than the Chas-Sullivan product. In particular the coproduct is not homotopy-invariant and it seems much harder to compute. In this article we give an overview over the string topology coproduct and use the notion of intersection multiplicity of homology classes in loop spaces to show that the str…

Intersection Multiplicity in Loop Spaces and the String Topology Coproduct
Philippe Kupper, Maximilian Stegemeyer
https://arxiv.org/abs/2404.03460 https://…

Intersection Multiplicity in Loop Spaces and the String Topology Coproduct
The string topology coproduct is often perceived as a counterpart in string topology to the Chas-Sullivan product. However, in certain aspects the string topology coproduct is much harder to understand than the Chas-Sullivan product. In particular the coproduct is not homotopy-invariant and it seems much harder to compute. In this article we give an overview over the string topology coproduct and use the notion of intersection multiplicity of homology classes in loop spaces to show that the str…

The relative James construction and its application to homotopy groups
Zhu Zhongjian, Jin Tian
https://arxiv.org/abs/2402.07072 https://

The relative James construction and its application to homotopy groups
In this paper, we develop the new method to compute the homotopy groups of the mapping cone $C_f=Y\cup_{f}CX$ beyond the metastable range by analysing the homotopy of the $n$-th filtration of the relative James construction $J(X,A)$ for CW-pair $A\stackrel{i}\hookrightarrow X$, defined by B. Gray, which is homotopy equivalent to the homotopy fiber of the pinch map $X\cup_{i}CA\rightarrow ΣA$. As an application, we compute the 5 and 6-dim unstable homotopy groups of 3-dimensional mod $2^r$ Moor…

The pro-Nisnevich topology
Klaus Mattis
https://arxiv.org/abs/2404.17314 https://arxiv.org/pdf/2404.17314

The pro-Nisnevich topology
We construct the pro-Nisnevich topology, an analog of the pro-étale topology. We then show that the Nisnevich $\infty$-topos embeds into the pro-Nisnevich $\infty$-topos, and that the pro-Nisnevich $\infty$-topos is locally of homotopy dimension $0$.

This https://arxiv.org/abs/2403.18450 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Loop homology of moment-angle complexes in the flag case
We develop a general homological approach to presentations of connected graded associative algebras, and apply it to loop homology of moment-angle complexes $Z_K$ that correspond to flag simplicial complexes $K$. For arbitrary coefficient ring, we describe generators of the Pontryagin algebra $H_*(ΩZ_K)$ and defining relations between them. We prove that such moment-angle complexes are coformal over $\mathbb{Q},$ give a necessary condition for rational formality, and compute their homotopy gro…

This https://arxiv.org/abs/2402.16167 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A note on homotopy and pseudoisotopy of diffeomorphisms of $4$-manifolds
This note serves to record examples of diffeomorphisms of closed smooth $4$-manifolds $X$ that are homotopic but not pseudoisotopic to the identity, and to explain why there are no such examples when $X$ is orientable and its fundamental group is a free group.

This https://arxiv.org/abs/2403.18450 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Loop homology of moment-angle complexes in the flag case
We develop a general homological approach to presentations of connected graded associative algebras, and apply it to loop homology of moment-angle complexes $Z_K$ that correspond to flag simplicial complexes $K$. For arbitrary coefficient ring, we describe generators of the Pontryagin algebra $H_*(ΩZ_K)$ and defining relations between them. We prove that such moment-angle complexes are coformal over $\mathbb{Q},$ give a necessary condition for rational formality, and compute their homotopy gro…

This https://arxiv.org/abs/2210.11048 has been replaced.
link: https://scholar.google.com/scholar?q=a

The quadratic linking degree
By using motivic homotopy theory, we introduce a counterpart in algebraic geometry to oriented links and their linking numbers. After constructing the (ambient) quadratic linking degree -- our analogue of the linking number which takes values in the Witt group of the ground field -- and exploring some of its properties, we give a method to explicitly compute it. We illustrate this method on a family of examples which are analogues of torus links, in particular of the Hopf and Solomon links.

This https://arxiv.org/abs/2212.02444 has been replaced.
link: https://scholar.google.com/scholar?q=a

Homotopy type theory as internal languages of diagrams of $\infty$-logoses
We show that certain diagrams of $\infty$-logoses are reconstructed in internal languages of their oplax limits via lex, accessible modalities, which enables us to use plain homotopy type theory to reason about not only a single $\infty$-logos but also a diagram of $\infty$-logoses. This also provides a higher dimensional version of Sterling's synthetic Tait computability -- a type theory for higher dimensional logical relations. To prove the main result, we establish a precise correspondence b…

Natural homotopy of multipointed d-spaces
Philippe Gaucher
https://arxiv.org/abs/2404.18693 https://arxiv.org/pdf/2404.18693

Natural homotopy of multipointed d-spaces
We identify Grandis' directed spaces as a full reflective subcategory of the category of multipointed $d$-spaces. When the multipointed $d$-space realizes a precubical set, its reflection coincides with the standard realization of the precubical set as a directed space. The reflection enables us to extend the construction of the natural system of topological spaces in Baues-Wirsching's sense from directed spaces to multipointed $d$-spaces. In the case of a cellular multipointed $d$-space, there…

Obstruction Complexes in Grid Homology
Yan Tao
https://arxiv.org/abs/2404.19747 https://arxiv.org/pdf/2404.19747

Obstruction Complexes in Grid Homology
Recently, Manolescu-Sarkar constructed a stable homotopy type for link Floer homology, which uses grid homology and accounts for all domains that do not pass through a specific square. In doing so, they produced an obstruction chain complex of the grid diagram with that square removed. We define the obstruction chain complex of the full grid, without the square removed, and compute its homology. Though this homology is too complicated to immediately extend the Manolescu-Sarkar construction, we …

Natural homotopy of multipointed d-spaces
Philippe Gaucher
https://arxiv.org/abs/2404.18693 https://arxiv.org/pdf/2404.18693

Natural homotopy of multipointed d-spaces
We identify Grandis' directed spaces as a full reflective subcategory of the category of multipointed $d$-spaces. When the multipointed $d$-space realizes a precubical set, its reflection coincides with the standard realization of the precubical set as a directed space. The reflection enables us to extend the construction of the natural system of topological spaces in Baues-Wirsching's sense from directed spaces to multipointed $d$-spaces. In the case of a cellular multipointed $d$-space, there…

Motivic Toda brackets
Xiaowen Dong
https://arxiv.org/abs/2402.15906 https://arxiv.org/pdf/2402.15906…

Motivic Toda brackets
We construct Toda brackets in unstable motivic homotopy theory and prove some fundamental properties of them. Furthermore we construct some examples of motivic Toda brackets.

This https://arxiv.org/abs/1909.08865 has been replaced.
link: https://scholar.google.com/scholar?q=a

Elementary Methods for Persistent Homotopy Groups
In this paper, we study the basic properties of persistent homotopy groups. We show that the persistent fundamental group benefits from the Van Kampen theorem, and the interleaving distance between total spaces is less than or equal to the maximum of the interleaving distances between subspaces. We also prove excision and Hurewicz theorems for persistent homotopy groups. As an application, we analyze the sublevelset persistent homotopy groups of the energy landscape of alkane molecules.

Obstruction Complexes in Grid Homology
Yan Tao
https://arxiv.org/abs/2404.19747 https://arxiv.org/pdf/2404.19747

Obstruction Complexes in Grid Homology
Recently, Manolescu-Sarkar constructed a stable homotopy type for link Floer homology, which uses grid homology and accounts for all domains that do not pass through a specific square. In doing so, they produced an obstruction chain complex of the grid diagram with that square removed. We define the obstruction chain complex of the full grid, without the square removed, and compute its homology. Though this homology is too complicated to immediately extend the Manolescu-Sarkar construction, we …

On the homotopy links of stratified cell complexes
Lukas Waas
https://arxiv.org/abs/2403.06272 https://arxiv.org/pdf/2403.06272

On the homotopy links of stratified cell complexes
Homotopy links haven proven to be one of the most powerful tools of stratified homotopy theory. In previous work, we described combinatorial models for the generalized homotopy links of a stratified simplicial set. For many purposes, in particular to investigate the stratified homotopy hypothesis, a more general version of this result pertaining to stratified cell complexes is needed. Here we prove that, given a stratified cell complex $X$, the generalized homotopy links can be computed in term…

On the weak homotopy types of small finite spaces
Kango Matsushima, Shuichi Tsukuda
https://arxiv.org/abs/2404.12564 https://arxiv.or…

On the weak homotopy types of small finite spaces
We show that a connected finite topological space with $12$ or less points has a weak homotopy type of a wedge of spheres. In other words, we show that the order complex of a connected finite poset with $12$ or less points has a homotopy type of a wedge of spheres.

On the homotopy type of the space of fiberings of $S^1 \times S^2$ by simple closed curves
Yi Wang, Jingye Yang
https://arxiv.org/abs/2404.08545 https://…

On the homotopy type of the space of fiberings of $S^1 \times S^2$ by simple closed curves
For most aspherical Seifert-fibered 3-manifolds $M$, the space of Seifert fiberings $SF(M)$ is known to have contractible components. It is also known that the space of Hopf fiberings of the three-sphere is noncontractible. We provide the second example of a non-aspherical 3-manifold $M$ such that $SF(M)$ has noncontractible components. In particular, we show that certain components of $SF(S^1 \times S^2)$ are homotopy equivalent to a subspace homeomorphic to the identity-based loop space $ΩSO…

This https://arxiv.org/abs/2209.11257 has been replaced.
link: https://scholar.google.com/scholar?q=a

Linear actions of $\mathbb{Z}/p\times\mathbb{Z}/p$ on $S^{2n-1}\times S^{2n-1}$
For an odd prime $p$, we consider free actions of $(\mathbb{Z}/p)^2$ on $S^{2n-1}\times S^{2n-1}$ given by linear actions of $(\mathbb{Z}/p)^2$ on $\mathbb{R}^{4n}$. Simple examples include a lens space cross a lens space, but $k$-invariant calculations show that other quotients exist. Using the tools of Postnikov towers and surgery theory, the quotients are classified up to homotopy by the $k$-invariants and up to homeomorphism by the Pontrjagin classes. We will present these results and demon…

$\mathbb{E}_{\infty}$-coalgebras and $p$-adic homotopy theory
Tom Bachmann, Robert Burklund
https://arxiv.org/abs/2402.15850 https://…

$\mathbb{E}_{\infty}$-coalgebras and $p$-adic homotopy theory
We show that for any separably closed field $k$ of characteristic $p>0$, the canonical functor from nilpotent $p$-adic spaces to $\mathbb{E}_{\infty}$-coalgebras over $k$ (given by singular chains with coefficients in $k$) is fully faithful. We also identify the essential image of simply connected spaces inside coalgebras. This dualizes and removes finiteness assumptions from a theorem of Mandell.

Homotopy BV-algebras in Hermitian geometry
Joana Cirici, Scott O. Wilson
https://arxiv.org/abs/2403.12314 https://arxiv.org/pdf/2403.…

Homotopy BV-algebras in Hermitian geometry
We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative $BV_\infty$-algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar resu…

This https://arxiv.org/abs/2301.05030 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A General Blue-Shift Phenomenon
In chromatic homotopy theory, there is a well-known conjecture called blue-shift phenomenon (BSP). In this paper, we propose a general blue-shift phenomenon (GBSP) which unifies BSP and a new variant of BSP introduced by Balmer-Sanders under one framework. To explain GBSP, we use the roots of $p^j$-series of the formal group law of a complex-oriented spectrum $E$ in the homotopy group of the generalized Tate spectrum of $E$. We also incorporate the relationship between roots and coefficients of…

Path spaces of pushouts
David W\"arn
https://arxiv.org/abs/2402.12339 https://arxiv.org/pdf/2402.12339

Path spaces of pushouts
Given a span of spaces, one can form the homotopy pushout and then take the homotopy pullback of the resulting cospan. We give a concrete description of this pullback as the colimit of a sequence of approximations, using what we call the zigzag construction. We also obtain a description of loop spaces of homotopy pushouts. Using the zigzag construction, we reproduce generalisations of the Blakers-Massey theorem and fundamental results from Bass-Serre theory. We also describe the loop space of a…

This https://arxiv.org/abs/2401.16336 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Computational Synthetic Cohomology Theory in Homotopy Type Theory
This paper discusses the development of synthetic cohomology in Homotopy Type Theory (HoTT), as well as its computer formalisation. The objectives of this paper are (1) to generalise previous work on integral cohomology in HoTT by the current authors and Brunerie (2022) to cohomology with arbitrary coefficients and (2) to provide the mathematical details of, as well as extend, results underpinning the computer formalisation of cohomology rings by the current authors and Lamiaux (2023). With res…

New infinite families in the stable homotopy groups of spheres
Prasit Bhattacharya, Irina Bobkova, J. D. Quigley
https://arxiv.org/abs/2404.10062 https://<…

New infinite families in the stable homotopy groups of spheres
We identify seven new $192$-periodic infinite families of elements in the $2$-primary stable homotopy groups of spheres, whose images are nontrivial in the $\mathrm{K}(2)$- as well as the $\mathrm{T}(2)$-local stable stems. We also obtain new information about $2$-torsion and $2$-divisibility of some of the known $192$-periodic infinite families in the stable stems.

This https://arxiv.org/abs/2303.12213 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Alpha shapes in kernel density estimation
For every Gaussian kernel density estimator $f(x)=\sum_i a_i \exp(-\lVert x-x_i\rVert^2/2h^2)$ associated to a point cloud $\mathcal{D}=\{x_1,...,x_N\}\subset \mathbb{R}^d$, we define a nested family of closed subspaces $\mathcal{S}(a)\subset\mathbb{R}^d$, which we interpret as a continuous version of an alpha shape. Using arguments based on Fenchel duality, we prove that $\mathcal{S}(a)$ is homotopy equivalent to the superlevel set $\mathcal{L}(a)=f^{-1}[e^{-a},\infty)$, and that $\mathcal{L}(…

This https://arxiv.org/abs/2303.12213 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Alpha shapes in kernel density estimation
For every Gaussian kernel density estimator $f(x)=\sum_i a_i \exp(-\lVert x-x_i\rVert^2/2h^2)$ associated to a point cloud $\mathcal{D}=\{x_1,...,x_N\}\subset \mathbb{R}^d$, we define a nested family of closed subspaces $\mathcal{S}(a)\subset\mathbb{R}^d$, which we interpret as a continuous version of an alpha shape. Using arguments based on Fenchel duality, we prove that $\mathcal{S}(a)$ is homotopy equivalent to the superlevel set $\mathcal{L}(a)=f^{-1}[e^{-a},\infty)$, and that $\mathcal{L}(…

This https://arxiv.org/abs/2212.01885 has been replaced.
link: https://scholar.google.com/scholar?q=a

André-Quillen Cohomology and $k$-invariants of simplicial categories
Using the Harpaz-Nuiten-Prasma interpretation of the Dwyer-Kan-Smith cohomology of a simplicial category $\mathcal{X}$, we obtain a cochain complex for the André-Quillen cohomology groups in which the $k$-invariants for $\mathcal{X}$ take value. Given a map of simplicial categories $ϕ:\mathcal{Y}\rightarrow P^{(n-1)} \mathcal{X}$ into a Postnikov section of $\mathcal{X}$, we use a homotopy colimit decomposition of $\mathcal{Y}$ to study the obstruction to lifting $ϕ$ to $P^{(n)}\mathcal{X}$.…

This https://arxiv.org/abs/2212.01885 has been replaced.
link: https://scholar.google.com/scholar?q=a

André-Quillen Cohomology and $k$-invariants of simplicial categories
Using the Harpaz-Nuiten-Prasma interpretation of the Dwyer-Kan-Smith cohomology of a simplicial category $\mathcal{X}$, we obtain a cochain complex for the André-Quillen cohomology groups in which the $k$-invariants for $\mathcal{X}$ take value. Given a map of simplicial categories $ϕ:\mathcal{Y}\rightarrow P^{(n-1)} \mathcal{X}$ into a Postnikov section of $\mathcal{X}$, we use a homotopy colimit decomposition of $\mathcal{Y}$ to study the obstruction to lifting $ϕ$ to $P^{(n)}\mathcal{X}$.…

This https://arxiv.org/abs/2104.10206 has been replaced.
link: https://scholar.google.com/scholar?q=a

Homotopy, homology, and persistent homology using closure spaces
We develop persistent homology in the setting of filtrations of (Cech) closure spaces. Examples of filtrations of closure spaces include metric spaces, weighted graphs, weighted directed graphs, and filtrations of topological spaces. We use various products and intervals for closure spaces to obtain six homotopy theories, six cubical singular homology theories, and three simplicial singular homology theories. Applied to filtrations of closure spaces, these homology theories produce persistence …

Bounding the $K(p-1)$-local exotic Picard group at $p>3$
Irina Bobkova, Andrea Lachmann, Ang Li, Alicia Lima, Vesna Stojanoska, Adela YiYu Zhang
https://arxiv.org/abs/2403.15572

Bounding the $K(p-1)$-local exotic Picard group at $p>3$
In this paper, we bound the descent filtration of the exotic Picard group $κ_n$, for a prime number $p>3$ and $n=p-1$. Our method involves a detailed comparison of the Picard spectral sequence, the homotopy fixed point spectral sequence, and an auxiliary $β$-inverted homotopy fixed point spectral sequence whose input is the Farrell-Tate cohomology of the Morava stabilizer group. Along the way, we deduce that the $K(n)$-local Adams-Novikov spectral sequence for the sphere has a horizontal vani…

C_2-Equivariant Stable Stems
Bertrand J. Guillou, Daniel C. Isaksen
https://arxiv.org/abs/2404.14627 https://arxiv.org/pdf/2404.14627…

C_2-Equivariant Stable Stems
We compute the 2-primary $C_2$-equivariant stable homotopy groups $π^{C_2}_{s,c}$ for stems between 0 and 25 (i.e., $0 \leq s \leq 25$) and for coweights between -1 and 7 (i.e., $-1 \leq c \leq 7)$. Our results, combined with periodicity isomorphisms and sufficiently extensive $\mathbb{R}$-motivic computations, would determine all of the $C_2$-equivariant stable homotopy groups for all stems up to 20. We also compute the forgetful map $π^{C_2}_{s,c} \rightarrow π^{\mathrm{cl}}_s$ to the clas…

Strong minimal model theorem and Massey products
Martin Markl
https://arxiv.org/abs/2404.19607 https://arxiv.org/pdf/2404.19607

Strong minimal model theorem and Massey products
Kadeishvili's minimal model theorem establishes the existence of an $A_\infty$-structure, unique up to isomorphism, on the cohomology of a dg associative algebra, which captures its homotopy type. In this note we prove the existence of minimal models that are unique up to isotopy, a stronger result obviously known to T. Kadeishvili and certainly to others, yet seemingly overlooked by mankind. We will explore how this stronger result can help in the study of Massey products. First, we show that …

Strong minimal model theorem and Massey products
Martin Markl
https://arxiv.org/abs/2404.19607 https://arxiv.org/pdf/2404.19607

Strong minimal model theorem and Massey products
Kadeishvili's minimal model theorem establishes the existence of an $A_\infty$-structure, unique up to isomorphism, on the cohomology of a dg associative algebra, which captures its homotopy type. In this note we prove the existence of minimal models that are unique up to isotopy, a stronger result obviously known to T. Kadeishvili and certainly to others, yet seemingly overlooked by mankind. We will explore how this stronger result can help in the study of Massey products. First, we show that …

This https://arxiv.org/abs/2402.10568 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Examples and cofibrant generation of effective Kan fibrations
We will show make two contributions to the theory of effective Kan fibrations, which are a more explicit version of the notion of a Kan fibration, a notion which plays a fundamental role in simplicial homotopy theory. We will show that simplicial Malcev algebras are effective Kan complexes and that the effective Kan fibrations can be seen as the right class in an algebraic weak factorisation system. In addition, we will introduce two strengthenings of the notion of an effective Kan fibration, t…

Smooth Structures on $M^n\times\mathbb{S}^k$
Samik Basu, Ramesh Kasilingam, Ankur Sarkar
https://arxiv.org/abs/2402.18914 https://arx…

Smooth Structures on $M^n\times\mathbb{S}^k$
This paper explores various differentiable structures on the product manifold $M \times \mathbb{S}^k$, where $M$ is either a 4-dimensional closed oriented manifold or a simply connected 5-dimensional closed manifold. We identify the possible stable homotopy types of $M$ and use it to calculate the concordance inertia group and the concordance structure set of $M\times\mathbb{S}^k$ for $1\leq k\leq 10.$ These calculations enable us to further classify all manifolds that are homeomorphic to $\mat…

Presenting the topological stratified homotopy hypothesis
Lukas Waas
https://arxiv.org/abs/2403.07686 https://arxiv.org/pdf/2403.0768…

Presenting the topological stratified homotopy hypothesis
This article is concerned with three different homotopy theories of stratified spaces: The one defined by Douteau and Henriques, the one defined by Haine, and the one defined by Nand-Lal. One of the central questions concerning these theories has been how precisely they connect with geometric and topological examples of stratified spaces, such as piecewise linear pseudomanifolds, Whitney stratified spaces, or more recently Ayala, Francis and Tanaka's conically smooth stratified spaces. More pre…

Top cell attachment for a Poincare Duality complex
Stephen Theriault
https://arxiv.org/abs/2402.13775 https://arxiv.org/pdf/2402.1377…

Top cell attachment for a Poincare Duality complex
Let M be a simply-connected closed Poincare Duality complex of dimension n. Then M is obtained by attaching a cell of highest dimension to its (n-1)-skeleton M'. Conditions are given for when the skeletal inclusion i:M' --> M has the property that the based loops on i has a right homotopy inverse. This is an integral version of the rational statement that such a right homotopy inverse always exists provided the rational cohomology of M is not generated by a single element. New methods are devel…