A cubical approach to homotopy orbits of circle actions

定向拓扑II的立方逼近

Cubical Approximation for Directed Topology II

本文通过立方逼近方法，建立了定向拓扑与立方集之间的等价关系，并探讨了其在同调单形和分类空间中的应用。

立方集和图的同伦n-型

Homotopy $n$-types of cubical sets and graphs

本文提出了一种基于右传递的模型结构构造方法，用于在立方集和图的范畴中定义同伦n-型的纤维范畴结构。

The Uniform Homotopy Category

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes of weak equivalences, special cases of classical weak equivalences, appropriate for the respective Lipschitz and uniform settings. Cubical sets and uniform spaces admit the additional compatible structures of categories of (co)fibrant objects. A categorical equivalence between classical homotopy categories of cubical sets and spaces lifts to a full and faithful embedding from an associated Lipschitz homotopy category of cubical sets into an associa…

立方同调的Hurewicz定理

The Hurewicz theorem for cubical homology

本文利用立方集环路空间概念，为立方Kan复形给出了Hurewicz定理的初等证明，揭示了同伦群与同调群之间的关系。

Varieties of Cubical Sets

We define a variety of notions of cubical sets, based on sites organized using substructural algebraic theories presenting PRO(P)s or Lawvere theories. We prove that all our sites are test categories in the sense of Grothendieck, meaning that the corresponding presheaf categories of cubical sets model classical homotopy theory. We delineate exactly which ones are even strict test categories, meaning that products of cubical sets correspond to products of homotopy types.

Abstract cubical homotopy theory

Triangulations and higher triangulations axiomatize the calculus of derived cokernels when applied to strings of composable morphisms. While there are no cubical versions of (higher) triangulations, in this paper we use coherent diagrams to develop some aspects of a rich cubical calculus. Applied to the models in the background, this enhances the typical examples of triangulated and tensor-triangulated categories. The main players are the cardinality filtration of n-cubes, the induced interpolation between cocartesian and strongly cocartesian n-cubes, and the yoga of iterated cone construction…

Homotopy Types of Random Cubical Complexes

We study the topology of a random cubical complex associated to Bernoulli site percolation on a cubical grid. We begin by establishing a limit law for homotopy types. More precisely, looking within an expanding window, we define a sequence of normalized counting measures (counting connected components according to homotopy type), and we show that this sequence of random probability measures converges in probability to a deterministic probability measure. We then investigate the dependence of the limiting homotopy measure on the coloring probability $p$, and our results show a qualitative chang…

有向拓扑的立方逼近 I

Cubical approximation for directed topology I

本文研究有向拓扑中的立方逼近理论，证明了立方集和有向空间的几何实现诱导等价，并探讨了有向映射同伦关系的相等条件。

比较立方与球状有向路径

Comparing cubical and globular directed paths

本文构造了一个不依赖于余纤维替代选择的新实现函子，将预立方集映射为流，并证明其执行路径空间与几何实现中的非恒定 d-路径空间同伦等价。

从立方图和同伦余极限构造单子

Constructing monads from cubical diagrams and homotopy colimits

本文将 Hess–Johnson 基于相容余单子的广义微积分构造对偶到单子情形，利用 $\mathcal{P}(n)$-模和同伦余极限构造单子，并证明其推广 McCarthy 对偶微积分。

Spaces of directed paths on pre-cubical sets II

For a given pre-cubical set ($\square$--set) $K$ with two distinguished vertices $\bO$, $\bI$, we prove that the space $\vP(K)_\bO^\bI$ of d-paths on the geometric realization of $K$ with source $\bO$ and target $\bI$ is homotopy equivalent to its subspace $\vP^t(K)_\bO^\bI$ of tame d-paths. When $K$ is the underlying $\square$--set of a Higher Dimensional Automaton $A$, tame d-paths on $K$ represent step executions of $A$. Then, we define the cube chain category of $K$ and prove that its nerve is weakly homotopy equivalent to $\vP(K)_\bO^\bI$.

Digital homotopy relations and digital homology theories

In this paper we prove results relating to two homotopy relations and four homology theories developed in the topology of digital images. We introduce a new type of homotopy relation for digitally continuous functions which we call "strong homotopy." Both digital homotopy and strong homotopy are natural digitizations of classical topological homotopy: the difference between them is analogous to the difference between digital 4-adjacency and 8-adjacency in the plane. We also consider four different digital homology theories: a simplicial homology theory by Arslan et al which is the homology of …

超图同调理论的立方体方法

A cubical approach to homology theories for hypergraphs

本文提出超图的三种同调理论（Γ、□、×），证明其互异且不同于嵌入同调，并引入扩展图离散同伦的超图同伦，发现仅□-同调具同伦不变性，且Γ-同调有独特结构行为。

de Rham theory and cocycles of cubical sets from smooth quandles

We show a de Rham theory for cubical manifolds, and study rational homotopy type of the classifying spaces of smooth quandles. We also show that secondary characteristic classes in \cite{Dup2,DK} produce cocycles of quandles.

在 Cubical Agda 中形式化与计算三维球面的第四同伦群

Formalising and Computing the Fourth Homotopy Group of the $3$-Sphere in Cubical Agda

本文在 Cubical Agda 中完整形式化了三维球面第四同伦群为 $\mathbb{Z}/2\mathbb{Z}$ 的证明，通过修改Brunerie的证明避免关键技术难点，并提出新证明使特定数迅速归一化为 $-2$，实现了计算机辅助验证。

Homotopy Theory of Labelled Symmetric Precubical Sets

This paper is the third paper of a series devoted to higher dimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition systems. In this sequel, it is proved that there exists a model category of labelled symmetric precubical sets which is Quillen equivalent to the Bousfield localization of this left determined model category by the cubification functor. The realization functor from labelled symmetric precubical sets to cubical transition systems which was introduced in the first paper of this series is use…

$c_1$-数字图像的离散立方奇异性状同调的可计算性

Computability of digital cubical singular homology of $c_1$-digital images

本文研究了$c_1$-数字图像的离散立方奇异性状同调的计算问题，通过与更易计算的$c_1$-立方同调进行比较，探索简化计算的方法，并证明了$c_1$-同调的函子性和同调类型不变性。

相对优雅性与带一种连接的笛卡尔立方体

Relative elegance and cartesian cubes with one connection

本文在带一种连接的笛卡尔立方体集合上建立Quillen等价，得到第二个同时构造性模拟同伦类型论并呈现无穷群胚的模型结构，首个实例为等变笛卡尔模型。

The homotopy fixed points of the circle action on Hochschild homology

We show that Connes' B-operator on a cyclic differential graded k-module M is a model for the canonical circle action on the geometric realization of M. This implies that the negative cyclic homology and the periodic cyclic homology of a differential graded category can be identified with the homotopy fixed points and the Tate fixed points of the circle action on its Hochschild complex.

在笛卡尔立方集上的等变模型结构

The equivariant model structure on cartesian cubical sets

本文基于笛卡尔立方范畴的预层发展了构造性同调类型论模型，通过引入等变条件改进立方Kan纤维化，主要贡献是用计算机证明助手形式化了关键技术结果。

On the Whitehead spectrum of the circle

The seminal work of Waldhausen, Farrell and Jones, Igusa, and Weiss and Williams shows that the homotopy groups in low degrees of the space of homeomorphisms of a closed Riemannian manifold of negative sectional curvature can be expressed as a functor of the fundamental group of the manifold. To determine this functor, however, it remains to determine the homotopy groups of the topological Whitehead spectrum of the circle. The cyclotomic trace of B okstedt, Hsiang, and Madsen and a theorem of Dundas, in turn, lead to an expression for these homotopy groups in terms of the equivariant homotopy …

映射的同伦相似性。圆圈上的映射

Homotopy similarity of maps. Maps of the circle

本文研究了圆圈映射的同伦相似性与有限阶不变量之间的关系。

Some non-trivial PL knots whose complements are homotopy circles

We show that there exist non-trivial piecewise-linear (PL) knots with isolated singularities $S^{n-2}\subset S^n$, $n\geq 5$, whose complements have the homotopy type of a circle. This is in contrast to the case of smooth, PL locally-flat, and topological locally-flat knots, for which it is known that if the complement has the homotopy type of a circle, then the knot is trivial.

$S^{1}$上$S^{1}$-丛上纤维映射的同伦极小周期

Homotopy minimal periods for fiber maps on circle bundles over circle

研究纤维丛上纤维映射的同伦极小周期，给出了基和纤维均为$S^{1}$的纤维丛上同伦极小周期的完整描述。

Calculating the Fundamental Group of the Circle in Homotopy Type Theory

Recent work on homotopy type theory exploits an exciting new correspondence between Martin-Lof's dependent type theory and the mathematical disciplines of category theory and homotopy theory. The category theory and homotopy theory suggest new principles to add to type theory, and type theory can be used in novel ways to formalize these areas of mathematics. In this paper, we formalize a basic result in algebraic topology, that the fundamental group of the circle is the integers. Though simple, this example is interesting for several reasons: it illustrates the new principles in homotopy type …

On bipartite circle graphs and Khovanov homology

We prove that independence complex of a bipartite circle graph is homotopy equivalent to a wedge of spheres, resolving a conjecture posed by Przytycki and Silvero. As a corollary, we obtain that extreme Khovanov spectrum, $\mathcal{X}_{j_{extreme}}$ is homotopy equivalent to a wedge of spheres. In particular, the extreme Khovanov homology has no torsion.

The Vietoris-Rips complexes of a circle

Given a metric space X and a distance threshold r>0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and r is sufficiently small, then the Vietoris-Rips complex is homotopy equivalent to the original manifold. Little is known about the behavior of Vietoris-Rips complexes for larger values of r, even though these complexes arise naturally in applications using persistent homology. We show that as r increases, the Vietoris-Rips complex of the circle obtains the h…

Homotopy theory of C*-algebras

In this work we construct from ground up a homotopy theory of C*-algebras. This is achieved in parallel with the development of classical homotopy theory by first introducing an unstable model structure and second a stable model structure. The theory makes use of a full fledged import of homotopy theoretic techniques into the subject of C*-algebras. The spaces in C*-homotopy theory are certain hybrids of functors represented by C*-algebras and spaces studied in classical homotopy theory. In particular, we employ both the topological circle and the C*-algebra circle of complex-valued continuous…

Vietoris-Rips Metric Thickenings of the Circle

Vietoris-Rips metric thickenings have previously been proposed as an alternate approach to understanding Vietoris-Rips simplicial complexes and their persistent homology. Recent work has shown that for totally bounded metric spaces, Vietoris-Rips metric thickenings have persistent homology barcodes that agree with those of Vietoris-Rips simplicial complexes, ignoring whether endpoints of bars are open or closed. Combining this result with the known homotopy types and barcodes of the Vietoris-Rips simplicial complexes of the circle, the barcodes of the Vietoris-Rips metric thickenings of the ci…

Higher symmetries in abstract stable homotopy theories

This survey offers an overview of an on-going project on uniform symmetries in abstract stable homotopy theories. This project has calculational, foundational, and representation-theoretic aspects, and key features of this emerging field on abstract representation theory include the following. First, generalizing the classical focus on representations over fields, it is concerned with the study of representations over rings, differential-graded algebras, ring spectra, and in more general abstract stable homotopy theories. Second, restricting attention to specific shapes, it offers an explanati…

Grafting Boardman's Cherry Trees to Quantum Field Theory

Michael Boardman has been a major contributor to the theory of infinite loop spaces and higher homotopy algebra. Indeed Boardman was the first to refer to `homotopy everything'. One particular contribution which has had major progeny is his use of `geometric' trees, combinatorial tress with lengths attached to edges. Here is a modified version of the talk given to honor Mike on the occassion of his 60th birthday. It is an idiosyncratic survey of parts of homotopy algebra from Poincaré to the present day, with emphasis on Boardman's original ideas, starting with his cubical subdivision of the a…

On the homotopy decomposition for the quotient of a moment-angle complex and its applications

In this paper we prove that the quotient of any real or complex moment-angle complex by any closed subgroup in the naturally acting compact torus on it is equivariantly homotopy equivalent to the homotopy colimit of a certain toric diagram. For any quotient we prove an equivariant homeomorphism generalizing the well-known Davis-Januszkiewicz construction for quasitoric manifolds and small covers. We deduce formality of the corresponding Borel construction space under the natural assumption on the group action in the complex case leading to the new description of the equivariant cohomology for …

On fibering and splitting of 5-manifolds over the circle

Our main result is a generalization of Cappell's 5-dimensional splitting theorem. As an application, we analyze, up to internal s-cobordism, the smoothable splitting and fibering problems for certain 5-manifolds mapping to the circle. For example, these maps may have homotopy fibers which are in the class of finite connected sums of certain geometric 4-manifolds. Most of these homotopy fibers have non-vanishing second mod 2 homology and have fundamental groups of exponential growth, which are not known to be tractable by Freedman--Quinn topological surgery. Indeed, our key technique is topolog…

Some smooth circle and cyclic group actions on exotic spheres

Classical work of Lee, Schultz, and Stolz relates the smooth transformation groups of exotic spheres to the stable homotopy groups of spheres. In this note, we apply recent progress on the latter to produce smooth circle and cyclic group actions on certain exotic spheres.

某些单连通7维流形上的自由圆作用

Free circle actions on certain simply connected $7-$manifolds

研究确定哪些非负整数k,l和同调7球面Σ使kS²×S⁵#lS³×S⁴#Σ具有自由光滑圆作用

圆上的Goodwillie塔的阿贝尔模型

An abelian model for the Goodwillie tower of the circle

本文通过Behrens-Kuhn的Whitehead猜想结果，建立了圆上恒等函子Goodwillie塔的p-局部等价，将其逼近识别为无穷环空间，并引入Goodwillie演算的Gray序列类比以简化同伦群计算。

Persistent homology with non-contractible preimages

For a fixed $N$, we analyze the space of all sequences $z=(z_1,\dots,z_N)$, approximating a continuous function on the circle, with a given persistence diagram $P$, and show that the typical components of this space are homotopy equivalent to $S^1$. We also consider the space of functions on $Y$-shaped (resp., star-shaped) trees with a 2-point persistence diagram, and show that this space is homotopy equivalent to $S^1$ (resp., to a bouquet of circles).

Construction of the Circle in UniMath

We show that the type $\mathrm{T}\mathbb{Z}$ of $\mathbb{Z}$-torsors has the dependent universal property of the circle, which characterizes it up to a unique homotopy equivalence. The construction uses Voevodsky's Univalence Axiom and propositional truncation, yielding a stand-alone construction of the circle not using higher inductive types.

最简单 Morse-Bott 叶状结构在透镜空间上的微分同胚群的同伦类型，2

Homotopy types of diffeomorphisms groups of simplest Morse-Bott foliations on lens spaces, 2

本文证明透镜空间上最简单 Morse-Bott 叶状结构的保叶微分同胚群到不交换奇异圆的叶状微分同胚群的包含映射是同伦等价。

Homotopy dimension of orbits of Morse functions on surfaces

Let $f$ be a real- or circle-valued Morse function on a compact surface M having exactly $n>0$ critical points. Denote by $O$ the orbit of $f$ with respect to the right action of the group of diffeomorphisms of $M$. We show that the connected components of $O$ have the homotopy type of a finite-dimensional CW-complex. Actually, these connected components are homotopy equivalent to a certain covering space of the $n$-th configuration space of the interior of $M$. As a consequence we obtain that the fundamental group of $O$ is a subgroup of the $n$-th braid group of $M$.

Intersection cohomology of circle actions

A classical result says that a free action of the circle $\Bbb{S}^1$ on a topological space $X$ is geometrically classified by the orbit space $B$ and by a cohomological class ${H}^{^{2}}{(B,\Bbb{Z})}$, the Euler class. When the action is not free we have a difficult open question: $Π$ : "Is the space $X$ determined by the orbit space $B$ and the Euler class?" The main result of this work is a step towards the understanding of the above question in the category of unfolded pseudomanifolds. We prove that the orbit space $B$ and the Euler class determine: * the intersection cohomology of $X$, * …

四维流形上的非动力学同伦相干作用

Non-kinetic homotopy coherent actions on four-manifolds

该研究给出闭单连通光滑四维流形上二阶非动力学光滑同伦相干作用的首个例子，通过限制离散圆周群在稳定化\(K3\)曲面上的作用获得，还构造相关扩张并展示特殊边界对合，证明了关于整同调球上自由对合光滑扩张的定理。