Gregory Lupton, Samuel Bruce Smith
We give a general method that may be effectively applied to the question of whether two components of a function space have the same homotopy type. We describe certain group-like actions on function spaces. Our basic results assert that if two maps are in the same orbit under such an action, then the components of the function space that contain these maps have the same homotopy type.
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Are two finite $H$-spaces homotopy equivalent?
本文提出一种算法，判定由有限 $k$-型且有限维 $d$ 的有限单纯集合表示的两个单连通空间是否同伦等价，并在等价时找到维 $d$ 的 Postnikov 阶段间的同伦等价，进而推得稳定同伦等价的判定算法。
The reader of doi:10.1016/j.topol.2010.08.006 might conjecture that Δ(\barΠ_n)/G is homotopy equivalent to a wedge of spheres for any n>=3 and any subgroup G<S_n. We disprove this by showing that Δ(\barΠ_p)/C_p is not homotopy equivalent to a wedge of spheres for any prime number p>=5.
When are two spaces homotopy equivalent?
本文研究了从算法角度确定两个有限单纯复形同伦等价的充分必要条件，基于Postnikov塔的底层数据和有效同调。
The state spaces of machines admit the structure of time. A homotopy theory respecting this additional structure can detect machine behavior unseen by classical homotopy theory. In an attempt to bootstrap classical tools into the world of abstract spacetime, we identify criteria for classically homotopic, monotone maps of pospaces to future homotope, or homotope along homotopies monotone in both coordinates, to a common map. We show that consequently, a hypercontinuous lattice equipped with its Lawson topology is future contractible, or contractible along a future homotopy, if its underlying s…
Generalized Chevalley criteria in simplicial homotopy type theory
本研究在Riehl–Shulman单纯同伦类型论框架下，将（余）笛卡尔箭头、纤维化与函子的经典条件推广为以任意形状包含替代端点包含的广义形式，为合成内蕴$(\u221e,1)$-范畴论研究提供了新的处理方式。
We classify, up to homeomorphism, all closed manifolds having the homotopy type of a connected sum of two copies of real projective n-space.
Given a simplicial group G, there are two known classifying simplicial set constructions, the Kan classifying simplicial set Wbar G and Diag N G, where N denotes the dimensionwise nerve. They are known to be weakly homotopy equivalent. We will show that Wbar G is a strong simplicial deformation retract of Diag N G. In particular, Wbar G and Diag N G are simplicially homotopy equivalent.
Graphs and Their Clique Complexes Are Pseudotopological Weakly Homotopy Equivalent
研究图与其团复形几何实现的关系，通过建立联系，证明具有特定结构的图与团复形几何实现弱同伦等价。
Homotopy rigidity for quasitoric manifolds over a product of simplices
本文研究上同调刚性问题的变体，证明当两个拟环流流形的上同调环同构且其中一个的多胞体为单纯形乘积时，它们在局部化掉显式素数列表后同伦等价。
This paper gives an introduction to homotopy descent, and its applications in algebraic $K$-theory computations for fields. On the étale site of a field, a fibrant model of a simplicial presheaf can be constructed from naive Galois cohomological objects given by homotopy fixed point constructions, but only up to pro-equivalence. The homotopy fixed point spaces define finite Galois descent for simplicial presheaves (and their relatives) over a field, but a pro-categorical construction is a necessary second step for passage from finite descent conditions to full homotopy descent in a Galois coho…
We present a counterexample to the conjecture on the homotopy invariance of configuration spaces. More precisely, we consider the lens spaces $L_{7,1}$ and $L_{7,2}$, and prove that their configuration spaces are not homotopy equivalent by showing that their universal coverings have different Massey products.
Finite type nilpotent spaces are weakly equivalent if and only if their singular cochains are quasi-isomorphic as E-infinity algebras. The cochain functor from the homotopy category of finite type nilpotent spaces to the homotopy category of E-infinity algebras is faithful but not full.
We introduce a new class of poset edge labelings for locally finite lattices which we call $SB$-labelings. We prove for finite lattices which admit an $SB$-labeling that each open interval has the homotopy type of a ball or of a sphere of some dimension. Natural examples include the weak order, the Tamari lattice, and the finite distributive lattices.
In this paper, we investigate the Quillen model structure defined by Bisson and Tsemo in the category of directed graphs Gph. In particular, we give a precise description of the homotopy category of graphs associated to this model structure. We endow the categories of N-sets and Z-sets with related model structures, and show that their homotopy categories are Quillen equivalent to the homotopy category Ho(Gph). This enables us to show that Ho(Gph) is equivalent to the category cZSet of periodic Z-sets, and to show that two finite directed graphs are almost-isospectral if and only if they are h…
Smooth manifolds homotopy equivalent to products of spheres
分类了同伦等价于特定球面乘积的光滑闭定向流形，通过球面丛、Milnor 管道连接和 Novikov 球面丛等显式构造实现了正常不变映射的像。
We prove that the homotopy type of a map from a Peano continuum into a planar or one-dimensional space is determined by the induced homomorphism of fundamental groups. This provides a new proof that planar sets are aspherical and is used to prove that two planar or one-dimensional Peano continua are homotopy equivalent if and only if they have isomorphic fundamental groups.
In this article, we show that the Fredholm Lagrangian Grassmannian is homotopy equivalent with the space of compact perturbations of a fixed lagrangian. As a corollary, we obtain that the Maslov index with respect to a lagrangian is a isomorphism between the fundamental group of the Fredholm Lagrangian Grassmannian and the integers.
Homotopy rigidity for quasitoric manifolds over a product of $d$-simplices
本文证明了在适当局部化条件下，若两个定义在d-单形乘积上的拟环面流形具有同构的整系数上同调环，则它们必然同伦等价，从而揭示了这类流形的同伦刚性。
Rational Homotopy in Pseudotopological Spaces
本文证明 Rieser 的伪拓扑空间模型范畴与单纯集 Quillen 等价，并利用与拓扑 CW 复形的弱等价关系，为单连通伪拓扑空间建立有理同伦，同时证明伪拓扑 CW 复形的 Cartesian 积封闭。
Homotopy Theory for Ordered Simplicial Complexes
本文构建了有序单纯复形的模型结构，与标准的单纯集模型结构Quillen等价，证明了单纯复形为空间同伦论提供了新的模型。
Generalised doubles and simple homotopy types of high dimensional manifolds
本文刻画了高维流形中存在 h-配边但非简单同伦等价例子的基本群条件，并基于加厚体二重结构及 Whitehead 挠率公式，证明当 n≥12 为偶数且 Wh(G) 对合非平凡时必存在此类例子。
We show that the free construction from multicategories to permutative categories is a categorically-enriched non-symmetric multifunctor. Our main result then shows that the induced functor between categories of algebras is an equivalence of homotopy theories. We describe an application to ring categories.
Some Properties Of Proximal Homotopy Theory
本文探究邻近空间及描述性邻近空间中的若干同伦论概念，揭示了邻近理论与同伦论通过度量空间及点邻近性建立的紧密关联。
We prove that the homotopy theory of cofibration categories is equivalent to the homotopy theory of cocomplete quasicategories. This is achieved by presenting both homotopy theories as fibration categories and constructing an explicit equivalence between them.
In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify small contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every small contravariant functor from spaces to spaces which takes coproducts to products up to homotopy and takes homotopy pushouts to homotopy pullbacks is naturally weekly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the …
We construct a cofibrantly generated model structure on the category of flows such that any flow is fibrant and such that two cofibrant flows are homotopy equivalent for this model structure if and only if they are S-homotopy equivalent. This result provides an interpretation of the notion of S-homotopy equivalence in the framework of model categories.
We study the homotopy types of moment-angle complexes, or equivalently, of complements of coordinate subspace arrangements. The overall aim is to identify the simplicial complexes K for which the corresponding moment-angle complex Z_K has the homotopy type of a wedge of spheres or a connected sum of sphere products. When K is flag, we identify in algebraic and combinatorial terms those K for which Z_K is homotopy equivalent to a wedge of spheres, and give a combinatorial formula for the number of spheres in the wedge. This extends results of Berglund and Joellenbeck on Golod rings and homotopy…
An algebraic theory $T$ is a category with objects $t_0,t_2...$ such that for each $n$ the object $t_n$ is an $n$-fold categorical product of $t_1$. A strict $T$-algebra is a product preserving functor $A: T\to Spaces$. Lawvere showed that for a suitable choice of T giving such an algebra amounts to providing the space $A(t_1)$ with a familiar structure of a monoid group, ring, Lie algebra... Given a functor $X: T\to Spaces$ which preserves products up to a weak equivalence we show that $X$ is more or less canonically weakly equivalent to a strict $T$-algebra $LX$. Thus any `homotopy' algebrai…
In this article, we construct a cofibrantly generated model structure on the category of spaces stratified over a fixed poset, and show that it is Quillen-equivalent to a category of diagrams of simplicial sets. Then, considering all those model structures together, we construct a cofibrantly generated model structure on the category of all stratified spaces. In both model categories, weak-equivalences are characterized by stratified homotopy groups.
Crossed modules are known to be a model of pointed connected homotopy 2-types; formally, the homotopy category of crossed modules is equivalent to the category of pointed connected homotopy 2-types. In forming the homotopy category of crossed modules, one must resort to computing derived morphisms using non-constructive topological methods, but Behrang Noohi was able to find an algebraic model of these derived morphisms called butterflies. The result is a completely algebraic model of pointed connected homotopy 2-types. Reduced crossed complexes are a generalization of crossed modules and mode…
On smooth and bundle structures on topological manifolds homotopy equivalent to $S^{4k-1}$-bundles over $S^{4k}$
本文验证了 $S^{4k}$ 上 $S^{4k-1}$ 丛全空间的拓扑结构集计算，并研究了其中元素能否实现为丛，确定了光滑到拓扑结构集遗忘映射的像，像中元素底层流形均为该丛。
Simple homotopy types of even dimensional manifolds
本文研究偶维闭流形的简单同伦类型，利用代数K理论、手术障碍映射和同伦自同构体构造了首个例子，展示了同伦等价但非简单同伦等价的流形族。
We show that the independence complexes of generalised Mycielskian of complete graphs are homotopy equivalent to a wedge sum of spheres, and determine the number of copies and the dimensions of these spheres. We also prove that the independence complexes of categorical product of complete graphs are wedge sum of circles, upto homotopy. Further, we show that if we perturb a graph $G$ in a certain way, then the independence complex of this new graph is homotopy equivalent to the suspension of the independence complex of $G$.
This article explains and extends semialgebraic homotopy theory (developed by H. Delfs and M. Knebusch) to o-minimal homotopy theory (over a field). The homotopy category of definable CW-complexes is equivalent to the homotopy category of topological CW-complexes (with continuous mappings). If the theory of the o-minimal expansion of a field is bounded, then these categories are equivalent to the homotopy category of weakly definable spaces. Similar facts hold for decreasing systems of spaces. As a result, generalized homology and cohomology theories on pointed weak polytopes uniquely correspo…
We lift Charles Rezk's complete Segal space model structure on the category of simplicial spaces to a Quillen equivalent one on the category of relative categories.
In previous work, the first author defined homotopy theories for stratified spaces from a simplicial and a topological perspective. In both frameworks stratified weak-equivalences are detected by suitable generalizations of homotopy links. These two frameworks are connected through a stratified version of the classical adjunction between the realization and the functor of singular simplices. Using a modified version of this adjunction, the first author showed that over a fixed poset of strata the two homotopy theories were equivalent. Building on this result we now show that the unmodified adj…
In this paper, we give a new construction of a Khovanov homotopy type. We show that this construction gives a space stably homotopy equivalent to the Khovanov homotopy types constructed in [LS14a] and [HKK] and, as a corollary, that those two constructions give equivalent spaces. We show that the construction behaves well with respect to disjoint unions, connected sums and mirrors, verifying several conjectures from [LS14a]. Finally, combining these results with computations from [LS14c] and the refined s-invariant from [LS14b] we obtain new results about the slice genera of certain knots.
One proves that the category of globular CW-complexes up to dihomotopy is equivalent to the category of flows up to weak dihomotopy. This theorem generalizes the classical theorem which states that the category of CW-complexes up to homotopy is equivalent to the category of topological spaces up to weak homotopy. This note is the second one presenting some of the results of math.AT/0201252.
Shape theory works nice for (Hausdorff) paracompact spaces, but for spaces with no separation axioms, it seems to be quite poor. However, for finite and locally finite spaces their weak homotopy type is rather rich, and is equivalent to the weak homotopy type of finite and locally finite polynedra, respectively. In the paper there is proposed a variant of shape theory called quasi-shape, which suits both paracompact and locally finite spaces, i.e. the quas-shape is isomorphic to the weak homotopy type for locally finite spaces, and is \natural-equivalent to the ordinary shape in the case of pa…
Vietoris thickenings and complexes are weakly homotopy equivalent
本文解决了 Vietoris--Rips 复形与其度量加厚之间同伦类型的关系问题，证明了对于任意度量空间 $X$ 和阈值 $r>0$，两者间的自然双射构成弱同伦等价。
We give a homotopy theoretic characterization of stacks on a site $\cC$ as the {\it homotopy sheaves} of groupoids on $\cC$. We use this characterization to construct a model category in which stacks are the fibrant objects. We compare different definitions of stacks and show that they lead to Quillen equivalent model categories. In addition, we show that these model structures are Quillen equivalent to the $S^2$-nullification of Jardine's model structure on sheaves of simplicial sets on $\cC$.
Complexes equivalent to $S^{2k-1}$-fibrations over $S^{2k}$
本文研究了CW复形顶细胞的粘贴映射 $f$ 具有 $S^{2k-1}$-纤维 over $S^{2k}$ 的同伦类型条件，并确定了总空间的粘贴映射阶数，当 $k\ne 2,4$ 时，通过稳定同伦类对总空间进行分类。
For an arbitrary simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CW-complexes whose integral cohomology rings are isomorphic to the Stanley-Reisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction (here called c(K)), which they showed to be homotopy equivalent to Davis and Januszkiewicz's examples. It is therefore natural to investigate the extent to which the homotopy type of a space is determined by having such a cohomology ring. We begin this study here, in the context of model category theory. In particular, we ext…
We decompose the K-theory space of a Waldhausen category in terms of its Dwyer-Kan simplicial localization. This leads to a criterion for functors to induce equivalences of K-theory spectra that generalizes and explains many of the criteria appearing in the literature. We show that under mild hypotheses, a weakly exact functor that induces an equivalence of homotopy categories induces an equivalence of K-theory spectra.
We compare two models for $\infty$-operads: the complete Segal operads of Barwick and the complete dendroidal Segal spaces of Cisinski and Moerdijk. Combining this with comparison results already in the literature, this implies that all known models for $\infty$-operads are equivalent - for instance, it follows that the homotopy theory of Lurie's $\infty$-operads is equivalent to that of dendroidal sets and that of simplicial operads.
The framed little 2-discs operad is homotopy equivalent to a cyclic operad. We show that the derived modular envelope of this cyclic operad (i.e., the modular operad freely generated in a homotopy invariant sense) is homotopy equivalent to the modular operad made from classifying spaces of diffeomorphism groups of 3-dimensional handlebodies with marked discs on their boundaries. A modification of the argument provides a new and elementary proof of K. Costello's theorem that the derived modular envelope of the associative operad is homotopy equivalent to the ``open string'' modular operad made …
We consider a certain class of simplicial complexes which includes the independence complexes of forests. We show that if a simplicial complex $K$ belongs to this class, then the polyhedral join $\mathcal{Z}^*_{K}(\underline{X}, \emptyset)$ is homotopy equivalent to a wedge sum of CW complexes of the form $Σ^r X_{i_1} * X_{i_2} * \cdots * X_{i_k}$, where $\underline{X}$ is a family $\{X_i\}_{i \in V(K)}$ of CW complexes and $Σ$ denotes the unreduced suspension. This result is applied to study the homotopy type of the independence complex of the lexicographic product $G[H]$ of a graph $H$ over …
In the seminal monograph "Theory of retracts", Borsuk raised the following question: suppose two compact ANR's are $h$--equal, i.e. mutually homotopy dominate each other, are they homotopy equivalent? The current paper approaches this question in two ways. On one end, we provide conditions on the fundamental group which guarantee a positive answer to the Borsuk question. On the other end, we construct various examples of compact $h$--equal, not homotopy equivalent continua, with distinct properties. The first class of these examples has trivial all known algebraic invariants (such as homology,…
The configuration space $F_2 (M)$ of ordered pairs of distinct points in a manifold $M$, also known as the deleted square of $M$, is not a homotopy invariant of $M$: Longoni and Salvatore produced examples of homotopy equivalent lens spaces $M$ and $N$ of dimension three for which $F_2 (M)$ and $F_2 (N)$ are not homotopy equivalent. In this paper, we study the natural question whether two arbitrary $3$-dimensional lens spaces $M$ and $N$ must be homeomorphic in order for $F_2 (M)$ and $F_2 (N)$ to be homotopy equivalent. Among our tools are the Cheeger--Simons differential characters of delete…
Manifold structures on highly connected Poincaré complexes
本文构造了同伦等价于拓扑流形却不同伦等价于光滑流形的高连通庞加莱复形，并确定了k=7、15、31时特定Kervaire不变量1的闭带边流形的同伦型。
Let $f:T^2\to\mathbb{R}$ be a Morse function on $2$-torus $T^2$ such that its Kronrod-Reeb graph $Γ(f)$ has exactly one cycle, i.e. it is homotopy equivalent to $S^1$. Under some additional conditions we describe a homotopy type of the orbit of $f$ with respect to the action of the group of diffeomorphism of $T^2$. This result holds for a larger class of smooth functions $f:T^2\to\mathbb{R}$ having the following property: for every critical point $z$ of $f$ the germ of $f$ at $z$ is smoothly equivalent to a homogeneous polynomial $\mathbb{R}^2\to\mathbb{R}$ without multiple factors.
Let p be a fibration over a finite simplicial complex, whose fibers have the homotopy type of finite simplicial complexes. Then p is equivalent to an approximate fibration whose total space is a compact ENR. The proof uses homotopy coherent diagrams and their homotopy colimits. We also comment on the simple homotopy type of the total space and give an application to the fibering of Hilbert cube manifolds.
Homotopy reflectivity is equivalent to the weak Vopěnka principle
本文证明单纯集、谱及可展示 ∞-范畴中的任意映射类局部化等反射性陈述均等价于弱 Vopěnka 原理，并借助 Wilson 2020 年的相对一致性结果完成论证。
We prove that the homotopy prederivator of a cofibration category is equivalent to the homotopy prederivator of its associated quasi-category of frames, as introduced by Szumiło. We use this comparison result to deduce various abstract properties of the obtained prederivators.
Coarse Homotopy Theory and Shape Theory
建立欧几里得锥的粗同伦理论与紧度量空间形状理论的联系，通过逆极限序列和强形状等价刻画粗同伦等价，并证明粗Whitehead定理及其反例。
We prove that the space of gauge equivalence classes of U(1)-invariant connections on some SU(2)-principle bundles over the 4-sphere S^4 is weakly homotopy equivalent to a component of the second loop space of the 2-sphere S^2.
We show that the rational homotopy type of the complement of a toric arrangement is completely determined by two sets of combinatorial data. This is obtained by introducing a differential graded algebra over Q whose minimal model is equivalent to the Sullivan minimal model of the arrangement.
We prove that the homotopy theory of Picard 2-categories is equivalent to that of stable 2-types.
The complement of the codimension 2 complex coordinate subspace arrangement is shown to be homotopy equivalent to a wedge of spheres.
Theorem (after Giraud, SGA 4): Suppose $A$ is a simplicial category. The following conditions are equivalent: (i) There is a cofibrantly generated closed model category $M$ such that $A$ is equivalent to the Dwyer-Kan simplicial localization $L(M)$; (ii) $A$ admits all small homotopy colimits, and there is a small subset of objects of $A$ which are $A$-small, and which generate $A$ by homotopy colimits; (iii) There exists a small 1-category $C$ and a morphism $g:C\to A$ sending objects of $C$ to $A$-small objects, which induces a fully faithful inclusion $i:A\to \hat{C}$, such that $i$ admits …
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