The homotopy theory of equivalence relations

Nilpotency of self homotopy equivalences with coefficients

In this paper we study the nilpotency of certain groups of self homotopy equivalences. Our main goal is to extend, to localized homotopy groups and/or homotopy groups with coefficients, the general principle of Dror and Zabrodsky by which a group of self homotopy equivalences of a finite space which acts nilpotently on the homotopy groups is itself nilpotent.

Triangular homotopy equivalences

A map $f:X\to Y$ to a simplicial complex $Y$ is called a $Y$-triangular homotopy equivalence if it has a homotopy inverse $g$ and homotopies $h_1:f\circ g\simeq \mathrm{id}_Y$, $h_2:g\circ f\simeq \mathrm{id}_X$ such that for all simplices $σ\in Y$, $f|_σ:f^{-1}(σ) \to σ$ is a homotopy equivalence with inverse $g|_σ:σ\to f^{-1}(σ)$ and homotopies $h_1|_σ$ and $h_2|_σ$. In this paper we prove that for all pairs $X,Y$ of finite-dimensional locally finite simplicial complexes there is an $ε(X,Y)>0$ such that any $ε$-controlled homotopy equivalence $f:X\to Y$ for $ε<ε(X,Y)$ is homotopic to a $Y$-t…

富集范畴的 Dwyer-Kan 同伦理论

Dwyer-Kan homotopy theory of enriched categories

本文在满足幺半群公理的组合闭对称幺半模范畴上的富集小范畴范畴中，建立了以 Dwyer-Kan 等价为弱等价的模型结构。

同伦自等价的群是一个Lax函子

The group of homotopy self-equivalences is a Lax functor

本文证明了拓扑空间的同伦自等价群 $\E(X)$ 构成一个从拓扑空间范畴 $\mathscr Top$ 到群对应严格2-范畴 $\op{Corr}_{\mathscr Gr}$ 的Lax函子。

On the group of self-homotopy equivalences of an $F_{0}$-Space

In \cite{G}, G. Lupton conjectured that the group of self-homotopy equivalences of an $F_0$-space inducing the identity on the homotopy groups is finite. Thus, the aim of this paper is to establish this conjecture.

T-homotopy and refinement of observation (V) : Strom model structure for branching and merging homologies

We check that there exists a model structure on the category of flows whose weak equivalences are the S-homotopy equivalences. As an application, we prove that the generalized T-homotopy equivalences preserve the branching and merging homology theories of a flow. The method of proof is completely different from the one of the third part of this series of papers.

Left fibrations and homotopy colimits II

For a small simplicial category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the homotopy-coherent nerve of A provides a left Quillen equivalence between the projective model structure on the former category and the covariant model structure on the latter. We compare this Quillen equivalence to the straightening-unstraightening equivalence previously established by Lurie, where the left adjoint goes in the opposite direction. The existence of left Quillen functors in both directions considerably simplifies t…

椭圆空间的自同伦等价群及其在一般线性群中的嵌入

Self-Homotopy Equivalence Group of an Elliptic Space and Its Embedding in general Linear Groups

本文研究有理椭圆空间的同伦群与自同伦等价群的关系，并探讨该群在一般线性群中的嵌入。

Multifunctorial $K$-Theory is an Equivalence of Homotopy Theories

We show that each of the three $K$-theory multifunctors from small permutative categories to $\mathcal{G}_*$-categories, $\mathcal{G}_*$-simplicial sets, and connective spectra, is an equivalence of homotopy theories. For each of these $K$-theory multifunctors, we describe an explicit homotopy inverse functor. As a separate application of our general results about pointed diagram categories, we observe that the right-induced homotopy theory of Bohmann-Osorno $\mathcal{E}_*$-categories is equivalent to the homotopy theory of pointed simplicial categories.

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the "homotopy theory" of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a "homotopy theory of homotopy theories." In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to …

T-homotopy and refinement of observation (IV) : Invariance of the underlying homotopy type

This series explores a new notion of T-homotopy equivalence of flows. The new definition involves embeddings of finite bounded posets preserving the bottom and the top elements and the associated cofibrations of flows. In this fourth part, it is proved that the generalized T-homotopy equivalences preserve the underlying homotopy type of a flow. The proof is based on Reedy model category techniques.

Monoids related to self homotopy equivalences of fibred product

We introduce a nested sequence of monoids related to self-homotopy equivalences of fibrewise pointed spaces, such that the limit is the group of homotopy classes of fibrewise pointed self-equivalences. We explore this monoid for the fibred product in terms of individual spaces. Further we study two related invariants associated to these monoids: self closeness number and self length.

有限群作为有限空间的同伦自等价

Finite groups as homotopy self-equivalences of finite spaces

本文研究有限群作为有限空间同伦自等价类群的实现问题，利用无穷族两两非弱同伦的反对称空间，构造出同伦自等价类群同构于给定有限群$G$的有限空间。

Fibrations up to an equivalence, homotopy colimits and pullbacks

We gather conditions on a class H of continuous maps of topological spaces that allow a reasonable theory of fibrations up to an equivalence (a map from this class) which we call H-fibrations. The weak homotopy equivalences recover quasifibrations and homology equivalences yield homology fibrations. We study local H-fibrations that behave nicely with respect to homotopy colimits together with universal H-fibrations that behave nicely with respect to pullbacks. We then proceed to classify H-fibrations up to a natural notion of equivalence.

理性同调论的现代视角

A modern perspective on rational homotopy theory

本文通过现代方法扩展了Quillen的模型结构，将理性同调等价和理性空间的概念推广至更广泛的空间范畴。

Subgroups of the group of self-homotopy equivalences

Denote by E(Y) the group of homotopy classes of self-homotopy equivalences of a finite-dimensional complex Y. We give a selection of results about certain subgroups of E(Y). We establish a connection between the Gottlieb groups of Y and the subgroup of E(Y) consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of Y, denoted by E_#(Y). We give an upper bound for the solvability class of E_#(Y) in terms of a cone decomposition of Y. We dualize the latter result to obtain an upper bound for the solvability class of the subgroup of E(Y) consist…

Topological model categories generated by finite complexes

Our main result states that for each finite complex L the category ${\bf TOP}$ of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov's notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them -- describing ho…

Spaces which invert weak homotopy equivalences

It is well known that if $X$ is a CW-complex, then for every weak homotopy equivalence $f:A\to B$, the map $f_*:[X,A]\to [X,B]$ induced in homotopy classes is a bijection. For which spaces $X$ is $f^*:[B,X]\to [A,X]$ a bijection for every weak equivalence $f$? This question was considered by J. Strom and T. Goodwillie. In this note we prove that a non-empty space inverts weak equivalences if and only if it is contractible.

Realizability of the group of rational self-homotopy equivalences

For a 1-connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalences of $X$. The aim of this paper is to prove that, for every $n\in\Bbb N$, there exists a 1-connected rational CW-complex $X_{n}$ such that $\mathcal{E}(X_{n})\cong \underset{2^{n+1}\mathrm{. times}}{\underbrace{\Bbb Z_{2}\oplus... \Bbb \oplus \Bbb Z_{2}}}$.

Simplicial and categorical comma categories

We consider four categories: the category of diagrams of small categories indexed by a given small category O, the (comma) category of small categories over O, the category of diagrams of simplicial sets indexed by O, and the category of simplicial sets over the nerve of O. Fritsch and Golasinski claimed that these four categories have equivalent homotopy categories but, in fact, their proof contains an error and the homotopy categories are not equivalent with the weak equivalences they use in the comma categories. We show here that the correct weak equivalences are the ``weak fibre homotopy e…

Path categories and quasi-categories

This paper gives an introduction to the homotopy theory of quasi-categories. Weak equivalences between quasi-categories are characterized as maps which induce equivalences on a naturally defined system of groupoids. These groupoids effectively replace higher homotopy groups in quasi-category homotopy theory.

Explicit Homotopy Equivalences Between Some Operads

In this work we present an explicit operad morphism that is also a homotopy equivalence between the operad given by the real Fulton MacPherson compactification of configuration spaces and the little $n$-disks operad. In particular, the construction gives an operadic homotopy equivalence between the associahedra and the little intervals explicitly. It can also be extended to the case of Kontsevich compactification and Voronov swiss-cheese operad.

有理同调等价

Rational Homotopy Equivalence

本文提出算法构建Sullivan极小模型，用于判断两个连通空间是否具有相同的有理同调类型。

Rational homotopy equivalences and singular chains

Bousfield and Kan's $\mathbb{Q}$-completion and fiberwise $\mathbb{Q}$-completion of spaces lead to two different approaches to the rational homotopy theory of non-simply connected spaces. In the first approach, a map is a weak equivalence if it induces an isomorphism on rational homology. In the second, a map of connected and pointed spaces is a weak equivalence if it induces an isomorphism between fundamental groups and higher rationalized homotopy groups; we call these maps $π_1$-rational homotopy equivalences. In this paper, we compare these two notions and show that $π_1$-rational homotop…

Equivariant self-homotopy equivalences of product spaces

Let G be a finite group. We study the group of G-equivariant self-homotopy equivalences of product of G-spaces. For a product of n-spaces, we represent it as product of n-subgroups under the assumption of equivariant reducibility. Further we describe each factor as a split short exact sequence. Also, we obtain an another kind of factorisation, called $LU$ type decomposition, as product of two subgroups.

Left fibrations and homotopy colimits

For a small category A, we prove that the homotopy colimit functor from the category of simplicial diagrams on A to the category of simplicial sets over the nerve of A establishes a left Quillen equivalence between the projective (or Reedy) model structure on the former category and the covariant model structure on the latter. We compare this equivalence to a Quillen equivalence in the opposite direction previously established by Lurie. From our results we deduce that a categorical equivalence of simplicial sets induces a Quillen equivalence on the corresponding over-categories, equipped with …

The rational homotopy type of the space of self-equivalences of a fibration

Let Aut(p) denote the topological monoid of self-fibre-homotopy equivalences of a fibration p:E\to B. We make a general study of this monoid, especially in rational homotopy theory. When E and B are simply connected CW complexes with E finite, we identify the rational Samelson Lie algebra of the identity component of Aut(p) as the homology of a certain DG Lie algebra of derivations arising from the Koszul-Sullivan model of p. We obtain related identifications for the rational homotopy groups of fibrewise mapping spaces and for the rationalization of a natural nilpotent subgroup of Aut(p).

Rational visibility of a Lie group in the monoid of self-homotopy equivalences of a homogeneous space

Let M be a homogeneous space admitting a left translation by a connected Lie group G. The adjoint to the action gives rise to a map from G to the monoid of self-homotopy equivalences of M.The purpose of this paper is to investigate the injectivity of the homomorphism which is induced by the adjoint map on the rational homotopy. In particular, the visible degrees are determined explicitly for all the cases of simple Lie groups and their associated homogeneous spaces of rank one which are classified by Oniscik.

The group of self-homotopy equivalences of $A_n^2$-polyhedra

Let $X$ be a finite type $A_n^2$-polyhedron, $n \geq 2$. In this paper we study the quotient group $\mathcal{E}(X)/\mathcal{E}_*(X)$, where $\mathcal{E}(X)$ is the group of self-homotopy equivalences of $X$ and $\mathcal{E}_*(X)$ the subgroup of self-homotopy equivalences inducing the identity on the homology groups of $X$. We show that not every group can be realised as $\mathcal{E}(X)$ or $\mathcal{E}(X)/\mathcal{E}_*(X)$ for $X$ an $A_n^2$-polyhedron, $n\ge 3$, and specific results are obtained for $n=2$.

Certain maps preserving self-homotopy equivalences

Let $\mathcal{E}(X)$ be the group of homotopy classes of self homotopy equivalences for a connected CW complex $X$. We observe two classes of maps $\mathcal{E}$-maps and co-$\mathcal{E}$-maps. They are defined as the maps $X\to Y$ that induce the homomorphisms $\mathcal{E}(X)\to \mathcal{E}( Y)$ and $\mathcal{E}(Y)\to \mathcal{E}(X)$, respectively. We give some rationalized examples related to spheres, Lie groups and homogeneous spaces by using Sullivan models. Furthermore, we introduce an $\mathcal{E}$-equivalence relation between rationalized spaces $X_{\mathbb{Q}}$ and $Y_{\mathbb{Q}}$ as a…

The de Rham homotopy theory and differential graded category

This paper is a generalization of arXiv:0810.0808. We develop the de Rham homotopy theory of not necessarily nilpotent spaces, using closed dg-categories and equivariant dg-algebras. We see these two algebraic objects correspond in a certain way. We prove an equivalence between the homotopy category of schematic homotopy types and a homotopy category of closed dg-categories. We give a description of homotopy invariants of spaces in terms of minimal models. The minimal model in this context behaves much like the Sullivan's minimal model. We also provide some examples. We prove an equivalence be…

Rational Homotopy Type of the Classifying Space for Fibrewise Self-Equivalences

Let p be a fibration of simply connected CW complexes with finite base B and fibre F. Let aut_1(p) denote the identity component of the space of all fibre-homotopy self-equivalences of p and Baut_1(p) the classifying space for this topological monoid. We give a differential graded Lie algebra model for Baut_1(p). We use this model to give classification results for the rational homotopy types represented by Baut_1(p) and also to obtain conditions under which the monoid aut_1(p) is a double loop-space after rationalization.

每个群都是有限维CW复形的自同伦等价群

Every group is the group of self homotopy equivalences of finite dimensional CW-complex

本研究证明任意群都可作为有限维CW复形的自同伦等价群，推广了Costoya和Viruel关于有限群可作为有理椭圆空间自同伦等价群的经典定理。

Every finite group is the group of self homotopy equivalences of an elliptic space

In this paper we prove that every finite group $G$ can be realized as the group of self-homotopy equivalences of infinitely many elliptic spaces $X$. Moreover, $X$ can be chosen to be the rationalization of an inflexible compact simply connected manifold.

Homotopy equivalence of spaces of metrics with invertible Dirac operator

We prove that for cobordant closed spin manifolds of dimension $n\geq 3$ the associated spaces of metrics with invertible Dirac operator are homotopy equivalent. This is the spinorial counterpart of a similar result on positive scalar curvature of Chernysh/Walsh and generalizes the surgery result of Ammann-Dahl-Humbert on the existence of metrics with invertible Dirac operator under surgery. We also give a relative statement of this homotopy equivalence.

Bar-cobar duality for operads in stable homotopy theory

We extend bar-cobar duality, defined for operads of chain complexes by Getzler and Jones, to operads of spectra in the sense of stable homotopy theory. Our main result is the existence of a Quillen equivalence between the category of reduced operads of spectra (with the projective model structure) and a new model for the homotopy theory of cooperads of spectra. The crucial construction is of a weak equivalence of operads between the Boardman-Vogt W-construction for an operad P, and the cobar-bar construction on P. This weak equivalence generalizes a theorem of Berger and Moerdijk that says the…

Rational self-homotopy equivalences and Whitehead exact sequence

For a simply connected CW-complex $X$, let $\mathcal{E}(X)$ denote the group of homotopy classes of self-homotopy equivalence of $X$ and let $\mathcal{E}_{\sharp}(X)$ be its subgroup of homotopy classes which induce the identity on homotopy groups. As we know, the quotient group $\frac{\mathcal{E}(X)}{\mathcal{E}_{\sharp}(X)}$ can be identified with a subgroup of $Aut(π_{*}(X))$. The aim of this work is to determine this subgroup for rational spaces. We construct the Whitehead exact sequence associated with the minimal Sullivan model of $X$ which allows us to define the subgroup $\mathrm{Coh.A…

Homotopy self-equivalences of 4-manifolds

We establish a braid of interlocking exact sequences containing the group of homotopy self-equivalences of a smooth or topological 4-manifold. The braid is computed for manifolds whose fundamental group is finite of odd order.

Homotopy equivalences of localized aspherical complexes

By studying the group of self homotopy equivalences of the localization (at a prime $p$ and/or zero) of some aspherical complexes, we show that, contrary to the case when the considered space is a nilpotent complex, $\mathcal{E}_{\#}^m (X_{p})$ is in general different from $\mathcal{E}_{\#}^m (X)_{p}$. That is the case even when $X=K(G,1)$ is a finite complex and/or $G$ satisfies extra finiteness or nilpotency conditions, for instance, when $G$ is finite or virtually nilpotent.

s-Cobordism classification of $4$-manifolds through the group of homotopy self-equivalences

The aim of this paper is to give an $s$-cobordism classification of topological $4$-manifolds in terms of the standard invariants using the group of homotopy self-equivalences. Hambleton and Kreck constructed a braid to study the group of homotopy self-equivalences of $4$-manifolds. Using this braid together with the modified surgery theory of Kreck, we give an $s$-cobordism classification for certain $4$-manifolds with fundamental group $π$, such that cd $π\leq 2$.

On the homotopy groups of the self equivalences of linear spheres

Let $S(V)$ be a complex linear sphere of a finite group $G$. %the space of unit vectors in a complex representation $V$ of a finite group $G$. Let $S(V)^{*n}$ denote the $n$-fold join of $S(V)$ with itself and let $\aut_G(S(V)^*)$ denote the space of $G$-equivariant self homotopy equivalences of $S(V)^{*n}$. We show that for any $k \geq 1$ there exists $M>0$ which depends only on $V$ such that $|π_k \aut_G(S(V)^{*n})| \leq M$ is for all $n \gg 0$.

Monoidal uniqueness theorems for stable homotopy theory

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. As an application we show that with an added assumption about underlying model structures Margolis' axioms uniquely determine the stable homotopy category of spectra up to monoidal equivalence. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, \W-spaces, orthogonal sp…

Constructing homotopy equivalences of chain complexes of free ZG-modules

We describe a general method for algorithmic construction of G-equivariant chain homotopy equivalences from non-equivariant ones. As a consequence, we obtain an algorithm for computing equivariant (co)homology of Eilenberg-MacLane spaces K(pi,n), where pi is a finitely generated ZG-module. This result will be used in a forthcoming paper to construct equivariant Postnikov towers of simply connected spaces with free actions of a finite group G and further to compute stable equivariant homotopy classes of maps between such spaces. The methods of this paper work for modules over any non-negatively…

Homotopy coherent theorems of Dold-Kan type

We establish a large class of homotopy coherent Morita-equivalences of Dold-Kan type relating diagrams with values in any weakly idempotent complete additive $\infty$-category; the guiding example is an $\infty$-categorical Dold-Kan correspondence between the $\infty$-categories of simplicial objects and connective coherent chain complexes. Our results generalize many known 1-categorical equivalences such as the classical Dold-Kan correspondence, Pirashvili's Dold-Kan type theorem for abelian $Γ$-groups and, more generally, the combinatorial categorical equivalences of Lack and Street.

同伦等价的阻碍序列

Obstruction sequences to homotopy equivalences

研究完备微分分次李代数中规范等价存在性的阻碍理论，用于刻画特定微分分次代数间同伦等价，应用此理论在代数拓扑和几何中建立新结果，尤其关注高连通簇极小模型。

Dendroidal sets and simplicial operads

We establish a Quillen equivalence relating the homotopy theory of Segal operads and the homotopy theory of simplicial operads, from which we deduce that the homotopy coherent nerve functor is a right Quillen equivalence from the model category of simplicial operads to the model category structure for infinity-operads on the category of dendroidal sets. By slicing over the monoidal unit, this also gives the Quillen equivalence between Segal categories and simplicial categories proved by J. Bergner, as well as the Quillen equivalence between quasi-categories and simplicial categories proved by …

On the group of self-homotopy equivalences of a 2-connected and 6-dimensional CW-complex

Let $X$ be a \text{\rm{2}}-connected and \text{\rm{6}}-dimensional CW-complex $X$ such that $H_{3}(X)\otimes\Z_2=0$. This paper aims to describe the group $\E(X)$ of the self-homotopy equivalences of $X$ modulo its normal subgroup $\E_{*}(X)$ of the elements that induce the identity on the homology groups. Making use of the Whitehead exact sequence of $X$, denoted by WES$(X)$, we define the group $Γ\mathcal{S}(X)$ of $Γ$-automorphisms of WES$(X)$ and we prove that $\E(X)/\E_*(X)\cong Γ\mathcal{S}(X).$

Weak equivalences of simplicial presheaves

The usual way of defining weak equivalences for simplicial presheaves is to require an isomorphism on all sheaves of homotopy groups. We unravel some of the machinery here, and give a more concrete description in terms of local homotopy lifting properties. This characterization is used to prove some basic facts about the local homotopy theory of simplicial presheaves.

On the homotopy theory of $\mathbf{G}$ - spaces

The aim of this paper is to show that the most elementary homotopy theory of $\mathbf{G}$-spaces is equivalent to a homotopy theory of simplicial sets over $\mathbf{BG}$, where $\mathbf{G}$ is a fixed group. Both homotopy theories are presented as Relative categories. We establish the equivalence by constructing a strict homotopy equivalence between the two relative categories. No Model category structure is assumed on either Relative Category.

Stable A^1-homotopy and R-equivalence

We prove that existence of a k-rational point can be detected by the stable A^1-homotopy category of S^1-spectra, or even a "rationalized" variant of this category.

楔和空间的自同伦等价群的阶

The order of the group of self-homotopy equivalence of wedge spaces

本文研究楔和空间自同伦等价群的阶，在可约性条件下将一般有限楔和的等价群分解为子群乘积，并给出相应悬挂楔和的阶的计算公式。

Extending homotopy theories across adjunctions

Constructions of spectra from symmetric monoidal categories are typically functorial with respect to strict structure-preserving maps, but often the maps of interest are merely lax monoidal. We describe conditions under which one can transport the weak equivalences from one category to another with the same objects and a broader class of maps. Under mild hypotheses this process produces an equivalence of homotopy theories. We describe examples including algebras over an operad, such as symmetric monoidal categories and $n$-fold monoidal categories; and diagram categories, such as $Γ$-categorie…

非本质有向映射与有向同伦等价

Inessential directed maps and directed homotopy equivalences

本文定义非本质有向映射与有向同伦等价，证明其满足2-out-of-3性质，并保持有向拓扑复杂度及诱导配对分量范畴同构。

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。