Loop structures on the homotopy type of S^3 revisited

模型范畴的同伦极限再探

Homotopy limits of model categories, revisited

本文回顾模型范畴左Quillen函子图表同伦极限的定义及应用，探讨改用右Quillen函子的可行性，指出无法同时使用左右Quillen函子的原因。

重探 Adams 同构

The Adams isomorphism revisited

本文在等变可展示且等变半加法的全局范畴中建立抽象 Adams 同构，统一推广稳定同伦论、Mackey 函子、$G$-global 同伦论与 Kasparov 范畴中的既有结果。

离散同伦理论的立方设定再探

Cubical setting for discrete homotopy theory, revisited

本文提出了一种将图转换为Kan复杂性的立方集构造方法，并证明了图的A群与同伦群的关系，同时验证了一个长期猜想并加强了Hurewicz定理的证明。

Floer homotopy theory, revisited

In 1995 the author, Jones, and Segal introduced the notion of "Floer homotopy theory". The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, "When is the Floer homology isomorphic to the (singular) homology of a naturally occuring (pro)spectrum defined from the properties of the moduli spaces inherent in the Floer theory?". A proposal for how to construct such a spectrum was given in terms of a "framed flow category", and some rather simple examples were described. Years passed before this noti…

The homotopy type of the space of algebraic loops on a toric variety

We investigate the homotopy type of the space of tuples of polynomials inducing base-point preserving algebraic maps from the circle S1 to a toric variety XΣ. In particular, we prove a homotopy stability result for this space by combining the Vassiliev spectral sequence and the scanning map.

A Homotopy Theoretic Proof of the BV Identity in Loop Homology

Chas and Sullivan proved the existence of a Batalin-Vilkovisky algebra structure in the homology of free loop spaces on closed finite dimensional smooth manifolds using chains and chain homotopies. This algebraic structure involves an associative product called the loop product, a Lie bracket called the loop bracket, and a square 0 operator called the BV operator. Cohen and Jones gave a homotopy theoretic description of the loop product in terms of spectra. In this paper, we give an explicit homotopy theoretic description of the loop bracket and, using this description, we give a homological p…

On the homotopy classification of spaces by the fixed loop space homology

Let $R\subseteq \Bbb Q$ be a subring of the rationals and let $p$ be the least prime (if none, $p=\infty $) which is not invertible in $R.$ For an $R$-local $r$-connected $CW$-complex $X$ of dimension $\leq \min(r+2p-3,rp-1), r\geq 1, $ a complete homotopy invariant is constructed in terms of the loop space homology $H_*(ΩX).$ This allows us to classify all such $R$-local spaces up to homotopy with a fixed loop space homology.

Homotopy commutativity in Hermitian symmetric spaces

Ganea proved that the loop space of $\mathbb{C}P^n$ is homotopy commutative if and only if $n=3$. We generalize this result to that the loop spaces of all irreducible Hermitian symmetric spaces but $\mathbb{C}P^3$ are not homotopy commutative. The computation also applies to determining the homotopy nilpotency of the loop spaces of flag manifolds.

The homotopy type of the loops on $(n-1)$-connected $(2n+1)$-manifolds

For $n\geq 2$ we compute the homotopy groups of $(n-1)$-connected closed manifolds of dimension $(2n+1)$. Away from the finite set of primes dividing the order of the torsion subgroup in homology, the $p$-local homotopy groups of $M$ are determined by the rank of the free Abelian part of the homology. Moreover, we show that these $p$-local homotopy groups can be expressed as a direct sum of $p$-local homotopy groups of spheres. The integral homotopy type of the loop space is also computed and shown to depend only on the rank of the free Abelian part and the torsion subgroup.

Homotopy groups of certain highly connected manifolds via loop space homology

For $n\geq 2$ we consider $(n-1)$-connected closed manifolds of dimension at most $(3n-2)$. We prove that away from a finite set of primes, the $p$-local homotopy groups of $M$ are determined by the dimension of the space of indecomposable elements in the cohomology ring $H^\ast(M)$. Moreover, we show that these $p$-local homotopy groups can be expressed as direct sum of $p$-local homotopy groups of spheres. This generalizes some of the results of our earlier work.

Loop spaces and homotopy operations

The question of whether a given H-space X is, up to homotopy, a loop space has been studied from a variety of viewpoints. Here we address this question from the aspect of homotopy operations, in the classical sense of operations on homotopy groups. First, we show how an H-space structure on X can be used to define the action of the primary homotopy operations on the shifted homotopy groups π_{*-1} X (which are isomorphic to π_* Y, if X=Ω\Y. This action will behave properly with respect to composition of operations if X is homotopy-associative, and will lift to a topological action of the monoi…

Big fundamental groups: generalizing homotopy and big homotopy

The concept of big homotopy theory was introduced by J. Cannon and G. Conner using big intervals of arbitrarily large cardinality to detect big loops. We find, for each space, a canonical cardinal that is sufficient to detect all big loops and all big homotopies in the space.

Homotopy Decompositions of Looped Stiefel manifolds, and their Exponents

Let $p$ be an odd prime, and fix integers $m$ and $n$ such that $0<m<n\leq (p-1)(p-2)$. We give a $p$-local homotopy decomposition for the loop space of the complex Stiefel manifold $W_{n,m}$. Similar decompositions are given for the loop space of the real and symplectic Stiefel manifolds. As an application of these decompositions, we compute upper bounds for the $p$-exponent of $W_{n,m}$. Upper bounds for $p$-exponents in the stable range $2m<n$ and $0<m\leq (p-1)(p-2)$ are computed as well.

Homotopy Colimits of Algebras Over Cat-Operads and Iterated Loop Spaces

We extend Thomason's homotopy colimit construction in the category of permutative categories to categories of algebras over an arbitrary $\Cat$ operad and analyze its properties. We then use this homotopy colimit to prove that the classifying space functor induces an equivalence between the category of $n$-fold monoidal categories and the category of $\mathcal{C}_n$-spaces after formally inverting certain classes of weak equivalences, where $\mathcal{C}_n$ is the little $n$-cubes operad. As a consequence we obtain an equivalence of the categories of $n$-fold monoidal categories and the categor…

Crossed modules and the homotopy 2-type of a free loop space

The question was asked by Niranjan Ramachandran: how to describe the fundamental groupoid of LX, the free loop space of a space X? We give an answer by assuming X to be the classifying space of a crossed module over a group, and then describe completely a crossed module over a groupoid determining the homotopy 2-type of LX. The method requires detailed information on the monoidal closed structure on the category of crossed complexes.

"Parallel" transport - revisited

Parallel transport in a fibre bundle with respect to smooth paths in the base space B have recently been extended to representations of the smooth singular simplicial set Sing_{smooth}(B). Inspired by these extensions,I revisit the development of a notion of `parallel' transport in the topological setting of fibrations with the homotopy lifting property and then extend it to representations of Sing(B) on such fibrations. Closely related is the notion of (strong or `infty') homotopy action, which has variants under a variety of names.

自由环路空间有理同调与挠同伦群中的指数增长

Exponential growth in the rational homology of free loop spaces and in torsion homotopy groups

本文利用积分方法推广了自由环路空间有理同调群增长的结果，构造出同伦群$p$-挠指数增长的新空间族，并提出有理同伦群与$p$-挠同伦群指数增长间存在紧密联系的猜想。

Homotopy nilpotent groups

We study the connection between the Goodwillie tower of the identity and the lower central series of the loop group on connected spaces. We define the simplicial theory of homotopy n-nilpotent groups. This notion interpolates between infinite loop spaces and loop spaces. We prove that the set-valued algebraic theory obtained by applying $π_0$ is the theory of ordinary n-nilpotent groups and that the Goodwillie tower of a connected space is determined by a certain homotopy left Kan extension. We prove that n-excisive functors of the form $ΩF$ have values in homotopy n-nilpotent groups.

对称空间中的同伦交换性

Homotopy commutativity in symmetric spaces

本文推广了关于Hermitian对称空间环路空间同伦交换性的已有结果，证明除$\mathbb{C}P^3$外所有不可约对称空间的环路空间均非同伦交换。

Decomposition of loop spaces and periodic problem on $π_*$

We provide a family of spaces localized at 2, whose stable homotopy groups are summands of their unstable homotopy groups. Application to mod 2 Moore spaces are given.

Bott-Chern 和 Aeppli 同伦

Bott-Chern and Aeppli homotopy

本文针对双单纯集的可纤维类引入 Bott-Chern 和 Aeppli 同伦集，建立其基本性质，并证明与 Bott-Chern 同调群的同构关系。

连通复形中带有球对的同伦

Homotopy of Simply Connected Complexes with a Spherical Pair

研究了具有球对的连通复形中循环空间分解及其对同调理论的影响，揭示了不同惰性层次之间的差异及高维中同伦等价的多样性。

Rational homotopy -- Sullivan models

This chapter is a short introduction to Sullivan models. In particular, we find the Sullivan model of a free loop space and use it to prove the Vigué-Poirrier-Sullivan theorem on the Betti numbers of a free loop space.

A Torus Theorem for homotopy nilpotent groups

Nilpotency for discrete groups can be defined in terms of central extensions. In this paper, the analogous definition for spaces is stated in terms of principal fibrations having infinite loop spaces as fibers, yielding a new invariant we compare with classical cocategory, but also with the more recent notion of homotopy nilpotency introduced by Biedermann and Dwyer. This allows us to characterize finite homotopy nilpotent loop spaces in the spirit of Hubbuck's Torus Theorem, and corresponding results for $p$-compact groups and $p$-Noetherian groups.

Mod p decompositions of the loop spaces of compact symmetric spaces

We give p-local homotopy decompositions of the loop spaces of compact, simply-connected symmetric spaces for quasi-regular primes. The factors are spheres, sphere bundles over spheres, and their loop spaces. As an application, upper bounds for the homotopy exponents are determined.

K-theory of toric varieties revisited

After surveying higher K-theory of toric varieties, we present Totaro's old (c. 1997) unpublished result on expressing the corresponding homotopy theory via singular cohomology. It is a higher analog of the rational Chern character isomorphism for general toric schemes. In the special case of a projective simplicial toric scheme over a regular ring one obtains a rational isomorphism between the homotopy K-theory and the direct sum of m copies of the K-theory of the ground ring, m being the number of maximal cones in the underlying fan. Apart from its independent interest, in retrospect, Totaro…

Two-loop part of the rational homotopy of spaces of long embeddings

Arone and Turchin defined graph-complexes computing the rational homotopy of the spaces of long embeddings. The graph-complexes split into a direct sum by the number of loops in graphs. In this paper we compute the homology of its two-loop part.

Kan's combinatorial spectra and their sheaves revisited

We define a right Cartan-Eilenberg structure on the category of Kan's combinatorial spectra, and the category of sheaves of such spectra, assuming some conditions. In both structures, we use the geometric concept of homotopy equivalence as the strong equivalence. In the case of sheaves, we use local equivalence as the weak equivalence. This paper is the first step in a larger-scale program of investigating sheaves of spectra from a geometric viewpoint.

环路化后爆破的同伦论

Homotopy of blow ups after looping

本文结合纤维状手术理论与纯同伦论方法，研究爆破后基点环路空间的 p-局部、有理及整数同伦分解，并给出焦点亏格2流形等应用。

朴素离散同伦论中的映射纤维、环路与悬垂图

Mapping fiber, loop and suspension graphs in naive discrete homotopy theory

本文通过定义并研究映射纤维图及其性质，在技术性条件下构造出朴素离散同伦论中的 Puppe 序列类比，以揭示 A-同伦论与普通同伦论的异同。

The Loop Space Homotopy Type of Simply-connected Four-manifolds and their Generalizations

We determine loop space decompositions of simply-connected four-manifolds, $(n-1)$-connected $2n$-dimensional manifolds provided $n\notin\{4,8\}$, and connected sums of products of two spheres. These are obtained as special cases of a more general loop space decomposition of certain torsion-free $CW$-complexes with well-behaved skeleta and some Poincaré duality features.

$6$-流形在$4$-流形上的环路同伦

Loop homotopy of $6$-manifolds over $4$-manifolds

本文研究了定义在单连通闭$4$-流形上秩$3$向量丛球丛全空间的$6$-流形$M$，证明其环路化后一般同伦等价于球面环路乘积，具有上同调刚性，且在Berglund意义下满足Koszul性质。

An Interpolation between Homology and Stable Homotopy

By considering labeled configurations of ``bounded multiplicity'', one can construct a functor that fits between homology and stable homotopy. Based on previous work, we are able to give an equivalent description of this labeled construction in terms of loop space functors and symmetric products. This yields a direct generalization of the May-Milgram model for iterated loop spaces, and answers questions of Carlsson and Milgram posed in the handbook. We give a classifying space formulation of our results hence extending an older result of Segal. We finally relate our labeled construction to a t…

The Homotopy Type of a Poincaré Duality Complex after Looping

We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincaré $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n-1}(X;\mathbb{Z}_2)=0$. Then its loop space homotopy type is uniquely determined by the action of higher Bockstein operations on $H^{n-1}(X;\mathbb{Z}_p)$ for each odd prime $p$. A stronger result is obtained when localized at odd primes.

A chain rule in the calculus of homotopy functors

We formulate and prove a chain rule for the derivative, in the sense of Goodwillie, of compositions of weak homotopy functors from simplicial sets to simplicial sets. The derivative spectrum dF(X) of such a functor F at a simplicial set X can be equipped with a right action by the loop group of its domain X, and a free left action by the loop group of its codomain Y = F(X). The derivative spectrum d(E o F)(X)$ of a composite of such functors is then stably equivalent to the balanced smash product of the derivatives dE(Y) and dF(X), with respect to the two actions of the loop group of Y. As an …

An algebraic model for the loop space homology of a homotopy fiber

Let F denote the homotopy fiber of a map f:K-->L of 2-reduced simplicial sets. Using as input data the strongly homotopy coalgebra structure of the chain complexes of K and L, we construct a small, explicit chain algebra, the homology of which is isomorphic as a graded algebra to the homology of GF, the simplicial (Kan) loop group on F. To construct this model, we develop machinery for modeling the homotopy fiber of a morphism of chain Hopf algebras. Essential to our construction is a generalization of the operadic description of the category DCSH of chain coalgebras and of strongly homotopy c…

The Dual Polyhedral Product, Cocategory and Nilpotence

The notion of a dual polyhedral product is introduced as a generalization of Hovey's definition of Lusternik-Schnirelmann cocategory. Properties established from homotopy decompositions that relate the based loops on a polyhedral product to the based loops on its dual are used to show that if X is a simply-connected space then the weak cocategory of X equals the homotopy nilpotency class of the based loops on X. This answers a fifty year old question posed by Ganea. The methods are applied to determine the homotopy nilpotency class of quasi-p-regular exceptional Lie groups and sporadic p-compa…

Homotopy groups of highly connected Poincare duality complexes

Methods are developed to relate the action of a principal fibration to relative Whitehead products in order to determine the homotopy type of certain spaces. The methods are applied to thoroughly analyze the homotopy type of the based loops on certain cell attachments. Key examples are (n-1)-connected Poincare Duality complexes of dimension 2n or 2n+1 with minor cohomological conditions.

连通和上同伦纤维化的有理同伦型

The Rational Homotopy Type of Homotopy Fibrations Over Connected Sums

本文研究连通和上拉回纤维化全空间的有理同伦型，给出较Jeffrey–Selick更弱的有理上同调条件，使其在取闭路后等价于连通和。

环路余积的同伦同调不变性

Simple homotopy invariance of the loop coproduct

本文证明了弦拓扑中 Goresky-Hingston 环路余积在流形同伦等价下的变换公式，指出其涉及 Whitehead 挠率的迹，并揭示了该余积在简单同伦等价下的不变性及其在 2 维 TQFT 中的拓扑起源。

Homotopy Algebra Structures on Twisted Tensor Products and String Topology Operations

Given a $C_\infty$ coalgebra $C_*$, a strict dg Hopf algebra $H_*$, and a twisting cochain $τ:C_* \rightarrow H_*$ such that $Im(τ) \subset Prim(H_*)$, we describe a procedure for obtaining an $A_\infty$ coalgebra on $C_* \otimes H_*$. This is an extension of Brown's work on twisted tensor products. We apply this procedure to obtain an $A_\infty$ coalgebra model of the chains on the free loop space $LM$ based on the $C_\infty$ coalgebra structure of $H_*(M)$ induced by the diagonal map $M \rightarrow M \times M$ and the Hopf algebra model of the based loop space given by $T(H_*(M)[-1])$. When …

连通和的环路空间分解及其在Vigué-Poirrier猜想中的应用

Loop Space Decompositions of Connected Sums and Applications to the Vigué-Poirrier Conjecture

本文扩展了分解高连通Poincaré对偶复形基点环路空间的方法，并应用于为有理同伦理论中长期存在的Vigué-Poirrier猜想提供新的支持实例。

The homotopy invariance of the string topology loop product and string bracket

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M_1 \to M_2 be a homotopy equivalence of closed, oriented n -manifolds. Then the induced equivalence, Lf : LM_1 \to LM_2 induces a ring isomorphism in homology, and an isomorphism of Lie algebra…

Typical representatives of free homotopy classes in a multi-punctured plane

We show that a uniform probability measure supported on a specific set of piecewise linear loops in a non-trivial free homotopy class in a multi-punctured plane is overwhelmingly concentrated around loops of minimal lengths. Our approach is based on extending Mogulskii's theorem to closed paths, which is a useful result of independent interest. In addition, we show that the above measure can be sampled using standard Markov Chain Monte Carlo techniques, thus providing a simple methods for approximating shortest loops.

Loop space construction of bigraphs and box complexes

Dochtermann introduced the loop space construction of a based graph $(G,v)$ whose basepoint is a looped vertex. He showed that the complex $C(Ω(G,v))$ is homotopy equivalent to the loop space $Ω(C(G),v)$ of $C(G)$. Here we write $C(G)$ to mean the clique complex of the maximal reflexive subgraph of $G$. In this paper, we consider its bigraph version. A bigraph is a graph equipped with its 2-coloring. We introduce the loop space construction $Ω_{/K_2}(X,x)$ of a based bigraph $(X,x)$. This is a graph such that $C(Ω_{/K_2}(X,x))$ is homotopy equivalent to the loop space of the box complex $B_{/K…

$\n\mathbb{C}^d$ 中点构型模空间的 Operad 再探

Operads of moduli spaces of points in $\mathbb{C}^d$ revisited

研究射影空间构型模空间同调上的 operad，用生成元和关系刻画，证明其同伦 Koszul 性及 Givental 型作用，并揭示其与框架化圆盘 operad 同伦商的关系，为 $d=1$ 情形提供新组合视角。

拟环面流形中的同伦交换性

Homotopy commutativity in quasitoric manifolds

本文证明了拟环面流形的环路空间同伦交换的充要条件，即底层多胞体为 $(Δ^3)^n$ 且特征矩阵属于特定类型，并构造了无限族同伦不等价的广义 Bott 流形，其中仅一半具有同伦交换性。

以射影空间稳定化的流形的同伦

Homotopy of manifolds stabilized by projective spaces

本文研究流形与射影空间连通和的同伦，提出了流形经射影空间稳定化后的环路同伦分解，并通过局部化方法追踪了特定乘积流形上外科手术的同伦影响。

H-Space and Loop Space Structures for Intermediate Curvatures

For dimensions $n\geq 3$ and $k\in\{2, \cdots, n\}$, we show that the space of metrics of $k$-positive Ricci curvature on the sphere $S^{n}$ has the structure of an $H$-space with a homotopy commutative, homotopy associative product operation. We further show, using the theory of operads and results of Boardman, Vogt and May that the path component of this space containing the round metric is weakly homotopy equivalent to an $n$-fold loop space.

Braid groups, free groups, and the loop space of the 2-sphere

The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor, and Habegger-Lin on "homotopy string links". The current article exploits Lie algebras associated to Vassiliev invariants in work of T. Kohno, and provides connections between these various topics. Two consequences are as follows: 1) the homotopy groups of spheres are identified as "natural" sub-quotients of free products of pure braid groups, and 2) an axiomatizatio…

通过参数化不动点理论探讨同伦不变性的障碍

Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory

本文通过参数化不动点理论探讨同伦不变性的障碍，定义Reidemeister迹并表征谱余乘积的失败，揭示简单同伦等价下谱余乘积的一致性。

Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds I. Invariants of homology 3-spheres

We give a generalization of Fukaya's Morse homotopy theoretic approach for 2-loop Chern--Simons perturbation theory to 3-valent graphs with arbitrary number of loops at least 2. We construct a sequence of invariants of integral homology 3-spheres with values in a space of 3-valent graphs (Jacobi diagrams or Feynman diagrams) by counting graphs in an integral homology 3-sphere satisfying certain condition described by a set of ordinary differential equations.

H-Spaces, Loop Spaces and the Space of Positive Scalar Curvature Metrics on the Sphere

For dimensions n greater than or equal to 3, we show that the space of metrics of positive scalar curvature on the n-sphere is homotopy equivalent to a subspace which takes the form of a H-space with a homotopy commutative, homotopy associative product operation. This product operation is based on the connected sum construction. We then exhibit an action of the little n-disks operad on this subspace which, using results of Boardman, Vogt and May implies that when n=3 or n is at least 5, the space of metrics of positive scalar curvature on the n-sphere is weakly homotopy equivalent to an n-fold…

Homotopy algebras and noncommutative geometry

We study cohomology theories of strongly homotopy algebras, namely $A_\infty, C_\infty$ and $L_\infty$-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of $C_\infty$-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a $C_\infty$-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\infty$-algebra (an $\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the `string topology' oper…