Configuration spaces are not homotopy invariant

Representation stability for homotopy groups of configuration spaces

We prove that the dual rational homotopy groups of the configuration spaces of a 1-connected manifold of dimension at least 3 are uniformly representation stable in the sense of Church, and that their derived dual integral homotopy groups are finitely-generated as FI-modules in the sense of Church-Ellenberg-Farb. This is a consequence of a more general theorem relating properties of the cohomology groups of a 1-connected co-FI-space to properties of its dual homotopy groups. We also discuss several other applications.

An elementary proof of the homotopy invariance of stabilized configuration spaces

In this paper we give an elementary proof of the proper homotopy invariance of the equivariant stable homotopy type of the configuration space $F(M,k)$ for a topological manifold $M$. Our technique is to compute the Spanier-Whitehead dual of $Σ^\infty_+ F(M,k)$ and use the results of Spivak and Wall on normal spherical fibrations to deduce that the Spanier-Whitehead dual is a proper homotopy invariant. This stable invariance was recently proved by Knudsen using factorization homology. Aside from being elementary, our proof has the advantage that it readily extends to ``generalized configuratio…

Lie coalgebras and rational homotopy theory II: Hopf invariants

We give a new solution of the "homotopy periods" problem, as highlighted by Sullivan, which places explicit geometrically meaningful formulae first dating back to Whitehead in the context of Quillen's formalism for rational homotopy theory and Koszul-Moore duality. Geometrically, we show that homotopy groups are rationally given by "generalized linking/intersection invariants" of cochain data. Moreover, we give a method for determining when two maps from $S^n$ to $X$ are homotopic after allowing for multiplication by some integer. For applications, we investigate wedges of spheres and homogene…

Higher homotopy invariants for spaces and maps

For a pointed topological space $X$, we use an inductive construction of a simplicial resolution of $X$ by wedges of spheres to construct a "higher homotopy structure" for $X$ (in terms of chain complexes of spaces). This structure is then used to define a collection of higher homotopy invariants which suffice to recover $X$ up to weak equivalence. It can also be used to distinguish between different maps $f$ from $X$ to $Y$ which induce the same morphism on homotopy groups $f_*$ from $π_* X$ to $π_* Y$.

配置空间的l-adic同伦型

On the l-adic homotopy type of configuration spaces

本文为Tate型代数簇的配置空间给出了驯顺同伦型的代数模型，并利用étale上同调中的权理论，推广到更一般的排列补空间。

Configuration categories and homotopy automorphisms

Let M be a smooth compact manifold with boundary. Under some geometric conditions on M, a homotopical model for the pair (M,boundary of M) can be recovered from the configuration category of the interior of M. The grouplike monoid of derived homotopy automorphisms of the configuration category of the interior of M then acts on the homotopical model of the pair (M,boundary of M), in a way which is compatible with a known homotopical action of the homeomorphism group of M minus boundary on (M,boundary of M).

Straight homotopy invariants

Let $X$ and $Y$ be spaces and $M$ be an abelian group. A homotopy invariant $f\colon [X,Y]\to M$ is called straight if there exists a homomorphism $F\colon L(X,Y)\to M$ such that $f([a])=F(\langle a\rangle)$ for all $a\in C(X,Y)$. Here $\langle a\rangle\colon\langle X\rangle\to\langle Y\rangle$ is the homomorphism induced by $a$ between the abelian groups freely generated by $X$ and $Y$ and $L(X,Y)$ is a certain group of `admissible' homomorphisms. We show that all straight invariants can be expressed through a `universal' straight invariant of homological nature.

Geometric approach to stable homotopy groups of spheres. I. The Hopf invariant

A geometric approach to the stable homotopy groups of spheres is developed in this paper, based on the Pontryagin-Thom construction. The task of this approach is to obtain an alternative proof of the Hill-Hopkins-Ravenel theorem [H-H-R] on Kervaire invariants in all dimensions, except, possibly, a finite number of dimensions. In the framework of this approach, the Adams theorem on the Hopf invariant is studied, for all dimensions with the exception of 15, 31, 63, 127. The new approach is based on the methods of geometric topology.

Axiomatic Description of Lefschetz Type Equivariant Homotopy Invariants

We describe generators of universal Lefschetz groups consisting of self-maps of equivariant 1-spheres. This allows to formulate a normalization axiom which, together with the usual axioms, determines an equivariant Lefschetz number uniquely. We apply this technique to identify the fixed orbit detecting equivariant Lefschetz number of Chorny with the fixed orbit index of Dzedzej.

积分同调群的配置空间线性稳定范围

Linear stable ranges for integral homotopy groups of configuration spaces

研究通过同调方法证明配置空间的FI模块线性稳定范围，利用广义Blakers-Massey定理推导表示稳定性结果，并推广到FI_G模块和轨道配置空间。

Order of a homotopy invariant in the stable case

Let X and Y be CW-complexes, U be an abelian group, and f:[X,Y]->U be a map (a homotopy invariant). We say that f has order at most r if the characteristic function of the r'th Cartesian power of the graph of a continuous map a:X->Y Z-linearly determines f([a]). Suppose that the CW-complex X is finite and we are in the stable case: dim X<2n-1 and Y is (n-1)-connected. We prove that then the order of f equals its degree with respect to the Curtis filtration of the group [X,Y].

The Whitehead exact sequence and the classification problem of homotopy types

This paper defines an invariant associated to Whitehead's certain exact sequence of a simply connected CW-complex which is much more elementary - and less powerful - than the boundary invariant of Baues. Nevertheless, in good cases, it classifies the homotopy types of CW-complexes.

Geometric approach to stable homotopy groups of spheres II. The Kervaire invariant

A solution to the Kervaire invariant problem is presented. We introduce the concepts of abelian structure on skew-framed immersions, bicyclic structure on $\Z/2^{[3]}$--framed immersions, and quaternionic-cyclic structure on $\Z/2^{[4]}$--framed immersions. Using these concepts, we prove that for sufficiently large $n$, $n=2^{\ell}-2$, an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant. Additionally, for $\ell \ge 12$ (i.e., for $n \ge 4094$) an arbitrary skew-framed immersion in Euclidean $n$-space $\R^n$ has zero Kervaire invariant if this skew-frame…

Some homotopy theoretical questions arising in Nielsen coincidence theory

Basic examples show that coincidence theory is intimately related to central subjects of differential topology and homotopy theory such as Kervaire invariants and divisibility properties of Whitehead products and of Hopf invariants. We recall some recent results and ask a few questions which seems to be important for a more comprehensive understanding.

Homotopy Diagrams of Algebras

In [math.AT/9907138] we proved that strongly homotopy algebras are homotopy invariant concepts in the category of chain complexes. Our arguments were based on the fact that strongly homotopy algebras are algebras over minimal cofibrant operads and on the principle that algebras over cofibrant operads are homotopy invariant. In our approach, algebraic models for colored operads describing diagrams of homomorphisms played an important role. The aim of this paper is to give an explicit description of these models. A possible application is an appropriate formulation of the `ideal' homological per…

Homotopy graph-complex for configuration and knot spaces

In the paper we prove that the primitive part of the Sinha homology spectral sequence E^2-term for the space of long knots is rationally isomorphic to the homotopy E^2-term. We also define natural graph-complexes computing the rational homotopy of configuration and of knot spaces.

通过离散Morse理论的持久简单同伦不变量

Persistent Simple-homotopy invariants via discrete Morse theory

本文提出一种改进的持久同调，用于检测同调不可见的简单同伦现象。通过定义Morse复杂度轮廓，证明其在简单同伦等价下不变，并提供高效的Vietoris-Rips滤波算法。

Hopf algebras and homotopy invariants

In this paper we explore new relations between Algebraic Topology and the theory of Hopf Algebras. For an arbitrary topological space $X$, the loop space homology $H_*(ΩΣX; \coefZ)$ is a Hopf algebra. We introduce a new homotopy invariant of a topological space $X$ taking for its value the isomorphism class (over the integers) of the Hopf algebra $H_*(ΩΣX; \coefZ)$. This invariant is trivial if and only if the Hopf algebra $H_*(ΩΣX; \coefZ)$ is isomorphic to a Lie-Hopf algebra, that is, to a primitively generated Hopf algebra. We show that for a given $X$ these invariants are obstructions to t…

On the homotopy groups of E(n)-local spectra with unusual invariant ideals

Let E(n) and T(m) for nonnegative integers n and m denote the Johnson-Wilson and the Ravenel spectra, respectively. Given a spectrum whose E(n)_*-homology is E(n)_*(T(m))/(v_1,...,v_{n-1}), then each homotopy group of it estimates the order of each homotopy group of L_nT(m). We here study the E(n)-based Adams E_2-term of it and present that the determination of the E_2-term is unexpectedly complex for odd prime case. At the prime two, we determine the E_{infty}-term for pi_*(L_2T(1)/(v_1)), whose computation is easier than that of pi_*(L_2T(1)) as we expect.

Rational model of the configuration space of two points in a simply connected closed manifold

Let $M$ be a simply connected closed manifold of dimension $n$. We study the rational homotopy type of the configuration space of 2 points in $M$, $F(M,2)$. When $M$ is even dimensional, we prove that the rational homotopy type of $F(M,2)$ depends only on the rational homotopy type of $M$. When the dimension of $M$ is odd, for every $x\in H^{n-2} (M, \mathbb{Q})$, we construct a commutative differential graded algebra $C(x)$. We prove that for some $x \in H^{n-2} (M; \mathbb{Q})$, $C(x)$ encodes completely the rational homotopy type of $F(M,2)$. For some class of manifolds, we show that we can…

Homotopy Invariants and Almost Non-Negative Curvature

This paper explores the relation between the structure of fibre bundles akin to those associated to a closed almost nonnegatively sectionally curved manifold and rational homotopy theory.

Configurations spaces and Theta_n

We demonstrate that Joyal's category Theta_n, which is central to numerous definitions of (infinity,n)-categories, naturally encodes the homotopy type of configuration spaces of marked points in R^n. This article is largely self-contained and uses only elementary techniques in combinatorics and homotopy theory.

Homotopy Groups of Diagonal Complements

For $X$ a connected finite simplicial complex we consider $Δ^d(X,n)$ the space of configurations of $n$ ordered points of $X$ such that no $d+1$ of them are equal, and $B^d(X,n)$ the analogous space of configurations of unordered points. These reduce to the standard configuration spaces of distinct points when $d=1$. We describe the homotopy groups of $Δ^d(X,n)$ (resp. $B^d(X,n)$) in terms of the homotopy (resp. homology) groups of $X$ through a range which is generally sharp. It is noteworthy that the fundamental group of the configuration space $B^d(X,n)$ abelianizes as soon as we allow poin…

$E_n$-Hopf invariants

The classical Hopf invariant is an invariant of homotopy classes of maps from $S^{4n-1} $ to $S^{2n}$, and is an important invariant in homotopy theory. The goal of this paper is to use the Koszul duality theory for $E_n$-operads to define a generalization of the classical Hopf invariant. One way of defining the classical Hopf invariant is by defining a pairing between the cohomology of the associative bar construction on the cochains of a space $X$ and the homotopy groups of $X$. In this paper we will give a generalization of the classical Hopf invariant by defining a pairing between the coho…

Geometric approach towards stable homotopy groups of spheres. Kervaire Invariant

It is proved that there exists an integer $L$ such that a framed manifold of dimension $2^l-2$, $l\le L$ has the trivial Kervaire Invariant.

有理空间的Postnikov不变量

Postnikov invariants for rational spaces

本文为完全同伦李代数构造Postnikov不变量理论，该理论被Robert-Nicoud与Vallette的积分函子保持，推导得从滤过Chevalley-Eilenberg同调群到群上同调群的积分函子，结果与Berglund定理相关。

迈向 $p=3$ Kervaire 不变量问题：计算 $π_*({E_6}^{hC_9})$ 的同伦不动点谱序列的 $E_2$-页

Towards the $p=3$ Kervaire Invariant Problem: The $E_2$-page for the homotopy fixed points spectral sequence computing $π_*({E_6}^{hC_9})$

针对 $p=3$ 的 Kervaire 不变量问题，本文证明了相关检测定理，并利用 $E_6$ 上 $C_9$ 作用的猜想形式，计算了 $π_* E_6^{hC_9}$ 同伦不动点谱序列在非零同调度处的 $E_2$ 页。

On Brunnian-type links and the link invariants given by homotopy groups of spheres

We introduce the (general) homotopy groups of spheres as link invariants for Brunnian-type links through the investigations on the intersection subgroup of the normal closures of the meridians of strongly nonsplittable links. The homotopy groups measure the difference between the intersection subgroup and symmetric commutator subgroup of the normal closures of the meridians and give the invariants of the links obtained in this way. Moreover the higher homotopy-group invariants can produce some links that could not be detected by the Milnor invariants. Furthermore all homotopy groups of spheres…

矩阵不变量作为有限 $T_0$-空间的同伦不变量

Matrix Invariants as Homotopy Invariants in Finite $T_0$-spaces

本文通过矩阵与有限 $T_0$-空间的双射关系，揭示了矩阵行列式和秩作为同伦不变量的应用，探讨了有限偏序集的进一步联系与问题。

Kervaire Invariant One [after M. A. Hill, M. J. Hopkins, and D. C. Ravenel]

The question of when the Kervaire invariant is nontrivial was the only question left unresolved by Kervaire and Milnor in their 1963 study of the relationship between groups of homotopy spheres and stable homotopy groups. In 2009, Mike Hill, Mike Hopkins, and Doug Ravenel resolved this question except in one dimension, by a highly innovative attack using large amounts of equivariant stable homotopy theory and small amounts of computation. The present paper is a Seminaire Bourbaki report on this work.

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.

等变同伦理论与不动点不变量

Isovariant homotopy theory and fixed point invariants

本文研究有限群作用下等变映射的不动点理论，建立等变 Whitehead 定理，并证明在适当维数条件下紧光滑 $G$-流形自映射的不动点可等变移除当且仅当其等变 Reidemeister 迹为零，同时探讨等变同伦理论在流形同胚分类中的作用。

Invariance homotopique de certains espaces de configurations

For a smooth manifold A, we consider the ordered configuration space F_k(AxR) of k distinct points in AxR. We obtain an explicit homotopy construction of the configuration space F_k(AxR) and of the (k-2)-fold suspension of F_k(A). Under certain conditions, we then show that the homotopy types of these two spaces depend only on the homotopy type of A. ----- Pour une variete lisse A, on considere F_k(AxR) l'espace des configurations ordonnees de k particules distinctes dans AxR. On effectue une construction explicite de l'espace de configurations F_k(AxR) et de la suspension (k-2)-ieme de F_k(A)…

Higher order generalization of Fukaya's Morse homotopy invariant of 3-manifolds II. Invariants of 3-manifolds with $b_1=1$

In this paper, it is explained that a topological invariant for 3-manifold $M$ with $b_1(M)=1$ can be constructed by applying Fukaya's Morse homotopy theoretic approach for Chern--Simons perturbation theory to a local system on $M$ of rational functions associated to the free abelian covering of $M$. Our invariant takes values in Garoufalidis--Rozansky's space of Jacobi diagrams whose edges are colored by rational functions. It is expected that our invariant gives a lot of nontrivial finite type invariants of 3-manifolds.

Morse theory for the Yang-Mills functional via equivariant homotopy theory

In this paper we show the existence of non minimal critical points of the Yang-Mills functional over a certain family of 4-manifolds with generic SU(2)-invariant metrics using Morse and homotopy theoretic methods. These manifolds are acted on fixed point freely by the Lie group SU(2) with quotient a compact Riemann surface of even genus. We use a version of invariant Morse theory for the Yang-Mills functional used by Parker and by Rade.

Link Homotopy in S^n x R^{m-n} and Higher Order mu-invariants

Given a suitable link map f into a manifold M, we constructed, in [10], link homotopy invariants kappa(f) and mu(f). In the present paper we study the case M=S^n x R^{m - n} in detail. Here mu(f) turns out to be the starting term of a whole sequence mu^(s)(f), s = 0, 1, ..., of higher mu-invariants which together capture all the information contained in kappa(f). We discuss the geometric significance of these new invariants. In several instances we obtain complete classification results. A central ingredient of our approach is the homotopy theory of wedges of spheres.

球面稳定同调群的几何方法 II；阿夫-凯瓦里不变量

Geometric approach to stable homotopy groups of spheres II; Arf-Kervaire Invariants

本文通过几何方法解决球面稳定同调群中的凯瓦里不变量问题，引入了新的对称结构并证明了相关定理。

Geometric approach towards stable homotopy groups of spheres. The Steenrod-Hopf invariant I

In this paper a geometric approach toward stable homotopy groups of spheres, based on the Pontrjagin-Thom construction is proposed. From this approach a new proof of Hopf Invariant One Theorem by J.F.Adams for all dimensions except $15,31,63,127$ is obtained. It is proved that for $n>127$ in the stable homotopy group of spheres $Π_n$ there is no elements with Hopf invariant one. The new proof is based on geometric topology methods. The Pontrjagin-Thom Theorem (in the form proposed by R.Wells) about the representation of stable homotopy groups of the real projective infinite-dimensional space (…

On the homotopy invariance of configuration spaces

For a closed PL manifold M, we consider the configuration space F(M,k) of ordered k-tuples of distinct points in M. We show that a suitable iterated suspension of F(M,k) is a homotopy invariant of M. The number of suspensions we require depends on three parameters: the number of points k, the dimension of M and the connectivity of M. Our proof uses a mixture of Poincare embedding theory and fiberwise algebraic topology.

Second homotopy and invariant geometry of flag manifolds

We use the Hopf fibration to explicitly compute generators of the second homotopy group of the flag manifolds of a compact Lie group. We show that these $2$-spheres have nice geometrical properties such as being totally geodesic surfaces with respect to any invariant metric on the flag manifold. We characterize when the generators with the same invariant geometry are in the same homotopy class. This is done by exploring the action of Weyl group on the irreducible components of isotropy representation of the flag manifold.

分级融合2-范畴与4维流形的量子同伦不变量

Graded-fusion 2-categories and quantum homotopy invariants of 4-manifolds

本文引入融合2-范畴的3-群分次扩张，构造了4维流形的非平凡量子同伦不变量，推广了Douglas-Reutter不变量，且在该语境下推广了Pachner定理。

Configuration spaces and directed paths on the final precubical set

The main goal of this paper is to prove that the space of directed loops on the final precubical set is homotopy equivalent to the "total" configuration space of points on the plane; by "total" we mean that any finite number of points in a configuration is allowed. We also provide several applications: we define new invariants of precubical sets, prove that directed path spaces on any precubical complex have the homotopy types of CW-complexes and construct certain presentations of configuration spaces of points on the plane as nerves of categories.

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.

等变同伦交错与持久Whitehead定理

Equivariant Homotopy Interleaving and Persistent Whitehead Theorem

本文针对有限群作用的持久空间，定义了$G$-同伦交错距离并证明其普适性，建立了等变持久Whitehead定理与神经引理，最终导出等变弱大数定律。

Homotopy invariants for $\overline{\mathcal{M}}_{0,n}$ via Koszul duality

We show that the integral cohomology rings of the moduli spaces of stable rational marked curves are Koszul. This answers an open question of Manin. Using the machinery of Koszul spaces developed by Berglund, we compute the rational homotopy Lie algebras of those spaces, and obtain some estimates for Betti numbers of their free loop spaces in case of torsion coefficients. We also prove and conjecture some generalisations of our main result.

Twisted Parametrized Stable Homotopy Theory

We introduce a framework, twisted parametrized stable homotopy theory, for describing semi-infinite homotopy types. A twisted parametrized spectrum is a section of a bundle whose fibre is the category of spectra. We define these bundles in terms of modules over a stack of parametrized spectra and in terms of diagrams of simplicial categories. We present a classification of bundles of categories of spectra and of the associated twisted parametrized spectra. Though twisted parametrized spectra do not have global homotopy types and therefore do not have generalized homology invariants in the usua…

Homotopy Invariant Commutative Algebra over fields

These notes illustrates the power of formulating ideas of commutative algebra in a homotopy invariant form. They can then be applied to derived categories of rings or ring spectra. These ideas are powerful in classical algebra, in representation theory of groups, in classical algebraic topology and elsewhere. The notes grew out of a series of lectures given during the `Interactions between Representation Theory, Algebraic Topology and Commutative Algebra' (IRTATCA) at the CRM (Barcelona) in Spring 2015.

Homotopy Brunnian links and the $κ$-invariant

We provide an alternative proof that Koschorke's $κ$-invariant is injective on the set of link homotopy classes of $n$-component homotopy Brunnian links $BLM(n)$. The existing proof (by Koschorke \cite{Koschorke97}) is based on the Pontryagin--Thom theory of framed cobordisms, whereas ours is closer in spirit to techniques based on Habegger and Lin's string links. We frame the result in the language of Fox's torus homotopy groups and the rational homotopy Lie algebra of the configuration space $\text{Conf}(n)$ of $n$ points in $\mathbb{R}^3$. It allows us to express the relevant Milnor's $μ$--…

Homotopy invariants of covers and KKM type lemmas

With any (open or closed) cover of a space T we associate certain homotopy classes of maps T into n-spheres. These homotopy invariants can be considered as obstructions for extensions of covers of a subspace A to a space X. We using these obstructions for generalizations of the classic KKM (Knaster-Kuratowski-Mazurkiewicz) and Sperner lemmas. In particular, we show that in the case when A is a k-sphere and X is a (k+1)-disk there exist KKM type lemmas for covers by n+2 sets if and only if the k-homotopy group of n-sphere is not zero.

The f-invariant and index theory

In this paper we prove a tertiary index theorem which relates a spectral geometric and a homotopy theoretic invariant of an almost complex manifold with framed boundary. It is derived from the index theoretic and homotopy theoretic versions of a complex elliptic genus and interestingly related with the structure of the stable homotopy groups of spheres.

Transversal homotopy theory

Implementing an idea due to John Baez and James Dolan we define new invariants of Whitney stratified manifolds by considering the homotopy theory of smooth transversal maps. To each Whitney stratified manifold we assign transversal homotopy monoids, one for each natural number. The assignment is functorial for a natural class of maps which we call stratified normal submersions. When the stratification is trivial the transversal homotopy monoids are isomorphic to the usual homotopy groups. We compute some simple examples and explore the elementary properties of these invariants. We also assign …

A1-homotopy invariants of corner skew Laurent polynomial algebras

In this note we prove some structural properties of all the A1-homotopy invariants of corner skew Laurent polynomial algebras. As an application, we compute de mod-l algebraic K-theory of Leavitt path algebras using solely the kernel/cokernel of the incidence matrix. This leads naturally to some vanishing and divisibility properties of the algebraic K-theory of these algebras.

同伦下辫子与焊接辫子的Milnor不变量

Milnor invariants of braids and welded braids up to homotopy

本文研究同伦下纯焊接辫子群的结构，通过将其商群分解为迭代半直积解决Andreadakis问题并给出生成元表示，同时关联Milnor不变量并讨论经典辫子的对应情况。

Configuration space integrals and the cohomology of the space of homotopy string links

Configuration space integrals have in recent years been used for studying the cohomology of spaces of (string) knots and links in $\mathbb{R}^n$ for $n>3$ since they provide a map from a certain differential algebra of diagrams to the deRham complex of differential forms on the spaces of knots and links. We refine this construction so that it now applies to the space of homotopy string links -- the space of smooth maps of some number of copies of $\mathbb{R}$ in $\mathbb{R}^n$ with fixed behavior outside a compact set and such that the images of the copies of $\R$ are disjoint -- even for $n=3…