Michael A. Mandell
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.
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The K-theory cochains of H-spaces and height 1 chromatic homotopy theory
本文研究了H空间的K理论余弦和高度1色染 homotopy 理论，通过生成元和关系描述了KU_p余弦谱，并证明了其同伦类型的周期性建模。
We give homotopy invariant definitions corresponding to three well known properties of complete intersections, for the ring, the module theory and the endomorphisms of the residue field, and we investigate them for the mod p cochains on a space, showing that suitable versions of the second and third are equivalent and that the first is stronger. We are particularly interested in classifying spaces of groups, and we give a number of examples. This paper follows on from arXiv:0906.4025 which considered the classical case of a commutative ring and arXiv:0906.3247 which considered the case of rati…
In this paper, we set up a rational homotopy theory for operads in simplicial sets whose term of arity one is not necessarily reduced to an operadic unit, extending results obtained by the author in the book "Homotopy of operads and Grothendieck-Teichmüller groups". In short, we prove that the rational homotopy type of such an operad is determined by a cooperad in cochain differential graded algebras (a cochain Hopf dg-cooperad for short) as soon as the Sullivan rational homotopy theory works for the spaces underlying our operad (e.g. when these spaces are connected, nilpotent, and have finite…
We construct a Frobenius algebra structure on the Hochschild cochains of a group ring k[G] that extends the known structure of a <1, 2> topological quantum field theory on HH^0(k[G]; k[G]), k a field and G a finite group. The convolution product extends to the homotopy commutative Gerstenhaber product on cochains, the Frobenius coproduct extends to a coproduct on the chain complex for Hochschild homology, and there is a pairing on Hochschild cocahins satisfying Frobenius associativity. The pairing, however, degenerates on a certain subcomplex of Hochschild cochains. The cochain complex for gro…
Given an algebraic structure on the homology of a chain complex, we define its realization space as a Kan complex whose vertices are the structures up to homotopy realizing this structure at the homology level. Our algebraic structures are parametrised by props and thus include various kinds of bialgebras. We give a general formula to compute subsets of equivalences classes of realizations as quotients of automorphism groups, and determine the higher homotopy groups via the cohomology of deformation complexes. As a motivating example, we compute subsets of equivalences classes of realizations …
Early in the history of higher homotopy algebra, it was realized that Massey products are homotopy invariants in a special sense, but it was the work of Tornike Kadeisvili that showed they were but a shadow of an A-infinity-structure on the homology of a differential graded algebra. Here we relate his work to that of Victor Gugenheim and K.T. (Chester) Chen. This paper is a personal tribute to Tornike and the Georgian school of homotopy theory as well as to Gugenheim and Chen, who unfortunately are not with us to appreciate this convergence.
Stabilizations of $\mathbb{E}_\infty$ Operads and $p$-Adic Stable Homotopy Theory
该研究构造了作为 $\u211e_\u221e$ 操作子稳定化版本的稳定操作子，建立其代数同伦与上同调运算理论，并证明谱上链赋予其结构后可作为 $p$ 进稳定同伦型的代数模型。
Using the $E_\infty-$structure on singular cochains, we construct a homotopy coherent map from the cyclic bar construction of the differential graded algebra of cochains on a space to a model for the cochains on its free loop space. This fills a gap in the paper "Cyclic homology and equivariant homology" by John D.S. Jones.
Banagl's method of intersection spaces allows to modify certain types of stratified pseudomanifolds near the singular set in such a way that the rational Betti numbers of the modified spaces satisfy generalized Poincaré duality in analogy with Goresky-MacPherson's intersection homology. In the case of one isolated singularity, we show that the duality isomorphism comes from a nondegenerate intersection pairing which depends on the choice of a chain representative of the fundamental class of the regular stratum. On the technical side, we use piecewise linear polynomial differential forms due to…
The commutative differential graded algebra $A_{\mathrm{PL}}(X)$ of polynomial forms on a simplicial set $X$ is a crucial tool in rational homotopy theory. In this note, we construct an integral version $A^{\mathcal{I}}(X)$ of $A_{\mathrm{PL}}(X)$. Our approach uses diagrams of chain complexes indexed by the category of finite sets and injections $\mathcal{I}$ to model $E_{\infty}$ differential graded algebras by strictly commutative objects, called commutative $\mathcal{I}$-dgas. We define a functor $A^{\mathcal{I}}$ from simplicial sets to commutative $\mathcal{I}$-dgas and show that it is a…
We introduce a new algebraic concept of an algebra which is "almost" commutative (more precisely "quasi-commutative differential graded algebra" or ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how we can recover the homotopy type of the topological realization of X from this algebraic structure (assuming some finiteness conditions). The theory is sufficiently general to include also ringed spaces X. The construction is by itself interesting since it uses the difference calculus (instead of the differential calculus of Sullivan's theory) and a new type…
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 introduce the notion of a strongly homotopy-comultiplicative resolution of a module coalgebra over a chain Hopf algebra, which we apply to proving a comultiplicative enrichment of a well-known theorem of Moore concerning the homology of quotient spaces of group actions. The importance of our enriched version of Moore's theorem lies in its application to the construction of useful cochain algebra models for computing multiplicative structure in equivariant cohomology. In the special cases of homotopy orbits of circle actions on spaces and of group actions on simplicial sets, we obtain small,…
In recent years, Benson, Iyengar and Krause have developed a theory of stratification for compactly generated triangulated categories with an action of a graded commutative Noetherian ring. Stratification implies a classification of localizing and thick subcategories in terms of subsets of the prime ideal spectrum of the given ring. In this paper two stratification results are presented: one for the derived category of a commutative ring-spectrum with polynomial homotopy and another for the derived category of cochains on certain spaces. We also give the stratification of cochains on a space a…
We show that the sum over planar trees formula of Kontsevich and Soibelman transfers C-infinity structures along a contraction. Applying this result to a cosimplicial commutative algebra A^* over a field of characteristic zero, we exhibit a canonical unital C-infinity structure on Tot(A^*), which is unital if A^* is; in particular, we obtain a canonical C-infinity structure on the cochain complex of a simplicial set.
The embedding Chains(R) into Cochains(R) as the compactly supported cochains might lead one to expect Chains(R) to carry a nonunital commutative Frobenius algebra structure, up to a degree shift and some homotopic weakening of the axioms. We prove that under reasonable "locality" conditions, a cofibrant resolution of the dioperad controlling nonunital shifted-Frobenius algebras does act on Chains(R), and in a homotopically-unique way. But we prove that this action does not extend to a homotopy Frobenius action at the level of properads or props. This gives an example of a geometrically meaning…
A homotopy Gerstenhaber structure on a differential graded algebra is essentially a family of operations defining a multiplication on its bar construction. We prove that the normalized singular cochain algebra of a Davis-Januszkiewicz space is formal as a homotopy Gerstenhaber algebra, for any coefficient ring. This generalizes a recent result by the author about classifying spaces of tori and also strengthens the well-known dga formality result for Davis-Januszkiewicz spaces due to the author and Notbohm-Ray. As an application, we determine the cohomology rings of free and based loop spaces o…
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…
The purpose of the present article is threefold. First of all, we rebuild the whole theory of cosimplicial models of mapping spaces by using systematically Kan adjunction techniques. Secondly, given two topological spaces X and Y, we construct a cochain algebra which is quasi-isomorphic (as an algebra) to the singular cochain algebra of the corresponding mapping space from X to Y. Here, X has to be homotopy equivalent to the geometric realization a finite simplicial set and of dimension less or equal to the connectivity of Y. At last, we apply these results to the study of finite group actions…
We construct E-infinity cell algebra models for the cochain algebras of the free and based loop spaces on a simply-connected topological space. Techniques from rational homotopy theory are exploited throughout.
Let X be a pseudomanifold. In this text, we use a simplicial blow-up to define a cochain complex whose cohomology with coefficients in a field, is isomorphic to the intersection cohomology of X, introduced by M. Goresky and R. MacPherson. We do it simplicially in the setting of a filtered version of face sets, also called simplicial sets without degeneracies, in the sense of C.P. Rourke and B.J. Sanderson. We define perverse local systems over filtered face sets and intersection cohomology with coefficients in a perverse local system. In particular, as announced above when X is a pseudomanifol…
The standard reduced bar complex B(A) of a differential graded algebra A inherits a natural commutative algebra structure if A is a commutative algebra. We address an extension of this construction in the context of E-infinity algebras. We prove that the bar complex of any E-infinity algebra can be equipped with the structure of an E-infinity algebra so that the bar construction defines a functor from E-infinity algebras to E-infinity algebras. We prove the homotopy uniqueness of such natural E-infinity structures on the bar construction. We apply our construction to cochain complexes of topol…
$\mathbb{Q}_p$-Homotopy Types and Applications to Topology and Algebraic Geometry
本文发展Q_p同伦理论，关联p完全空间与Q_p上的交换微分分次代数，证明其极小模型可还原同伦群等结构，并将该理论应用于拓扑与代数几何的多类问题。
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 …
First we describe a class of homotopy Frobenius algebras via cyclic operads which we call cyclic $A_\infty$ algebras. We then define a suitable new combinatorial operad which acts on the Hochschild cochains of such an algebra in a manner which encodes the homotopy BV structure. Moreover we show that this operad is equivalent to the cellular chains of a certain topological (quasi)-operad of CW complexes whose constituent spaces form a homotopy associative version of the Cacti operad of Voronov. These cellular chains thus constitute a chain model for the framed little disks operad, proving a cyc…
Algebraic models for equivariant rational homotopy theory for discrete groups
本文提出一种框架，将空间同调理论的代数模型推广到真正的equivariant情况。讨论了交换微分格雷德代数模型和完备微分格雷德李代数模型在有理空间中的应用，并与其他模型类别进行比较。
The primary interest of this paper is to discuss the role of twisting cochains in the theory of characteristic classes. We begin with the homological description of monodromy map, associated with a connection on a trivial bundle over a 1-connected manifold. We regard it as a homomorphism from the algebra of differential forms on the structure group to the algebra of differential forms on the based loopspace of the base, represented by the (reduced) bar-complex of differential forms on it. Next we discuss the notion of "twisting cochains", or more generally "twisting maps", their equivalence re…
A Bar-Natan homotopy type
本文为 Bar-Natan 同调构造了空间细化 CW-谱，其稳定同伦型为链环不变量，并猜想量子滤过可提升至空间层面，导向 $s$-不变量的上同伦细化。
We prove that the operad B of natural operations on the Hochschild cohomology has the homotopy type of the operad of singular chains on the little disks operad. To achieve this goal, we introduce crossed interval groups and show that B is a certain crossed interval extension of an operad T whose homotopy type is known. This completes the investigation of the algebraic structure on the Hochschild cochain complex that has lasted for several decades.
The singular cochain complex of a topological space is a classical object. It is a Differential Graded algebra which has been studied intensively with a range of methods, not least within rational homotopy theory. More recently, the tools of Auslander-Reiten theory have also been applied to the singular cochain complex. One of the highlights is that by these methods, each Poincare duality space gives rise to a Calabi-Yau category. This paper is a review of the theory.
We establish equalities between cochain and chain type levels of maps by making use of exact functors which connect appropriate derived and coderived categories. Relevant conditions for levels of maps to be finite are extracted from the equalities which we call duality on the levels. Moreover, we give a lower bound of the cochain type level of the diagonal map on the classifying space of a Lie group by considering the ghostness of a shriek map which appears in derived string topology. A variant of Koszul duality for a differential graded algebra is also discussed.
The notion of a derived A-infinity algebra, considered by Sagave, is a generalization of the classical notion of A-infinity algebra, relevant to the case where one works over a commutative ring rather than a field. We initiate a study of the homotopy theory of these algebras, by introducing a hierarchy of notions of homotopy between the morphisms of such algebras. We define r-homotopy, for non-negative integers r, in such a way that r-homotopy equivalences underlie E_r-quasi-isomorphisms, defined via an associated spectral sequence. We study the special case of twisted complexes (also known as…
The equalities of interleaving distances and cohomology interleavings of spaces over $BS^1$
本文在域上持续微分分次模范畴中定义上同调交错距离（CohID）并证明其与三类交错距离等价，将其应用于BS¹上空间，结合杯长研究其上下界，并通过复射影空间示例明确计算了该距离。
We construct for any algebra over an operad an Hochschild chain complex. In the case of the singular cochain complex of a topological space, considered as a commutative algebra up to homotopy, we show that this complex computes the singular cohomology of the free loop space over this topological space.
Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of $G$ and $K$ is invertible in a given principal ideal domain $k$. It is known that in this case the cohomology of the homogeneous space $G/K$ with coefficients in $k$ and the torsion product of $H^{*}(BK)$ and $k$ over $H^{*}(BG)$ are isomorphic as $k$-modules. We show that this isomorphism is multiplicative and natural in the pair $(G,K)$ provided that 2 is invertible in $k$. The proof uses homotopy Gerstenhaber algebras in an essential way. I…
We provide explicit quasi-isomorphisms between the following three algebraic structures associated to the unit interval: i) the commutative dg algebra of differential forms, ii) the non-commutative dg algebra of simplicial cochains and iii) the Whitney forms, equipped with a homotopy commutative and homotopy associative, i.e. $C_\infty$, algebra structure. Our main interest lies in a natural `discretization' $C_\infty$ quasi-isomorphism $φ$ from differential forms to Whitney forms. We establish a uniqueness result that implies that $φ$ coincides with the morphism from homotopy transfer, and ob…
Among its many corollaries, Poincare duality implies that the de Rham cohomology of a compact oriented manifold is a shifted commutative Frobenius algebra --- a commutative Frobenius algebra in which the comultiplication has cohomological degree equal to the dimension of the manifold. We study the question of whether this structure lifts to a "homotopy" shifted commutative Frobenius algebra structure at the cochain level. To make this question nontrivial, we impose a mild locality-type condition that we call "quasilocality": strict locality at the cochain level is unreasonable, but it is reaso…
Homotopy G-algebras and moduli space operad
本文强调了代数曲线模空间在结合代数与拓扑中的作用，证明了带乘法operad空间构成同伦Gerstenhaber代数，奇异上链复形自然构成operad，且装饰模空间operad在Kähler流形的de Rham复形上诱导出最一般的同伦G-代数结构。
A distance between maps via interleavings of relative Sullivan algebras
本文研究了通过相对Sullivan代数的交织构造映射间距离的方法，探讨了持久CDGA模型在同调集映射中的伪距离应用，以及持久CDGA与交织距离的关系。
Working in the context of symmetric spectra, we consider any higher algebraic structures that can be described as algebras over an operad O. We prove that the fundamental adjunction comparing O-algebra spectra with coalgebra spectra over the associated comonad K, via topological Quillen homology (or TQ-homology), can be turned into an equivalence of homotopy theories by replacing O-algebras with the full subcategory of 0-connected O-algebras. This resolves in the affirmative the 0-connected case of a conjecture of Francis-Gaitsgory. This derived Koszul duality result can be thought of as the s…
Let $A$ be a special homotopy G-algebra over a commutative unital ring $\Bbbk$ such that both $H(A)$ and $\operatorname{Tor}_{i}^{A}(\Bbbk,\Bbbk)$ are finitely generated $\Bbbk$-modules for all $i$, and let $τ_{i}(A)$ be the cardinality of a minimal generating set for the $\Bbbk$-module $\operatorname{Tor}_{i}^{A}(\Bbbk,\Bbbk).$ Then the set ${τ_{i}(A)} $ is unbounded if and only if $\tilde{H}(A)$ has two or more algebra generators. When $A=C^{\ast}(X;\Bbbk)$ is the simplicial cochain complex of a simply connected finite $CW$-complex $X,$ there is a similar statement for the "Betti numbers" of…
We determine the v1-periodic homotopy groups of all irreducible p-compact groups (BX,X). In the most difficult, modular, cases, we follow a direct path from their associated invariant polynomials to these homotopy groups. We show that, if p is odd, every irreducible p-compact group has X of the homotopy type of a product of explicit spaces related to p-completed Lie groups.
We show that the transfer map on Floer homotopy types associated to an exact Lagrangian embedding is an equivalence. This provides an obstruction to representing isotopy classes of Lagrangian immersions by Lagrangian embeddings, which, unlike previous obstructions, is sensitive to information that cannot be detected by Floer cochains. We show this by providing a concrete computation in the case of spheres.
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…
We analyse the homotopy types of gauge groups for principal $U(n)$-bundles over lens spaces.
We propose an analogue of the bounded derived category for an augmented ring spectrum, defined in terms of a notion of Noether normalization. In many cases we show this category is independent of the chosen normalization. Based on this, we define the singularity and cosingularity categories measuring the failure of regularity and coregularity and prove they are Koszul dual in the style of the BGG correspondence. Examples of interest include Koszul algebras and Ginzburg DG-algebras, $C^*(BG)$ for finite groups (or for compact Lie groups with orientable adjoint representation), cochains in ratio…
We describe the E-infinity algebra structure on the complex of singular cochains of a topological space, in the context of sheaf theory. As a first application, for any algebraic variety we define a weight filtration compatible with its E-infinity structure. This naturally extends the theory of mixed Hodge structures in rational homotopy to p-adic homotopy theory. The spectral sequence associated to the weight filtration gives a new family of multiplicative algebraic invariants of the varieties for any coefficient ring, carrying Steenrod operations. As a second application, we promote Deligne'…
We show some results of compact Kahler manifolds with elliptic homotopy type. In complex dimension 4 we list the Hodge diamonds of compact Kahler manifolds with elliptic homotopy type. In general dimension we obtain a partial characterization of the Hodge diamonds.
In the rational cohomology of a 1-connected space a structure of $C_{\infty}$-algebra is constructed and it is shown that this object determines the rational homotopy type
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。
We describe a collection of higher homotopy operations which determine the rational homotopy type of a simply-connected space X. These are described in terms of simplicial resolutions of successive approximations (L^k,α} to the Quillen DGL model for X. The operations lie in suitable cohomology groups H^*{L^{(k,α)}; π_* X\otimes Q} of these DGLs. To facilitate the recovery of an integral version of the operations from the rational description, we also define a differential graded non-associative algebra model for rational spaces.
We determine the number of distinct fibre homotopy types for the gauge groups of principal $Sp(2)$-bundles over a closed, simply-connected four-manifold.
This paper proves that the functor $C(*)$ that sends pointed, simply-connected CW-complexes to their chain-complexes equipped with diagonals and iterated higher diagonals, determines their integral homotopy type --- even inducing an equivalence of categories between the category of CW-complexes up to homotopy equivalence and a certain category of chain-complexes equipped with higher diagonals. Consequently, $C(*)$ is an algebraic model for integral homotopy types similar to Quillen's model of rational homotopy types. For finite CW complexes, our model is finitely generated. Our result implies …
Deformations and homotopy theory of Nijenhuis associative algebras
本文引入同伦Nijenhuis结合代数概念，证明其控制算子为极小模型，构造控制结合积与Nijenhuis算子同时形变的 $L_\infty$-代数，并导出形变复形与上同调理论。
We establish an upper bound for the cochain type level of the total space of a pull-back fibration. It explains to us why the numerical invariant for a principal bundle over the sphere are less than or equal to two. Moreover computational examples of the levels of path spaces and Borel constructions, including biquotient spaces and Davis-Januszkiewicz spaces, are presented. We also show that the chain type level of the homotopy fibre of a map is greater than the E-category in the sense of Kahl, which is an algebraic approximation of the Lusternik-Schnirelmann category of the map. The inequalit…
We define inductively a sequence of purely algebraic invariants - namely, classes in the Quillen cohomology of the Pi-algebra π_* X - for distinguishing between different homotopy types of spaces. Another sequence of such cohomology classes allows one to decide whether a given abstract Pi-algebra can be realized as the homotopy Pi-algebra of a space in the first place. The paper is written for a relatively general "resolution model category", so it also applies, for example, to rational homotopy types.
In this paper we address the classification problem for locally compact (n-1)-connected CW-complexes with dimension less or equal than n+2 up to proper homotopy type. We obtain complete classification theorems in terms of purely algebraic data in those cases where the representation type of the involved algebra is finite. For this we define new quadratic functors in controlled algebra and new homotopy and cohomology invariants in proper homotopy theory.
The notion of the \emph{homotopy type} of a topological stack has been around in the literature for some time. The basic idea is that an atlas $X \to \mathfrak{X}$ of a stack determines a topological groupoid $\mathbb{X}$ with object space $X$. The homotopy type of $\mathfrak{X}$ should be the classifying space $B \mathbb{X}$. The choice of an atlas is not part of the data of a stack and hence it is not immediately clear why this construction of a homotopy type is well-defined, let alone functorial. The purpose of this note is to give an elementary construction of such a homotopy-type functor.
In this paper, we determine the homotopy type of the quotient space $\mathcal{Z}_{Δ^k_m}/S^1_d$, given by the moment-angle complex $\mathcal{Z}_{Δ_m^k}$ under the diagonal circle action.
On the rational $C_2$-homotopy type of ${BSU_{\mathbb{R}}}_m$
本文通过等价Eilenberg-Maclane空间描述实数BSU_m的有理C_2等价同调类型，解决动机同调论中的相关问题。
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