80 pages. v2: Major revision. Improvements include comparison of string products, asymptotic existence of fundamental classes, and improved ring structure preservation. v3: Fix reference
我们证明了任何有限李型群在特征p≠ℓ下的模ℓ上同调可以作为自由循环空间BG的同调的模，通过由字符串拓扑（Chas-Sullivan风格）构造的环和模结构。如果有限群的同调中某个由G的基本类产生的类非平凡，则此模结构为自由秩一模，提供两个同调之间的高度结构同构。我们在多种情况下验证了该类的非平凡性，包括所有在q元素上的一般线性群，其中q≡1 mod ℓ。我们还展示了如何通过替换BG为依赖于q mod ℓ阶数的ℓ-紧固定点群来处理扭动并避免同余条件，而无需改变有限群。此修改后，我们尚未发现该类为平凡的例子，提出了关于Tezuka开放问题的可能一般性答案。
英文摘要：
We show that the mod $\ell$ cohomology of any finite group of Lie type in characteristic $p$ different from $\ell$ admits the structure of a module over the mod $\ell$ cohomology of the free loop space of the classifying space $BG$ of the corresponding compact Lie group $G$, via ring and module structures constructed from string topology, a la Chas-Sullivan. If a certain class in the homology of the finite group of Lie type, arising from the fundamental class of $G$, is nontrivial, then this module structure is free of rank one, providing a highly structured isomorphism between the two cohomologies. We verify the nontriviality of the class in a range of cases, including all simply connected untwisted classical groups over the field of $q$ elements, with $q$ congruent to 1 mod $\ell$. We also show how to deal with twistings and avoid the congruence condition by replacing $BG$ by a certain $\ell$-compact fixed point group depending on the order of $q$ mod $\ell$, without changing the finite group. With this modification, we know of no examples where the class is trivial, raising the possibility of a general structural answer to an open question of Tezuka, who speculated about the existence of an isomorphism between the two cohomology rings.
To open-closed cobordism surfaces, open-closed string topology associates topological quantum field theory (TQFT) operations, namely string operations, which depend only on homeomorphism types of surfaces and which satisfy the sewing property. We show that most TQFT string operations vanish in open-closed string topology. We describe those open-closed cobordisms with vanishing string operations, and give a short list of open-closed cobordisms with possibly nontrivial string operations.
We give a new proof of the string topology structure of a compact oriented surface of genus g greater than or equal to 2, using elementary algebraic topology. This reproves the result of Vaintrob.
This paper examines the string topology category of a manifold, defined by Blumberg, Cohen and Teleman. Since the string topology category is a subcategory of a compactly generated triangulated category, the machinery of stratification, constructed by Benson, Krause and Iyengar, can be applied in order to gain an understanding of the string topology category. It is shown that an appropriate stratification holds when the manifold in question is a simply connected compact Lie group. This last result is used to derive some properties of the relevant string topology categories.
A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and Batalin-Vilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes non simply-connected case, and therefore gives an algebraic and chain level model of Chas-Sullivan's String Topology.
This paper is an exposition of the new subject of String Topology. We present an introduction to this exciting new area, as well as a survey of some of the latest developments, and our views about future directions of research. We begin with reviewing the seminal paper of Chas and Sullivan, which started String Topology by introducing a BV-algebra structure on the homology of a loop space of a manifold, then discuss the homotopy theoretic approach to String Topology, using the Thom-Pontrjagin construction, the cacti operad, and fat graphs. We review quantum field theories and indicate how stri…
We construct a space of string diagrams, which are a type of fatgraph with some additional data, and show that there are string topology operations on the chains of the free loop space of a closed Riemannian manifold which are parameterized by the chains on the space of string diagrams. These operations are shown to recover known structure on homology of the free loop space.
The complete enriched Lie algebras constitue the natural extension of graded Lie algebras for connected spaces. Each complete enriched Lie algebra is the rational homotopy Lie algebra of a connected space. This text is the first part of a general study of those Lie algebras
We extend the structure of string topology from mapping spaces to embedding spaces $Emb(S^n,M)$. This extension comes from an action of the cleavage operad, a coloured $E_{n+1}$-operad. For all values of $n \in \mathbb{N}$, this gives an action of certain Thom spectra that provides an action on the level homology. We provide an analysis of the string topology structure derived from the embedding $Emb(S^n,M) \hookrightarrow Map(S^n,M)$. In the $1$-dimensional case it specializes a morphism of BV-algebras $\mathbb{H}_*(Emb(S^1,M)) \to \mathbb{H}_*(Map(S^1,M)$.
Examples of non-trivial higher string topology operations have been regrettably rare in the literature. In this paper, working in the context of string topology of classifying spaces, we provide explicit calculations of a wealth of non-trivial higher string topology operations associated to a number of different Lie groups. As an application of these calculations, we obtain an abundance of interesting homology classes in the twisted homology groups of automorphism groups of free groups, the ordinary homology groups of holomorphs of free groups, and the ordinary homology groups of affine groups…
Following Cieliebak, Fukaya, Latschev and Volkov, we construct an IBL-infinity chain model for equivariant string topology on cyclic Hochschild cochains of de Rham cohomology. We study its properties and perform explicit computations.
We use the computational power of rational homotopy theory to provide an explicit cochain model for the loop product and the string bracket of a 1-connected closed manifold M. We prove that the loop homology of M is isomorphic to the Hochschild cohomology of the commutative graded algebra A_{PL}(M) with coefficients in itself. Some explicit computations of the loop product and the string bracket are given.
Luc Menichi showed that the BV algebras on $H^\bullet(LS^2;Z_2)[-2]$ coming from string topology and the one on $HH^\bullet(H^\bullet(S^2;Z_2),H^\bullet(S^2;Z_2))$ using Poincaré duality on $H^\bullet(S^2;Z_2)$ are not isomorphic. In this note we show how one can obtain the string topology BV algebra on Hochschild cohomology using a Poincaré duality structure with higher homotopies. This Poincaré duality (with higher homotopies) on cohomology is induced by a local Poincaré duality (with higher homotopies) on the cochain level.
In this paper we continue to develop the topological method started in Santos-San Martin \cite{ariasm} to get semigroup generators of semi-simple Lie groups. Consider a subset $Γ\subset G$ that contains a semi-simple subgroup $G_{1}$ of $G$. Then $Γ$ generates $G$ if $\mathrm{Ad}\left( Γ\right) $ generates a Zariski dense subgroup of the algebraic group $\mathrm{Ad}\left( G\right) $. The proof is reduced to check that some specific closed orbits of $G_{1}$ in the flag manifolds of $G$ are not trivial in the sense of algebraic topology. Here, we consider three different cases of semi-simple Lie…
Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology built by M. Chas and D. Sullivan. We also study and build an action of the homology of reduced Sullivan's chord diagrams on the singular homology of free loop spaces, extending previous results of R. Cohen and V. Godin and unifying part of the rich algebraic structure of string topology as an algebra over the prop of these reduced diagrams. Some of these operations are extended to spaces of maps from a sphere to a compact manifold.
We construct an equivalence of $E_{2}$ algebras between two models for the Thom spectrum of the free loop space that are related by derived Koszul duality. To do this, we describe the functoriality and invariance properties of topological Hochschild cohomology.
In this paper we study the string topology (á la Chas-Sullivan) of an orbifold. We define the string homology ring product at the level of the free loop space of the classifying space of an orbifold. We study its properties (introducing an operad to do so) and do some explicit calculations.
Let G be a Poincare duality group of dimension n. For a given element g in G, let C_g denote its centralizer subgroup. Let L_G be the graded abelian group defined by (L_G)_p = oplus_{[g]}H_{p+n}(C_g) where the sum is taken over conjugacy classes of elements in G. In this paper we construct a multiplication on L_G directly in terms of intersection products on the centralizers. This multiplication makes L_G a graded, associative, commutative algebra. When G is the fundamental group of an aspherical, closed oriented n manifold M, then (L_G)_* = H_{*+n}(LM), where LM is the free loop space of M. W…
Given a smooth closed manifold M with a family {L_i} of closed submanifolds, we consider the free loop space LM and the spaces PM(L_i,L_j) of open strings (paths g:[0,1]->M with g(0) in L_i, and g(1) in L_j). We construct string topology operations resulting in an open-closed TQFT on the family (h_*(LM),h_*(PM(L_i,L_j)) which extends the known string topology TQFT on h_*(LM). Here, h_* is a multiplicative generalized homology theory supporting orientations for M and the L_i. To construct the operations, we introduce the notion of open-closed fat graph, generalizing fat graphs to the open-close…
Let M be a connected, simply connected, closed and oriented manifold, and G a finite group acting on M by orientation preserving diffeomorphisms. In this paper we show an explicit ring isomorphism between the orbifold string topology of the orbifold [M/G] and the Hochschild cohomology of the dg-ring obtained by performing the smash product between the group G and the singular cochain complex of M.
In this paper we prove that for an almost complex orbifold, its virtual orbifold cohomology [math.AT/0606573] is isomorphic as algebras to the Chen-Ruan orbifold cohomology of its cotangent orbifold.
In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, and evaluating cohomology classes on this fundamental class. By using similar constructions based on "fat" or ribbon graphs, we describe how to construct string topology operations on the loop space of a manifold, using Morse theoretic techniques. Finally, we discuss how to relate …
This note explores the interaction between cohomology operations in a generalized cohomology theory and a string topology loop coproduct dual to the Chas--Sullivan loop product. More precisely, we ask for a description for the failure of a given operation to commute with the loop coproduct, and will obtain a satisfactory answer in the case where the operation preserves both sums and products. Examples of such operations include the total Steenrod square in ordinary mod 2 cohomology and the Adams operations in K-theory.
The purpose of this paper is to describe a general and simple setting for defining $(g,p+q)$-string operations on a Poincaré duality space and more generally on a Gorenstein space. Gorenstein spaces include Poincaré duality spaces as well as classifying spaces or homotopy quotients of connected Lie groups. Our presentation implies directly the homotopy invariance of each $(g,p+q)$-string operation as well as it leads to explicit computations.
In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincaré duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading `string topology'. Our method is based on obstruction theory for $C_\infty$-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.
In this paper we study the topology of the cobordism category of open and closed strings. This is a 2-category in which the objects are compact one-manifolds whose boundary components are labeled by an indexing set (the set of "D-branes"), the 1-morphisms are cobordisms of manifolds with boundary, and the 2-morphisms are diffeomorphisms of the surface cobordisms. Our methods and techniques are direct generalizations of those used by U. Tillmann in her study of the category of closed strings. We input the striking theorem of Madsen and Weiss regarding the topology of the stable mapping class gr…
We study string topology for classifying spaces of connected compact Lie groups, drawing connections with Hochschild cohomology and equivariant homotopy theory. First, for a compact Lie group $G$, we show that the string topology prospectrum $LBG^{-TBG}$ is equivalent to the homotopy fixed-point prospectrum for the conjugation action of $G$ on itself, $G^{hG}$. Dually, we identify $LBG^{-ad}$ with the homotopy orbit spectrum $(DG)_{hG}$, and study ring and co-ring structures on these spectra. Finally, we show that in homology, these products may be identified with the Gerstenhaber cup product …
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…
This paper is a continuations of the project initiated in the book string topology for stacks. We construct string operations on the SO(2)-equivariant homology of the (free) loop space $L(X)$ of an oriented differentiable stack $X$ and show that $H^{SO(2)}_{*+dim(X) -2}(L(X))$ is a graded Lie algebra. In the particular case where $X$ is a 2-dimensional orbifold we give a Goldman-type description for the string bracket. To prove these results, we develop a machinery of (weak) group actions on topological stacks which should be of independent interest. We explicitly construct the quotient stack …
We define a coalgebra structure for open strings transverse to any framed codimension 2 submanifold. When the submanifold is a knot in R^3, we show this structure recovers a specialization of the Ng cord algebra, a non-trivial knot invariant which is not determined by a number of other knot invariants.
In 1999 Chas and Sullivan showed that the homology of the free loop space of an oriented manifold admits the structure of a Batalin-Vilkovisky algebra. In this paper we give a complete description of this Batalin-Vilkovisky algebra for complex projective spaces. This builds on a description of the ring structure that is due to Cohen, Jones and Yan. In the course of the proof we establish several new general results. These include a description of how symmetries of a manifold can be used to understand its string topology, and a relationship between characteristic classes and circle actions on s…
Let M be a smooth, simply-connected, closed oriented manifold, and LM the free loop space of M. Using a Poincare duality model for M, we show that the reduced equivariant homology of LM has the structure of a Lie bialgebra, and we construct a Hopf algebra which quantizes the Lie bialgebra.
The free loop space of a Lie group is homeomorphic to the product of the Lie group itself and its based loop space. We show that the coproduct on the homology of the free loop space that was introduced by Goresky and Hingston splits into the diagonal map on the group and a based coproduct on the homology of the based loop space. This result implies that the coproduct is trivial for even-dimensional Lie groups. Using results by Bott and Samelson, we show that the coproduct is trivial as well for a large family of simply connected Lie groups.
Chas and Sullivan have defined an intersection-type product on the homology of the free loop space LM of an oriented manifold M. In this paper we show how to extend this construction to a topological conformal field theory of degree d. In particular, we get operations on the homology of LM which are parameterized by the homology of the moduli space of open-closed Riemann surfaces.
Chas and Sullivan introduced string homology, which is the equivariant homology of the loop space with the $S^1$ action on loops by rotation. Craig Westerland computed the string homology for spheres with coefficients in $\mathbb{Z} /2\mathbb{Z}$, and in Somnath Basu's dissertation, he computes the string homology and string bracket for spheres over rational coefficients, and he finds that the bracket is trivial. In this paper, we compute string homology and the string bracket for spheres with integer coefficients, treating the odd- and even-dimensional cases separately. We use the Gysin seque…
In previous work with Schoenfeld, we considered a string-type chain complex of curves on surfaces, with differential given by resolving crossings, and computed the homology of this complex for discs. In this paper we consider the corresponding "string homology" of annuli. We find this homology has a rich algebraic structure which can be described, in various senses, as fermionic. While for discs we found an isomorphism between string homology and the sutured Floer homology of a related 3-manifold, in the case of annuli we find the relationship is more complex, with string homology containing f…
For M a closed, connected, oriented manifold, we obtain the Batalin-Vilkovisky (BV) algebra of its string topology through homotopy-theoretic constructions on its based loop space. In particular, we show that the Hochschild cohomology of the chain algebra C_*ΩM carries a BV algebra structure isomorphic to that of the loop homology $\mathbb{H}_*(LM)$. Furthermore, this BV algebra structure is compatible with the usual cup product and Gerstenhaber bracket on Hochschild cohomology. To produce this isomorphism, we use a derived form of Poincaré duality with C_*ΩM-modules as local coefficient syste…
We show that the reduced Hochschild homology of a DG open Frobenius algebra has the natural structure of a Batalin-Vilkovisky coalgebra, and the reduced cyclic homology has the natural structure of a gravity coalgebra. This gives an algebraic model for a Batalin-Vilkovisky coalgebra structure on the reduced homology of the free loop space of a simply connected closed oriented manifold, and a gravity coalgebra structure on the reduced equivariant homology.
In 1999 Chas and Sullivan discovered that the homology H_*(LX) of the space of free loops on a closed oriented smooth manifold X has a rich algebraic structure called string topology. They proved that H_*(LX) is naturally a Batalin-Vilkovisky (BV) algebra. There are several conjectures connecting the string topology BV algebra with algebraic structures on the Hochschild cohomology of algebras related to the manifold X, but none of them has been verified for manifolds of dimension n>1. In this work we study string topology in the case when X is aspherical (i.e. its homotopy groups π_i(X) vanish…
The generalized string topology construction of Gruher and Salvatore assigns to any bundle of $E_n$-algebras $A$ over a closed oriented manifold $M$ a collection of intersection-type operations on the homology of the total space. These operations are realized by an $H_n$-ring structure on the Thom spectrum $A^{-TM}$ under the Thom isomorphism. We rigidify and extend this construction to a functor connecting the homotopy theory of spaces and spectra parametrized by $M$ to the homotopy theory of module spectra over the Atiyah-Milnor-Spanier-Whitehead dual $M^{-TM} \simeq \bbD M$. Then, using an …
We study the string topology of a closed oriented Riemannian manifold M. We describe a compact moduli space of diagrams, and show how the cellular chain complex of this space gives algebraic operations on the singular chains of the free loop space LM of M. These operations are well-defined on the homology of a quotient of this moduli space, which has the homotopy type of a compactification of the moduli space of Riemann surfaces. In particular, our action of the 0-dimensional homology of the quotient space on the homology of the free loop space of M recovers the Cohen-Godin positive boundary T…
The loop product is an operation in string topology. Cohen and Jones gave a homotopy theoretic realization of the loop product as a classical ring spectrum $LM^{-TM}$ for a manifold $M$. Using this, they presented a proof of the statement that the loop product is isomorphic to the Gerstenhaber cup product on the Hochschild cohomology $HH^*(C^*(M)\,;C^*(M))$ for simply connected $M$. However, some parts of their proof are technically difficult to justify. The main aim of the present paper is to give detailed modification to a geometric part of their proof. To do so, we set up an "up to higher h…
We develop a variant of calculus of functors, and use it to relate the gauge group G(P) of a principal bundle P over M to the Thom ring spectrum (P^Ad)^{-TM}. If P has contractible total space, the resulting Thom ring spectrum is LM^{-TM}, which plays a central role in string topology. R.L. Cohen and J.D.S. Jones have recently observed that, in a certain sense, (P^Ad)^{-TM} is the linear approximation of G(P). We prove an extension of that relationship by demonstrating the existence of higher-order approximations and calculating them explicitly. This also generalizes calculations done by G. Ar…
Given a principal bundle over a closed manifold, G --> P --> M, let P^{Ad} --> M be the associated adjoint bundle. Gruher and Salvatore showed that the Thom spectrum (P^{Ad})^{-TM} is a ring spectrum whose corresponding product in homology is a Chas-Sullivan type string topology product. We refer to this spectrum as the `string topology spectrum of P", S (P). In the universal case when P is contractible, S(P) is equivalent to LM^{-TM} where LM is the free loop space of the manifold. This ring spectrum was introduced by the authors as a homotopy theoretic realization of the Chas-Sullivan string…
From an operad C with an action of a group G, we construct new operads using the homotopy fixed point and orbit spectra. These new operads are shown to be equivalent when the generalized G-Tate cohomology of C is trivial. Applying this theory to the little disk operad C_2 (which is an S^1 operad) we obtain variations on Getzler's gravity operad, which we show governs the Chas-Sullivan string bracket.
We show that a model of chain complex of the free loop space of a $C^\infty$-manifold, which is proposed in arxiv:1404.0153, admits an action of a certain dg operad. This is a chain level structure under the Chas-Sullivan BV structure on loop space homology. Our dg operad is a variant of the cacti operad, and we introduce combinatorial objects called "decorated cacti" to define it. We also define a chain level Gerstenhaber structure on Hochschild cochains of any differential graded algebra. Applied to the dga of differential forms, this structure is compatible with our chain level structure in…
Let $X$ be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$-equivariant homology $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ of the free loop space of $X$ preserves the Hodge decomposition of $\overline{H}_{\ast}^{S^1}(\mathcal{L}X,\mathbb{Q}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture proposed in …
$A_\infty$ categories are a mathematical structure that appears in topological field theory, string topology, and symplectic topology. This paper studies the cyclic homology of a Calabi-Yau $A_\infty$ category, and shows that it is naturally an equivariant topological conformal field theory, and in particular, contains an involutive Lie bialgebra. Applications of the theory to string topology and the Fukaya category are given; in particular, it is shown that there is a Lie bialgebra homomorphism from the cyclic cohomology of the Fukaya category of a symplectic manifold with contact type bounda…
We study the application of IBL-infinity-algebras to string topology and explicitly compute the case of spheres. This involves finding a Green kernel and computing integrals associated to trivalent ribbon graphs, which are similar to integrals from perturbative Chern-Simons theory. We generalize the computation and show that the twisted IBL-infinity-structure in question is in many cases trivial.
