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 introduce characteristics into chromatic homotopy theory. This parallels the prime characteristics in number theory as well as in our earlier work on structured ring spectra and unoriented bordism theory. Here, the K(n)-local Hopkins-Miller classes $ζ_n$ take the places of the prime numbers, and this allows us to discuss higher bordism theories. We prove that the K(2)-localizations of the spectrum of topological modular forms as well as the string bordism spectrum have characteristic $ζ_2$.
We show that $K(2)$-locally, the smash product of the string bordism spectrum and the spectrum $T_2$ splits into copies of Morava $E$-theories. Here, $T_2$ is related to the Thom spectrum of the canonical bundle over $ΩSU(4)$.
We compute the low dimensional String bordism groups of $BE_8$ and $BE_8\times BE_8$ using a combination of Adams spectral sequences together with comparisons to the Spin bordism cases.
Using spectral invariants of Dirac operators we construct a secondary version of the Witten genus, a bordism invariant of string manifolds in dimensions $4m-1$. We prove a secondary index theorem which relates this global-analytic construction with its homotopy-theoretic analog. The latter will be calculated through its factorization over topological modular forms.
This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle characteristic numbers arising in Borel-Hirzebruch Lie-group-theoretic calculations correspond precisely to the divisibility properties arising in the Hovey-Ravenel-Wilson BP-Hopf-ring-theoretic calculation of string bordism at primes >3.
It is one of the most important facts in 4-dimensional topology that not every spherical homology class of a 4-manifold can be represented by an embedded sphere. In 1978, M. Freedman and R. Kirby showed that in the simply connected case, many of the obstructions to constructing such a sphere vanish if one modifies the ambient 4-manifold by adding products of 2-spheres, a process which is usually called stabilisation. In this paper, we extend this result to non-simply connected 4-manifolds and show how it is related to the Spin^c-bordism groups of Eilenberg-MacLane spaces.
In this work we investigate the Swampland Cobordism Conjecture in the context of type IIB string theory geometries with non-trivial duality bundle. Quite remarkably, we find that many non-trivial bordism classes with duality bundles in Mp$(2,\mathbb{Z})$, a double cover of SL$(2,\mathbb{Z})$ related to fermions, correspond to asymptotic boundaries of well-known supersymmetric F-theory backgrounds. These include $[p,q]$-7-branes, non-Higgsable clusters, S-folds, as well as various lower-dimensional generalizations. These string theoretic objects break the global symmetries associated to the non…
This paper shows that $\mathrm{tmf}[1/6]$ is not a ring spectrum quotient of $\mathrm{MO}\langle8\rangle[1/6]$. In fact, for any prime $p>3$ and any sequence $X$ of homogeneous elements of $π_*\mathrm{MO}\langle8\rangle$, the $π_*\mathrm{MO}\langle8\rangle$-module $$π_*\big(\mathrm{MO}\langle8\rangle_{(p)}/X\big)$$ is not (even abstractly) isomorphic to $π_*\mathrm{tmf}_{(p)}$. It does so by showing that, for any commutative ring spectrum $R$ and any sequence $X$ of homogeneous elements of $π_*(R)$, there is an isomorphism of graded $\mathbf{Q}$-vector spaces $$π_*(R/X)\otimes\mathbf{Q} \cong …
The equivariant bordism classification of manifolds with group actions is an essential subject in the study of transformation groups. We are interesting in the action of 2-torus group $\mathbb{Z}_2^n$ and torus group $T^n$, and study the equivariant bordism of 2-torus manifolds and unitary toric manifolds. In this paper, we give a new description of the group $\mathcal{Z}_n(\mathbb{Z}_2^n)$ of 2-torus manifolds, and determine the dimention of $\mathcal{Z}_n(\mathbb{Z}_2^n)$ as a $\mathbb{Z}_2$-vector space. With the help of toric topology, Lü and Tan proved that the bordism groups $\mathcal{Z}…
We give a bordism-theoretic characterisation of those closed almost contact (2q+1)-manifolds (with q > 2) which admit a Stein fillable contact structure. Our method is to apply Eliashberg's h-principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.
We study and in some cases classify highly connected manifolds which admit a Riemannian metric with positive $p$-curvature. The $p$-curvature was defined and studied by the second author. It turns out that positivity of $p$-curvature could be preserved under surgeries of codimension at least $p+3$. This gives a key to reduce a geometrical classification problem to a topological one, in terms of relevant bordism groups and index theory. In particular, we classify 3-connected manifolds with positive 2-curvature in terms of the spin and string bordism groups, and by means of $α$-invariant and Wit…
We give a homotopy invariant construction of the Reidemeister trace for the coincidence of two maps between closed manifolds of not necessarily the same dimensions. It is realized as a homology class of the homotopy equalizer, which coincides with the Hurewicz image of Koschorke's stabilized bordism invariant. To define it, we use a kind of shriek maps appearing string topology. As an application, we compute the coincidence Reidemeister trace for the self-coincidence of the projections of $S^1$-bundles on $\mathbb{C}P^n$. We also mention how to relate our construction to the string topology op…
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.
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…
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…
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…
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…
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.
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…
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…
The aim of this paper is to define a chain level refinement of the Batalin-Vilkovisky (BV) algebra structure on the homology of the free loop space of a closed, oriented $C^\infty$-manifold. For this purpose, we define a (nonsymmetric) cyclic dg operad which consists of "de Rham chains" of free loops with marked points. A notion of de Rham chains, which is a certain hybrid of the notions of singular chains and differential forms, is a key ingredient in our construction. Combined with a generalization of cyclic Deligne's conjecture, this dg operad produces a chain model of the free loop space w…
In this paper, we investigate the behaviour of the Serre spectral sequence with respect to the algebraic structures of string topology in generalized homology theories, specificially with the Chas-Sullivan product and the corresponding coproduct and module structures. We prove compatibility for two kinds of fibre bundles: the fibre bundle $Ω^n M \to L^n M \to M$ for an $h_*$-oriented manifold $M$ and the looped fibre bundle $L^n F \to L^n E \to L^n B$ of a fibre bunde $F \to E \to B$ of $h_*$-oriented manifolds. Our method lies in the construction of Gysin morphisms of spectral sequences. We a…
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 direct description of this Batalin-Vilkovisky algebra in the case that the manifold is a compact Lie group G. Our answer is phrased in terms of the homology of G, the homology of the space of based loops on G, and the homology suspension. The result is applied to compute the Batalin-Vilkovisky algebra associated to the special orthogonal groups SO(n) with coefficients in the rational numbers and in the integers modulo tw…
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.
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 …
For a manifold N embedded inside euclidean space R^{n+1}, we produce a coloured operad that acts on the space of maps from N to M, where M is a compact, oriented, smooth manifold. For N the unit sphere, we indicate how this gives homological actions, generalizing the action of the Cacti operad and retrieving the Chas-Sullivan product by taking N to be the unit-circle in R^2. We go on to show that for S^n the unit sphere in R^{n+1}, the operad constructed is a coloured E_{n+1}-operad. This E_{n+1}-structure is finally twisted by SO(n+1) to homologically give actions of the framed little (n+1)-d…
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.
Let $M$ be a compact oriented $d$-dimensional smooth manifold. Chas and Sullivan have defined a structure of Batalin-Vilkovisky algebra on $\mathbb{H}_*(LM)$. Extending work of Cohen, Jones and Yan, we compute this Batalin-Vilkovisky algebra structure when $M$ is a sphere $S^d$, $d\geq 1$. In particular, we show that $\mathbb{H}_*(LS^2;\mathbb{F}_2)$ and the Hochschild cohomology $HH^{*}(H^*(S^2);H^*(S^2))$ are surprisingly not isomorphic as Batalin-Vilkovisky algebras, although we prove that, as expected, the underlying Gerstenhaber algebras are isomorphic. The proof requires the knowledge of…
We show that the Chas-Sullivan loop product, a combination of the Pontrjagin product on the fiber and intersection product on the base, makes sense on the total space homology of any fiberwise monoid E over a closed oriented manifold M. More generally the Thom spectrum E^{-TM} is a ring spectrum. Similarly a fiberwise module over E defines a module over E^{-TM}. Fiberwise monoids include adjoint bundles of principal bundles, and the construction is natural with respect to maps of principal bundles. This naturality implies homotopy invariance of the algebra structure on H_*(LM) arising from the…
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 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…
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.
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…
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…
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.
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.
For almost any compact connected Lie group $G$ and any field $\mathbb{F}\_p$, we compute the Batalin-Vilkoviskyalgebra $H^{*+\text{dim }G}(LBG;\mathbb{F}\_p)$ on the loop cohomology of the classifying space introduced byChataur and the second author.In particular, if $p$ is odd or $p=0$, this Batalin-Vilkovisky algebra is isomorphicto the Hochschild cohomology $HH^*(H\_*(G),H\_*(G))$. Over $\mathbb{F}\_2$, such isomorphism of Batalin-Vilkovisky algebrasdoes not hold when $G=SO(3)$ or $G=G\_2$.
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.
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.
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.
Let G be a compact Lie group. By work of Chataur and Menichi, the homology of the space of free loops in the classifying space of G is known to be the value on the circle in a homological conformal field theory. This means in particular that it admits operations parameterized by homology classes of classifying spaces of diffeomorphism groups of surfaces. Here we present a radical extension of this result, giving a new construction in which diffeomorphisms are replaced with homotopy equivalences, and surfaces with boundary are replaced with arbitrary spaces homotopy equivalent to finite graphs.…
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.
7-dimensional closed and simply-connected manifolds have been attractive as central and explicit objects in algebraic topology and differential topology of higher dimensional closed and simply-connected manifolds, which were studied actively especially in the 1950s--60s. Attractive studies of the class of these $7$-dimensional manifolds were started by the discovery of so-called exotic spheres by Milnor. It has influenced on the understanding of higher dimensional closed and simply-connected manifolds via algebraic and abstract objects. Recently this class is studied via more concrete notions …
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 …
