arXiv subjects

Ralph L. Cohen

Publications and source records attributed to Ralph L. Cohen.

At least 19 recordsLinked to original sources

arXiv2019

An algebraic geometric realization of the Chern character

Using symmetrized Grassmannians we give an algebraic geometric presentation, in the level of classifying spaces, of the Chern character and its relation to Chern classes. This allows one to define, for any projective variety $X$, a Chern character map from the "holomorphic K-theory of X to its morphic cohomology (introduced by Friedlander and Lawson). The holomorphic K-theory of $X$, introduced by Lawson, Lima-Filho and Michelsohn and also by Friedlander and Walker, is defined in terms a group-completion of the space of algebraic morphisms from X into BU. It has been further studied by the authors in a companion paper. Holomorphic K-theory sits between algebraic K-theory and topological K-theory in the same way that morphic cohomology sits between motivic cohomology and ordinary cohomology. Our constructions provide a bridge between these two worlds. We also realize Chern classes in the case where X is smooth, and establish a universal relation between the Chern character and the Chern classes. We use classical constructions with algebraic cycles and infinite symmetric products of projective spaces. The latter can be seen as the classifying space for motivic cohomology, and under this perspective our constructions are essentially motivic.

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arXiv2019

Floer homotopy theory, revisited

In 1995 the author, Jones, and Segal introduced the notion of "Floer homotopy theory". The proposal was to attach a (stable) homotopy type to the geometric data given in a version of Floer homology. More to the point, the question was asked, "When is the Floer homology isomorphic to the (singular) homology of a naturally occuring (pro)spectrum defined from the properties of the moduli spaces inherent in the Floer theory?". A proposal for how to construct such a spectrum was given in terms of a "framed flow category", and some rather simple examples were described. Years passed before this notion found some genuine applications to symplectic geometry and low dimensional topology. However in recent years several striking applications have been found, and the theory has been developed on a much deeper level. Here we summarize some of these exciting developments, and describe some of the new techniques that were introduced. Throughout we try to point out that this area is a very fertile ground at the interface of homotopy theory, symplectic geometry, and low dimensional topology.

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arXiv2018

Twisted Calabi-Yau ring spectra, string topology, and gauge symmetry

In this paper, we import the theory of "Calabi-Yau" algebras and categories from symplectic topology and topological field theories to the setting of spectra in stable homotopy theory. Twistings in this theory will be particularly important. There will be two types of Calabi-Yau structures in the setting of ring spectra: one that applies to compact algebras and one that applies to smooth algebras. The main application of twisted compact Calabi-Yau ring spectra that we will study is to describe, prove, and explain a certain duality phenomenon in string topology. This is a duality between the manifold string topology of Chas-Sullivan and the Lie group string topology of Chataur-Menichi. This will extend and generalize work of Gruher. Then, generalizing work of the first author and Jones, we show how the gauge group of the principal bundle acts on this compact Calabi-Yau structure, and compute some explicit examples. We then extend the notion of the Calabi-Yau structure to smooth ring spectra, and prove that Thom ring spectra of (virtual) bundles over the loop space, $ΩM$, have this structure. In the case when $M$ is a sphere we will use these twisted smooth Calabi-Yau ring spectra to study Lagrangian immersions of the sphere into its cotangent bundle. We recast the work of Abouzaid-Kragh to show that the topological Hochschild homology of the Thom ring spectrum induced by the $h$-principle classifying map of the Lagrangian immersion, detects whether that immersion can be Lagrangian isotopic to an embedding. We then compute some examples. Finally, we interpret these Calabi-Yau structures directly in terms of topological Hochschild homology and cohomology.

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arXiv2013

Homotopy automorphisms of R-module bundles, and the K-theory of string topology

Let $R$ be a ring spectrum and $ E\to X$ an $R$-module bundle of rank $n$. Our main result is to identify the homotopy type of the group-like monoid of homotopy automorphisms of this bundle, $hAut^R(E)$. This will generalize the result regarding $R$-line bundles previously proven by the authors. The main application is the calculation of the homotopy type of $BGL_n(End ((L))$ where $L \to X$ is any $R$-line bundle, and $End (L)$ is the ring spectrum of endomorphisms. In the case when such a bundle is the fiberwise suspension spectrum of a principal bundle over a manifold, $G \to P \to M$, this leads to a description of the $K$-theory of the string topology spectrum in terms of the mapping space from $M$ to $BGL (Σ^\infty (G_+))$.

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arXiv2013

Gauge theory and string topology

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 topology of M. The main purpose of this paper is to introduce an action of the gauge group of the principal bundle, G (P) on the string topology spectrum S(P), and to study this action in detail. Indeed we study the entire group of units and the induced representation G(P) --> GL_1(S (P)). We show that this group of units is the group of homotopy automorphisms of the fiberwise suspension spectrum of P. More generally we describe the homotopy type of the group of homotopy automorphisms of any E-line bundle for any ring spectrum E. We import some of the basic ideas of gauge theory, such as the action of the gauge group on the space of connections to the setting of E-line bundles over a manifold, and do explicit calculations. We end by discussing a functorial perspective, which describes a sense in which the string topology spectrum S(P) of a principal bundle is the "linearization" of the gauge group G(P).

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arXiv2010

Stability for closed surfaces in a background space

In this paper we present a new proof of the homological stability of the moduli space of closed surfaces in a simply connected background space $K$, which we denote by $S_g (K)$. The homology stability of surfaces in $K$ with an arbitrary number of boundary components, $S_{g,n} (K)$ was studied by the authors in \cite{cohenmadsen}. The study there relied on stability results for the homology of mapping class groups, $Γ_{g,n}$ with certain families of twisted coefficients. It turns out that these mapping class groups only have homological stability when $n$, the number of boundary components, is positive, or in the closed case when the coefficient modules are trivial. Because of this we present a new proof of the rational homological stability for $S_g(K)$, that is homotopy theoretic in nature. We also take the opportunity to prove a new stability theorem for closed surfaces in $K$ that have marked points.

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arXiv2009

Stability phenomena in the topology of moduli spaces

The recent proof by Madsen and Weiss of Mumford's conjecture on the stable cohomology of moduli spaces of Riemann surfaces, was a dramatic example of an important stability theorem about the topology of moduli spaces. In this article we give a survey of families of classifying spaces and moduli spaces where "stability phenomena" occur in their topologies. Such stability theorems have been proved in many situations in the history of topology and geometry, and the payoff has often been quite remarkable. In this paper we discuss classical stability theorems such as the Freudenthal suspension theorem, Bott periodicity, and Whitney's embedding theorems. We then discuss more modern examples such as those involving configuration spaces of points in manifolds, holomorphic curves in complex manifolds, gauge theoretic moduli spaces, the stable topology of general linear groups, and pseudoisotopies of manifolds. We then discuss the stability theorems regarding the moduli spaces of Riemann surfaces: Harer's stability theorem on the cohomology of moduli space, and the Madsen-Weiss theorem, which proves a generalization of Mumford's conjecture. We also describe Galatius's recent theorem on the stable cohomology of automorphisms of free groups. We end by speculating on the existence of general conditions in which one might expect these stability phenomena to occur.

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arXiv2009

Open-closed field theories, string topology, and Hochschild homology

In this expository paper we discuss a project regarding the string topology of a manifold, that was inspired by recent work of Moore-Segal, Costello, and Hopkins and Lurie, on "open-closed topological conformal field theories". Given a closed, oriented manifold M, we describe the "string topology category" S_M, which is enriched over chain complexes over a fixed field k. The objects of S_M are connected, closed, oriented submanifolds N of M, and the complex of morphisms between N_1 and N_2 is a chain complex homotopy equivalent to the singular chains C_*(P_{N_1, N_2}), where C_*(P_{N_1, N_2}) is the space of paths in M that start in N_1 and end in N_2. The composition pairing in this category is a chain model for the open string topology operations of Sullivan and expanded upon by Harrelson, and Ramirez. We will describe a calculation yielding that the Hochschild homology of the category S_M is the homology of the free loop space, LM. Another part of the project is to calculate the Hochschild cohomology of the open string topology chain algebras C_*(P_{N,N}) when M is simply connected, and relate the resulting calculation to H_*(LM). We also discuss a spectrum level analogue of the above results and calculations, as well as their relations to various Fukaya categories of the cotangent bundle T^*M.

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arXiv2008

A Morse theoretic description of string topology

Let M be a closed, oriented, n-dimensional manifold. In this paper we give a Morse theoretic description of the string topology operations introduced by Chas and Sullivan, and extended by the first author, Jones, Godin, and others. We do this by studying maps from surfaces with cylindrical ends to M, such that on the cylinders, they satisfy the gradient flow equation of a Morse function on the loop space, LM. We then give Morse theoretic descriptions of related constructions, such as the Thom and Euler classes of a vector bundle, as well as the shriek, or unkehr homomorphism.

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arXiv2008

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

Let M be a closed, oriented, n -manifold, and LM its free loop space. Chas and Sullivan defined a commutative algebra structure in the homology of LM, and a Lie algebra structure in its equivariant homology. These structures are known as the string topology loop product and string bracket, respectively. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M_1 \to M_2 be a homotopy equivalence of closed, oriented n -manifolds. Then the induced equivalence, Lf : LM_1 \to LM_2 induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds true for any generalized homology theory h_* that supports an orientation of the M_i 's.

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arXiv2008

Surfaces in a background space and the homology of mapping class groups

In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, S_{g,n} (X, γ). This is the space consisting of triples, (F_{g,n}, ϕ, f), where F_{g,n} is a Riemann surface of genus g and n-boundary components, ϕis a parameterization of the boundary, and f : F_{g,n} \to X is a continuous map that satisfies a boundary condition γ. We prove three theorems about these spaces. Our main theorem is the identification of the stable homology type of the space S_{\infty, n}(X; γ), defined to be the limit as the genus g gets large, of the spaces S_{g,n} (X; γ). Our result about this stable topology is a parameterized version of the theorem of Madsen and Weiss proving a generalization of the Mumford conjecture on the stable cohomology of mapping class groups. Our second result describes a stable range in which the homology of S_{g,n} (X; γ) is isomorphic to the stable homology. Finally we prove a stability theorem about the homology of mapping class groups with certain families of twisted coefficients. The second and third theorems are generalizations of stability theorems of Harer and Ivanov.

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arXiv2008

Floer homotopy theory, realizing chain complexes by module spectra, and manifolds with corners

In this paper we describe and continue the study begun by the author, Jones, and Segal, of the homotopy theory that underlies Floer theory. In that paper the authors addressed the question of realizing a Floer complex as the celluar chain complex of a CW -spectrum or pro-spectrum, where the attaching maps are determined by the compactified moduli spaces of connecting orbits. The basic obstructions to the existence of this realization are the smoothness of these moduli spaces, and the existence of compatible collections of framings of their stable tangent bundles. In this note we describe a generalization of this, to show that when these moduli spaces are smooth, and are oriented with respect to a generalized cohomology theory E^*, then a Floer E_* -homology theory can be defined. In doing this we describe a functorial viewpoint on how chain complexes can be realized by E -module spectra, generalizing the stable homotopy realization criteria given earlier by the author, Jones, and Segal. Since these moduli spaces, if smooth, will be manifolds with corners, we give a discussion about the appropriate notion of orientations of manifolds with corners.

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arXiv2008

Universal moduli spaces of surfaces with flat connections and cobordism theory

Given a semisimple, compact, connected Lie group G with complexification G^c, we show there is a stable range in the homotopy type of the universal moduli space of flat connections on a principal G-bundle on a closed Riemann surface, and equivalently, the universal moduli space of semistable holomorphic G^c-bundles. The stable range depends on the genus of the surface. We then identify the homology of this moduli space in the stable range in terms of the homology of an explicit infinite loop space. Rationally this says that the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller kappa-classes, and the ring of characteristic classes of principal G-bundles, H^*(BG). We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. We also explain how these results may be generalized to arbitrary compact connected Lie groups. Our methods combine the classical techniques of Atiyah and Bott, with the new techniques coming out of Madsen and Weiss's proof of Mumford's conjecture on the stable cohomology of the moduli space of Riemann surfaces.

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arXiv2007

Umkehr Maps

In this note we study umkehr maps in generalized (co)homology theories arising from the Pontrjagin-Thom construction, from integrating along fibers, pushforward homomorphisms, and other similar constructions. We consider the basic properties of these constructions and develop axioms which any umkehr homomorphism must satisfy. We use a version of Brown representability to show that these axioms completely characterize these homomorphisms, and a resulting uniqueness theorem follows. Finally, motivated by constructions in string topology, we extend this axiomatic treatment of umkehr homomorphisms to a fiberwise setting.

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arXiv2007

The Floer homotopy type of the cotangent bundle

Let M be a closed, oriented, n-dimensional manifold. In this paper we describe a spectrum in the sense of homotopy theory, Z(T^*M), whose homology is naturally isomorphic to the Floer homology of the cotangent bundle, T^*M. This Floer homology is taken with respect to a Hamiltonian H: S^1 x T^*M --> R, which is quadratic near infinity. Z(T^*M) is constructed assuming a basic smooth gluing result of J-holomorphic cylinders. This spectrum will have a C.W decomposition with one cell for every periodic solution of the equation defined by the Hamiltonian vector field X_H. Its induced cellular chain complex is exactly the Floer complex. The attaching maps in the C.W structure of Z(T^*M) are described in terms of the framed cobordism types of the moduli spaces of J -holomorphic cylinders in T^*M with given boundary conditions. This is done via a Pontrjagin-Thom construction, and an important ingredient in this is proving, modulo this gluing result, that these moduli spaces are compact, smooth, framed manifolds with corners. We then prove that Z(T^*M), which we refer to as the "Floer homotopy type" of T^*M, has the same homotopy type as the suspension spectrum of the free loop space, LM. This generalizes the theorem first proved by C. Viterbo that the Floer homology of T^*M is isomorphic to H_*(LM).

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arXiv2005

Morse field theory

In this paper we define and study the moduli space of metric-graph-flows in a manifold M. This is a space of smooth maps from a finite graph to M, which, when restricted to each edge, is a gradient flow line of a smooth (and generically Morse) function on M. Using the model of Gromov-Witten theory, with this moduli space replacing the space of stable holomorphic curves in a symplectic manifold, we obtain invariants, which are (co)homology operations in M. The invariants obtained in this setting are classical cohomology operations such as cup product, Steenrod squares, and Stiefel-Whitney classes. We show that these operations satisfy invariance and gluing properties that fit together to give the structure of a topological quantum field theory. By considering equivariant operations with respect to the action of the automorphism group of the graph, the field theory has more structure. It is analogous to a homological conformal field theory. In particular we show that classical relations such as the Adem relations and Cartan formulae are consequences of these field theoretic properties. These operations are defined and studied using two different methods. First, we use algebraic topological techniques to define appropriate virtual fundamental classes of these moduli spaces. This allows us to define the operations via the corresponding intersection numbers of the moduli space. Secondly, we use geometric and analytic techniques to study the smoothness and compactness properties of these moduli spaces. This will allow us to define these operations on the level of Morse-Smale chain complexes, by appropriately counting metric-graph-flows with particular boundary conditions.

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arXiv2005

Notes on 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 string topology fits into the general picture. Other topics include an open-closed version of string topology, a Morse theoretic interpretation, relation to Gromov-Witten invariants, and "brane'' topology, which deals with sphere spaces. The paper is a joint account of the lecture series given by each of us at the 2003 Summer School on String Topology and Hochschild Homology in Almeria, Spain.

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arXiv2004

Morse theory, graphs, and string topology

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 these string topology operations to the counting of J - holomorphic curves in the cotangent bundle. We end with speculations about the relationship between the absolute and relative Gromov-Witten theory of the cotangent bundle, and the open-closed string topology of the underlying manifold.

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