Ralph Cohen | Stanford University (original) (raw)

Papers by Ralph Cohen

Inventiones Mathematicae, Feb 1, 1985

Homology, Homotopy and Applications, 2009

arXiv (Cornell University), Apr 2, 2013

Given a principal bundle over a closed manifold, G → P → M , let P Ad → M be the associated adjoi... more Given a principal bundle over a closed manifold, G → P → M , let P Ad → M be the associated adjoint bundle. In Gruher and Salvatore showed that the Thom spectrum (P Ad ) -T M 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 ) ≃ LM -T M where LM is the free loop space of the manifold. This ring spectrum was introduced by the authors in [8] 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 ) → GL1(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 ). Contents 1 Parameterized spectra, Poincaré duality, and the loop space 7 2 Gauge groups and the units of string topology 11 * The first author was partially supported by a grant from the NSF. 1 3 Connections on an E-line bundle 15 4 Loop groups and string topology 17 5 String topology as the linearization of the gauge group 19

Communications in Mathematical Physics, Nov 1, 1993

Contemporary mathematics, 2000

arXiv (Cornell University), Oct 17, 2013

Journal of Topology, Apr 1, 2008

Asian Journal of Mathematics, 1999

arXiv (Cornell University), Jul 25, 2001

Mathematische Annalen, Dec 1, 2002

Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In... more Let M be a closed, oriented manifold of dimension d. Let LM be the space of smooth loops in M. In [2] Chas and Sullivan defined a product on the homology H * (LM) of degree −d. They then investigated other structure that this product induces, including a Batalin-Vilkovisky structure, and a Lie algebra structure on the S 1 equivariant homology H S 1 * (LM). These algebraic structures, as well as others, came under the general heading of the "string topology" of M. In this paper we will describe a realization of the Chas-Sullivan loop product in terms of a ring spectrum structure on the Thom spectrum of a certain virtual bundle over the loop space. We also show that an operad action on the homology of the loop space discovered by Voronov has a homotopy theoretic realization on the level of Thom spectra. This is the "cactus operad" defined in [6] which is equivalent to operad of framed disks in R 2. This operad action realizes the Chas-Sullivan BV structure on H * (LM). We then describe a cosimplicial model of this ring spectrum, and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the Hochschild cohomology, HH * (C * (M); C * (M)).

Proceedings of symposia in pure mathematics, 2009

Proceedings of the London Mathematical Society, 2006

In this paper, I give a generalized analogue of the string topology results of Chas and Sullivan,... more In this paper, I give a generalized analogue of the string topology results of Chas and Sullivan, and of Cohen and Jones. For a finite simplicial complex XXX and kgeq1k \geq 1kgeq1, I construct a spectrum Maps(Sk,X)S(X)Maps(S^k, X)^{S(X)}Maps(Sk,X)S(X), which is obtained by taking a generalization of the Spivak bundle on XXX (which however is not a stable sphere bundle unless XXX is a Poincaré space), pulling back to Maps(Sk,X)Maps(S^k, X)Maps(Sk,X) and quotienting out the section at infinity. I show that the corresponding chain complex is naturally homotopy equivalent to an algebra over the (k+1)(k + 1)(k+1)-dimensional unframed little disk operad mathcalCk+1\mathcal{C}_{k + 1}mathcalCk+1. I also prove a conjecture of Kontsevich, which states that the Quillen cohomology of a based mathcalCk\mathcal{C}_kmathcalCk-algebra (in the category of chain complexes) is equivalent to a shift of its Hochschild cohomology, as well as prove that the operad CastmathcalCkC_{\ast}\mathcal{C}_kCastmathcalCk is Koszul-dual to itself up to a shift in the derived category. This gives one a natural notion of (derived) Ko...

K-Theory, 1987

Let A(X) be the space defined by Waldhausen whose homotopy groups define the algebraic K-groups o... more Let A(X) be the space defined by Waldhausen whose homotopy groups define the algebraic K-groups of the space X and let B(X) = QZ(SE 1 x sIA(X)). Here A(X) denotes the free loop space of X and Q denotes the functor f~o~% For X = ZY, the suspension of a connected space Y, we shall prove that the homotopy fibers ,4(X),/](X) of the maps A(X) ~ A (point), B(X) ~ B (point) are equivalent as infinite loop spaces.

Advances in Mathematics, Jul 1, 2009

For a compact, connected Lie group G, we study the moduli of pairs (Σ, E), where Σ is a genus g R... more For a compact, connected Lie group G, we study the moduli of pairs (Σ, E), where Σ is a genus g Riemann surface and E → Σ is a flat G-bundle. Varying both the Riemann surface Σ and the flat bundle leads to a moduli space M G g , parametrizing families Riemann surfaces with flat G-bundles. We show that there is a stable range in which the homology of M G g is independent of g. 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, the stable cohomology of this moduli space is generated by the Mumford-Morita-Miller κ-classes, and the ring of characteristic classes of principal G-bundles, H * (BG). Equivalently, our theorem calculates the homology of the moduli space of semi-stable holomorphic bundles on Riemann surfaces. We then identify the homotopy type of the category of one-manifolds and surface cobordisms, each equipped with a flat G-bundle. 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.

arXiv (Cornell University), Jan 22, 2008

Given a semisimple, compact, connected Lie group G with complexification G c , we show there is a... more 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 κ-classes, and the ring of characteristic classes of principal G-bundles, H * (BG). We then identify the homotopy type of the category of onemanifolds 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.

Homology, Homotopy and Applications, 2011

In this paper we present a new proof of the homological stability of the moduli space of closed s... more 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 Sg(K). The homology stability of surfaces in K with an arbitrary number of boundary components, Sg,n(K) was studied by the authors in [4]. The study there relied on stability results for the homology of mapping class groups, Γg,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 Sg(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.

Notices of the American Mathematical Society, 2021

Tunisian Journal of Mathematics

In this paper we import the theory of "Calabi-Yau" algebras and categories from symplectic topolo... more 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 and Sullivan (1999) and the Lie group string topology of Chataur and Menichi (2012). This will extend and generalize work of Gruher (2007). Then, generalizing work of Cohen and Jones (2017), we show how the gauge group of the principal bundle acts on this compact Calabi-Yau structure, and we 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, M, 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 and Kragh (2016) 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.