arXiv subjects

Ralph L. Cohen

Publications and source records attributed to Ralph L. Cohen.

26 records · Page 2Linked to original sources

arXiv2004

Multiplicative properties of Atiyah duality

Let $M^n$ be a closed, connected $n$-manifold. Let $\mtm$ denote the Thom spectrum of its stable normal bundle. A well known theorem of Atiyah states that $\mtm$ is homotopy equivalent to the Spanier-Whitehead dual of $M$ with a disjoint basepoint, $M_+$. This dual can be viewed as the function spectrum, $F(M, S)$, where $S$ is the sphere spectrum. $F(M, S)$ has the structure of a commutative, symmetric ring spectrum in the sense of \cite{hss}, \cite{ship}. In this paper we prove that $\mtm$ also has a natural, geometrically defined, structure of a commutative, symmetric ring spectrum, in such a way that the classical duality maps of Alexander, Spanier-Whitehead, and Atiyah define an equivalence of symmetric ring spectra, $α: \mtm \to F(M, S)$. We discuss applications of this to Hochshield cohomology representations of the Chas-Sullivan loop product in the homology of the free loop space of $M$.

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arXiv2004

The topology of the category of open and closed strings

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 group to identify the homotopy type of the geometric realization of the category of open and closed strings as an infinite loop space.

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arXiv2004

A polarized view of string topology

Let M be a closed, connected manifold, and LM its loop space. In this paper we describe closed string topology operations in h_*(LM), where h_* is a generalized homology theory that supports an orientation of M. We will show that these operations give h_*(LM) the structure of a unital, commutative Frobenius algebra without a counit. Equivalently they describe a positive boundary, two dimensional topological quantum field theory associated to h_*(LM). This implies that there are operations corresponding to any surface with p incoming and q outgoing boundary components, so long as q >0. The absence of a counit follows from the nonexistence of an operation associated to the disk, D^2, viewed as a cobordism from the circle to the empty set. We will study homological obstructions to constructing such an operation, and show that in order for such an operation to exist, one must take h_*(LM) to be an appropriate homological pro-object associated to the loop space. Motivated by this, we introduce a prospectrum associated to LM when M has an almost complex structure. Given such a manifold its loop space has a canonical polarization of its tangent bundle, which is the fundamental feature needed to define this prospectrum. We refer to this as the "polarized Atiyah - dual" of LM . An appropriate homology theory applied to this prospectrum would be a candidate for a theory that supports string topology operations associated to any surface, including closed surfaces.

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arXiv2002

A homotopy theoretic realization of string topology

Let $M$ be a closed, oriented manifold of dimension $d$. Let $LM$ be the space of smooth loops in $M$. Chas and Sullivan recently 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 Voronov's "cactus operad", which is equivalent to the 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.

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arXiv2002

Fourier Decompositions of Loop Bundles

In this paper we investigate bundles whose structure group is the loop group LU(n). Our main result is to give a necessary and sufficient criterion for there to exist a Fourier type decomposition of such a bundle $ξ$. This is essentially a decomposition of $ξ$ as $ζ\otimes L\mathbb C$, where $ζ$ is a finite dimensional subbundle of $ξ$ and $L\mathbb C$ is the loop space of the complex numbers. The criterion is a reduction of the structure group to the finite rank unitary group U(n) viewed as the subgroup of LU(n) consisting of constant loops. Next we study the case where $ξ$ is the loop space of an $n$ dimensional bundle $ζ\to M$. The tangent bundle of $LM$ is such a bundle. We then show how to twist such a bundle by elements of the automorphism group of the pull back of $ζ$ over $LM$ via the map $LM \to M$ that evaluates a loop at a basepoint. Given a connection on $ζ$, we view the associated parallel transport operator as an element of this gauge group and show that twisting the loop bundle by such an operator satisfies the criterion and admits a Fourier decomposition.

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arXiv2002

The loop homology algebra of spheres and projective spaces

Chas and Sullivan recently defined an intersection product on the homology $H_*(LM)$ of the space of smooth loops in a closed, oriented manifold $M$. In this paper we will use the homotopy theoretic realization of this product described by the first two authors to construct a second quadrant spectral sequence of algebras converging to the loop homology multiplicatively, when $M$ is simply connected. The $E_2$ term of this spectral sequence is $H^*(M;H_*(ΩM))$ where the product is given by the cup product on the cohomology of the manifold $H^* (M)$ with coefficients in the Pontryagin ring structure on the homology of its based loop space $H_*(ΩM)$. We then use this spectral sequence to compute the ring structures of $H_* (LS^n)$ and $H_* (L\bcp^n)$.

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arXiv1999

Holomorphic K-theory, algebraic co-cycles, and loop groups

In this paper we study the "holomorphic K-theory" of a projective variety, which is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory was introduced by Lawson, Lima-Filho and Michelsohn, and also by Friedlander and Walker, and a related theory was considered by Karoubi. Using the Chern character studied by the authors in a companion paper, we give a rational isomorphism between holomorphic $K$-theory and the "morphic cohomology" defined by Lawson and Friedlander. In doing so, we describe a geometric model for rational morphic cohomology groups in terms of algebraic maps from the variety to the "symmetrized loop group" Loops(U(n)/S_n) where the symmetric group S_n acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K-theory by inverting the Bott class, then it is rationally isomorphic to topological K-theory. Finally we produce explicit obstructions to periodicity in holomorphic K-theory, and show that these obstructions vanish for generalized flag manifolds.

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arXiv1999

Stability for holomorphic spheres and Morse theory

In this paper we study the question of when does a closed, simply connected, integral symplectic manifold (W,omega) have the stability property for its spaces of based holomorphic spheres? This property states that in a stable limit under certain gluing operators, the space of based holomorphic maps from a sphere to X, becomes homotopy equivalent to the space of all continuous maps, lim_{->} Hol_{x_0}(P^1,X) = Omega^2 X. This limit will be viewed as a kind of stabilization of Hol_{x_0}(P^1,X). We conjecture that this stability holds if and only if an evaluation map E: lim_{->} Hol_{x_0}(P^1,X) -> X is a quasifibration. In this paper we will prove that in the presence of this quasifibration condition, then the stability property holds if and only if the Morse theoretic flow category (defined in [4]) of the symplectic action functional on the Z-cover of the loop space, L~X, defined by the symplectic form, has a classifying space that realizes the homotopy type of L~X. We conjecture that in the presence of this quasifibration condition, this Morse theoretic condition always holds. We will prove this in the case of X a homogeneous space, thereby giving an alternate proof of the stability theorem for holomorphic spheres for a projective homogeneous variety originally due to Gravesen [7].

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