Ralph L. Cohen

Floer's infinite dimensional Morse theory and homotopy theory

This paper is a progress report on our efforts to understand the homotopy theory underlying Floer homology. Its objectives are as follows: (A) to describe some of our ideas concerning what exactly the Floer homology groups compute; (B) to explain what kind of an object we think the «Floer homotopy type» of an infinite dimensional manifold should be; (C) to work out, in detail, the Floer homotopy type in some examples.

The topology of rational functions and divisors of surfaces

Segal studied the homotopy types of the spaces of holomorphic functions of the 2-sphere S 2 , of closed surfaces of higher genus, and of the spaces of divisors of these surfaces. We continue Segals program by describing the entire stable homotopy types of these spaces in terms of the homotopy types of more familiar spaces

The Loop Homology Algebra of Spheres and Projective Spaces

In [3] Chas and Sullivan 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 in [2] to construct a second quadrant spectral sequence of algebras converging to the loop homology multiplicatively, when M is simply connected. The E2 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 \( {H_*}\left( {L\mathbb{C}{\mathbb{P}^n}} \right). \))

Topological Hochschild homology of Thom spectra and the free loop space

We describe the topological Hochschild homology of ring spectra that arise as Thom spectra for loop maps f : X! BF, where BF denotes the classifying space for stable spherical fibrations. To do this, we consider sym- metric monoidal models of the category of spaces over BF and corresponding strong symmetric monoidal Thom spectrum functors. Our main result identi- fies the topological Hochschild homology as the Thom spectrum of a certain stable bundle over the free loop space L(BX). This leads to explicit calcula- tions of the topological Hochschild homology for a large class of ring spectra, including all of the classical cobordism spectra MO, MSO, MU, etc., and the Eilenberg-Mac Lane spectra HZ/p and HZ.

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

Let M n be a closed, oriented, n-manifold, and LM its free loop space. In [4] a commutative algebra structure in homology, H�(LM), and a Lie algebra structure in equivariant homology H S 1 � (LM), were defined. In this paper we prove that these structures are homotopy invariants in the following sense. Let f : M1 ! M2 be a homotopy equivalence of closed, oriented n-manifolds. Then the induced equivalence, Lf : LM1 ! LM2 induces a ring isomorphism in homology, and an isomorphism of Lie algebras in equivariant homology. The analogous statement also holds

ALGEBRAIC K-THEORY OF SPACES AND THE NOVIKOV CONJECTURE

LET x be a discrete group with the property that its classifying space Bn has the homotopy type of a finite CW-complex. The algebraic K-theory version of Novikov’s conjecture is the assertion that the rational algebraic K-groups K,(Z[n]) @I Q of the integral group ring Z [A] contain a direct summand isomorphic to H, (8~; Q) 6 K,(Z). In this article we will show how to reduce this conjecture to a single homotopy theoretic problem which is completely independent of the group n. We will construct a map, which is most naturally thought of as a character map or a trace map, ch: A(pt) + Map(CP”, QS’).

Algebraic Topology and Its Applications

A series of articles that describe recent advances in the research of algebraic topology and outline directions for future development. The main areas of concentration are homotopy theory, K-theory, and applications to geometric topology, gauge theory and moduli spaces.

Monopoles, braid groups, and the dirac operator

Using the relation between the space of rational functions on ℂ, the space ofSU(2)-monopoles on ℝ3, and the classifying space of the braid group, see [10], we show how the index bundle of the family of real Dirac operators coupled toSU(2)-monopoles can be described using permutation representations of Artins braid groups. We also show how this implies the existence of a pair consisting of a gauge fieldA and a Higgs field Φ on ℝ3 whose corresponding Dirac equation has an arbitrarily large dimensional space of solutions.

String topology of Poincaré duality groups

where the sum is taken over conjugacy classes of elements in G . In this paper we construct a multiplication on LG directly in terms of intersection products on the centralizers. This multiplication makes LG a graded, associative, commutative algebra. When G is the fundamental group of an aspherical, closed oriented n‐ manifold M , then .LG/ D HC n.LM/, where LM is the free loop space of M . We show that the product on LG corresponds to the string topology loop product on H .LM/ defined by Chas and Sullivan. 55P35; 20J06 Dedicated to Fred Cohen on the occasion of his 60th birthday

Holomorphic K-Theory, Algebraic Co-cycles, and Loop Groups

In this paper we study the “holomorphic K -theory” of a projective variety. This K theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory has been introduced in various places such as [12], [9], and a related theory was considered in [11]. This theory is built out of studying algebraic bundles over a variety up to “algebraic equivalence”. In this paper we will give calculations of this theory for “flag like varieties” which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors in [6], we will show that there is a rational isomorphism of graded rings between holomorphic K theory and the appropriate “morphic cohomology” groups, defined in [7] in terms of algebraic co-cycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the “symmetrized loop group” ΩU(n)/Σn where the symmetric group Σn 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 use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K theory by inverting the Bott class, then rationally this is isomorphic to topological K theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K theory, and show that these obstructions vanish for generalized flag manifolds.