John D. S. Jones

Differential forms on loop spaces and the cyclic bar complex

In this article, we present a model for the differential graded algebra (dga) of differential forms on the free loop space LX of a smooth manifold X and show how to construct certain important differential forms in terms of this model. Our motivation is an observation of Witten (described by Atiyah in [2]) that the index theorem for the Dirac operator can be thought of as an application of the localization (or fixed point) theorem in T-equivariant homology, suitably generalised to the infinite dimensional case of the free loop space; here, T is the circle group. We can summarise our main results as follows. We are really concerned with equivariant differential forms and equivariant currents and we show how to reformulate these geometric objects as cyclic chains and cochains over the differential graded algebra Ω(X) of differential forms on X. In fact, the cyclic chain complex of Ω(X), if it is normalized correctly, is a sub-complex of the complex of equivariant differential forms on LX, and is a good enough approximation that it allows us to compute the ordinary and equivariant cohomology of LX, and to write down explicitly certain important differential forms and currents. We begin by explaining the motivation more carefully. Let S be a Clifford module on X with Dirac operator D. Witten observed that it should be possible to associate to D an equivariantly closed (inhomogeneous) current μD on the free loop space of X. The basic property of this current is that the index of D is given by pairing μD with the differential form 1 ∈ Ω(LX):

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 cyclic homology of crossed product algebras

In this article, we give a new derivation of this spectral sequence, and generalize it to negative and periodic cyclic homology HC. (yl) and HP· (A). The method of proof is itself of interest, since it involves a natural generalization of the notion of a cyclic module, in which the condition that the morphism τ 6 Λ (n, n) is cyclic of order n + l is relaxed to the condition that it be invertible. We call this category the paracyclic category.

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 \( {H_*}\left( {L\mathbb{C}{\mathbb{P}^n}} \right). \))

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’).

Some generalized brown-gitler spectra

Brown-Gitler spectra for the homology theories associated with the spectra KZp, to, and bu are constructed. Complexes adapted to the new Brown- Gitler spectra are produced and a spectral sequence converging to stable maps into these spectra is constructed and examined.

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.

The cyclic homology of crossed product algebras. II. Topological algebras

This article is a sequel to [6], in which we constructed a spectral sequence for the cyclic homology of a crossed product algebra A * G, where A is an algebra and G is a discrete group acting on A. In this article, we will show how similar results hold when A is a topological algebra, and G is a Lie group acting differentiably on A. We assume the notation and results of [6], which we will refer to äs Part I.

String homology, and closed geodesics on manifolds which are elliptic spaces

Let