On Smooth Chas-sullivan Loop Product in Quillen’s Geometric Complex Cobordism of Hilbert Manifolds

In [1], by using Fredholm index we developed a version of Quillen’s geometric cobordism theory for infinite dimensional Hilbert manifolds. This cobordism theory has a graded group structure under topological union operation and has push-forward maps for complex orientable Fredholm maps. In [19], by using Quinn’s Transversality Theorem [23], it has been shown that this cobordism theory has a graded ring structure under transversal intersection operation and has pull-back maps for smooth maps. It has been shown that the Thom isomorphism in this theory was satisfied for finite dimensional vector bundles over separable Hilbert manifolds and the projection formula for Gysin maps has been proved. In [3], Chas and Sullivan described an intersection product on the homology of loop space LM . In [4], R. Cohen and J. Jones described 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. In this paper, we will extend this product on cobordism and bordism theories. 1. The Fredholm Index and Complex Cobordism of Hilbert Manifolds. In [22], Quillen gave a geometric interpretation of cobordism groups which suggests a way of defining the cobordism of separable Hilbert manifolds equipped with suitable structure. In order that such a definition be sensible, it ought to reduce to his for finite dimensional manifolds and smooth maps of manifolds and be capable of supporting reasonable calculations for important types of infinite dimensional manifolds such as homogeneous spaces of free loop groups of finite dimensional Lie groups. 1.1. Cobordism of separable Hilbert manifolds. By a manifold, we mean a smooth manifold modelled on a separable Hilbert space; see Lang [13] for details on infinite dimensional manifolds. The facts about Fredholm map can be found in [5]. Definition 1.1. Suppose that f : X → Y is a proper Fredholm map with even index at each point. Then f is an admissible complex orientable map if there is a smooth factorization f : X f̃ → ξ q → Y, where q : ξ → Y is a finite dimensional smooth complex vector bundle and f̃ is a smooth embedding endowed with a complex structure on its normal bundle ν(f̃). A complex orientation for a Fredholm map f of odd index is defined to be one for the map (f, ε) : X → Y × R given by (f, ε)(x) = (f(x), 0) for every x ∈ X. At x ∈ X, index(f, ε)x = (index fx) − 1. Also the finite dimensional complex vector bundle ξ in the smooth factorization will be replaced by ξ×R. Suppose that f is an admissible complex orientable map. Then since the map f is the Fredholm and ξ is a finite dimensional vector bundle, we see f̃ is also a Fredholm map. By the surjectivity of q, index f̃ = index f − dim ξ. Before we give a notion of equivalence of such factorizations f̃ of f , we want to give some definitions. Definition 1.2. Let X, Y be the smooth separable Hilbert manifolds and F : X×R → Y a smooth map. Then we will say that F is an isotopy if it satisfies the following conditions. (1) For every t ∈ R, the map Ft given by Ft(x) = F (x, t) is an embedding. (2) There exist numbers t0 < t1 such that Ft = Ft0 for all t 6 t0 and Ft = Ft1 for all t > t1. The closed interval [t0, t1] is called a proper domain for the isotopy. We say that two embeddings f : X → Y and g : X → Y are isotopic if there exists an isotopy Ft : X × R → Y with proper domain [t0, t1] such that f = Ft0 and g = Ft1 . Date: September 02,2003. 1991 Mathematics Subject Classification. Algebraic topology,Global Analysis.