  In \cite{baker-ozel}, 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 \cite{cenap-isr}, by using Quinn's Transversality Theorem \cite{Quinn}, 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 \cite{chas}, Chas and
Sullivan described an intersection product on the homology of loop space $LM$.
In \cite{cohen}, 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.
