  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 this work, by using Quinn's Transversality Theorem \cite{Quinn}, it will be
shown that this cobordism theory has a graded ring structure under transversal
intersection operation and has pull-back maps for smooth maps. It will be shown
that the Thom isomorphism in this theory will be satisfied for finite
dimensional vector bundles over separable Hilbert manifolds and the projection
formula for Gysin maps will be proved. After we discuss the relation between
this theory and classical cobordism, we describe some applications to the
complex cobordism of flag varieties of loop groups and we do some calculations.
