Dual homotopy invariants of G-foliations

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THIS PAPER develops a theory of dual homotopy invariants for G-foliations using the theory of minimal models. As an application of the theory which is constructed, we are able to extend the results of Heitsch on the independent variation of the secondary classes of foliations. A foliation with a non-zero rigid class is also shown to exist, based on an example of Schweitzer and Whitman. For Riemannian foliations, all of the indecomposable secondary classes are shown to be linearly independent in H*(FRI”). The work of Lazarov and Pasternack is used to show that all of the possibly variable indecomposable secondary classes are independently variable in H*(FR14). The third type of G-foliations considered are those with an integrable complex structure on their normal bundles. The results of Baum and Bott are used to establish that many of the secondary classes for these foliations are independently variable. For each of the three types of G-foliations considered, namely real, Riemannian and complex, it is shown that the homotopy groups of the corresponding classifying space BIG4 admit epimorphisms 7r~(BIo4)+RUn, where {u,} is a sequence depending on 4 and G, but which in general has a subsequence tending to infinity. If a manifold M is simply connected, then the invariants of a G-foliation 9 on M which we produce are functions on the homotopy groups of M. They can be viewed as generalizations of several other constructions of foliation invariants in the literature: There is a natural relation with the Chern-Simons invariants [9]. A means for producing such invariants was introduced by Haefliger in [18]. The various residue theorems for a G-foliation with singularities at a discrete set of points [2, 32 and 381 are special cases of this theory, where the residue is obtained by evaluating a dual homotopy class on the boundary of a disc about a singular point. For a Riemannian or complex foliation which is defined by a submersion[25], the secondary classes of the foliation are exactly cohomological representations of some of the dual homotopy invariants. The general theory of the invariants is developed in 02. We begin by showing that the algebra homotopy class of the truncated Chern-Weil homomorphism h(w) is a G-foliation invariant (Theorem 2.11); in fact, it is a universal invariant from which many other invariants of the foliation can be derived [23]. Applying the dual homotopy functor to h(o) yields a characteristic map h#: r*(I(G),)*r*(M) from the infinitedimensional vector space r*(l(G),) to the (pseudo) dual homotopy of the manifold (Theorem 2.12).