The Longitudinal Index Theorem for Foliations

In this paper , we use the bivariant K theory of Kasparov ([19]) as a basic tool to prove the K-theoretical version of the index theorem for longitudinal elliptic differential operators for foliations which is stated as a problem in [10], Section 10. When the foliation is by the fibers of a fibration, this theorem reduces to the Atiyah-Singer index theorem for families ([2], Theorem 3.1). It implies the index theorem for measured foliations ([9], Theorem, p. 136) and unlike the latter makes sense for arbitrary foliations, not necessarily gifted with a holonomy invariant transverse measure. The index in the Atiyah Singer theorem for families ([2]) is an element of the K-theory K(B) of the base space of the fibration. In the case of foliations the base B is the space of leaves of the foliation (V, F ). This space of leaves, as a topological space, is often degenerate (if the foliation is minimal there are no nontrivial open sets in V/F ). The algebra C(B) of continuous functions on B is replaced by a canonically defined C-algebra: C(V, F ), cf. [9], [10]. The K-theory K0(C (V, F )) of this C-algebra plays the role of K(B). In the case of a fibration C(V, F ) is (Morita) equivalent to C(B) so that K0(C (V, F )) = K(B). Let D be an elliptic differential operator along the leaves of the foliation (V, F ). Since D is elliptic it has an inverse modulo C(V, F ) hence it gives an element Inda(D) of K0(C (V, F )). Let us now describe the topological index. Let i be an auxiliary imbedding of the manifold V in R. Let N be the total space of the normal bundle to the leaves: Nx = (i∗(Fx)) ⊥ ⊂ R. Let us foliate Ṽ = V ×R by F , F(x,t) = Fx × {0}, so that the leaves of (Ṽ , F ) are just L = L × {t}, where L is a leaf of (V, F ) and t ∈ R. The map (x, ξ) → (x, i(x) + ξ) turns an open neighborhood of the 0-section in N into an open transversal T of the foliation (Ṽ , F ). For a suitable open neighborhood Ω of T in Ṽ , the C-algebra C(Ω, F ) of the restriction of F to Ω is (Morita) equivalent to C0(T ), hence the inclusion C (Ω, F ) ⊂ C(Ṽ , F ) yields a K-theory map: K(N) → K0(C (Ṽ , F )). Since C(Ṽ , F ) = C(V, F )⊗ C0(R ), one has, by Bott periodicity, the equality K0(C (Ṽ , F )) = K0(C (V, F )). Using the Thom isomorphism K(F ) is identified with K(N) so that one gets by the above construction, the topological index: Indt : K (F ) → K0(C (V, F )) . Our main result is the equality: Inda(D) = Indt([σD]) where σD is the longitudinal symbol of D and [σD] is its class in K(F ). In the first section, we formalize the elliptic pseudo-differential calculus for families of operators onX indexed by Y , in terms of the bivariant Kasparov theory. This gives a map of the K-theory with compact support K(T X × Y ) to the bivariant group KK(X,Y ). We then compute directly the Kasparov product of two such elements. In the second section we first recall the definition of [10] of the analytical element f ! ∈ KK(X,Y ) corresponding to a K-oriented map f from X to Y . We then prove that (idX)! is the unit of the ring KK(X,X). Using the computation of Section 1, we then prove the equality: (f ◦ g)! = g!⊗ f !. Computing f ! in the case