Topology of the space of conormal distributions

Abstract

Given a closed manifold M and a closed regular submanifold L, consider the correspondinglocally convexspace I = I(M, L)ofconormaldistributions,withitsnatural topology, and the strong dual I = I (M, L) = I(M,L; ) ofthespaceofconormal densities. It is shown that I is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and I is a complete Montel space, which is a projective limit of bornological barreled spaces. In the case of codimension one, similar properties and additional descriptions are proved for the subspace K ⊂ I of conormal distributions supported in L and for its strong dual K. We construct a locally convex Hausdoff space J and a continuous linear map I → J such that the sequence 0 → K → I → J → 0aswell as the transpose sequence 0 → J→ I→ K→ 0areshortexact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that I ∩I = C∞(M)inthespaceofdistributions.Inanotherpublication,theseresults are applied to prove a Lefschetz trace formula for a simple foliated flow φ ={φt} on a compact foliated manifold (M,F). It describes a Lefschetz distribution Ldis(φ) defined by the induced action φ∗ ={φt∗} on the reduced cohomologies ¯ H•I(F) and ¯ H•I (F) of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by φ.