Topology of the space of conormal distributions

Abstract Given a closed manifold M and a closed regular submanifold L, consider the corresponding locally convex space I=I (M, L) I = I (M, L) of conormal distributions, with its natural topology, and the strong dual I'=I' (M, L) =I (M, L;) ' I ′ = I ′ (M, L) = I (M, L ; Ω) ′ of the space of conormal densities. It is shown that I is a barreled, ultrabornological, webbed, Montel, acyclic LF-space, and I' 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 K ⊂ I of conormal distributions supported in L and for its strong dual K' K ′. We construct a locally convex Hausdoff space J and a continuous linear map I J I → J such that the sequence 0 K I J 0 0 → K → I → J → 0 as well as the transpose sequence 0 J' I' K' 0 0 → J ′ → I ′ → K ′ → 0 are short exact sequences in the category of continuous linear maps between locally convex spaces. Finally, it is shown that I I'=C^ (M) I ∩ I ′ = C ∞ (M) in the space of distributions. In another publication, these results are applied to prove a Lefschetz trace formula for a simple foliated flow =\ ᵗ\ ϕ = ϕ t on a compact foliated manifold (M, F) (M, F). It describes a Lefschetz distribution L₃₈ₒ () L dis (ϕ) defined by the induced action ^*=\ ^{t\, *\} ϕ ∗ = ϕ t ∗ on the reduced cohomologies H^ I (F) H ¯ ∙ I (F) and H^ I' (F) H ¯ ∙ I ′ (F) of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by ϕ.