Topology of the space of conormal distributions

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;Ω )' 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 φ =\φ ᵗ\ ϕ = { ϕ t } on a compact foliated manifold (M,F) ( M , F ) . It describes a Lefschetz distribution Ldis(φ ) L dis ( ϕ ) defined by the induced action φ ^*=\φ t\,*\ ϕ ∗ = { ϕ t ∗ } on the reduced cohomologies H̄^• I(F) H ¯ ∙ I ( F ) and H̄^• I'(F) H ¯ ∙ I ′ ( F ) of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by φ ϕ .