Topology of the space of conormal distributions

Abstract

AbstractGiven 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 $$I'=I'(M,L)=I(M,L;\Omega )'$$ 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 $$K\subset I$$ 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 $$I\rightarrow J$$ I → J such that the sequence $$0\rightarrow K\rightarrow I\rightarrow J\rightarrow 0$$ 0 → K → I → J → 0 as well as the transpose sequence $$0\rightarrow J'\rightarrow I'\rightarrow K'\rightarrow 0$$ 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 $$I\cap I'=C^\infty (M)$$ 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 $$\phi =\{\phi ^t\}$$ ϕ = { ϕ t } on a compact foliated manifold $$(M,\mathcal {F})$$ ( M , F ) . It describes a Lefschetz distribution $$L_{\textrm{dis}}(\phi )$$ L dis ( ϕ ) defined by the induced action $$\phi ^*=\{\phi ^{t\,*}\}$$ ϕ ∗ = { ϕ t ∗ } on the reduced cohomologies $${\bar{H}}^\bullet I(\mathcal {F})$$ H ¯ ∙ I ( F ) and $${\bar{H}}^\bullet I'(\mathcal {F})$$ H ¯ ∙ I ′ ( F ) of the complexes of leafwise currents that are conormal and dual-conormal at the leaves preserved by $$\phi $$ ϕ .