Let $ M$ be a manifold, $ P$ a closed subset and sub-manifold of $ M$ , and $ f : (M,\emptyset) \to (M,P)$ a smooth map (I note $ f : (M,P) \to (N,Q)$ a smooth map $ M \to N$ such that $ f(P) \subset Q$ ). Its pullback $ f^* : \Omega_{dR,c}(M,P)Read more