550. Math Homework 8. Due 11/10/03
Problem 1. Let U, V be open subsets of R2 . Prove that the coboundary map : H 0 (U V ) H 1 (U V ) is a homomorphism of vector spaces. Problem 2. Let F : U V be a smooth map from the open set U R2 into the open set V R2 . In a previous homework set we showed that F induces a pull-back operator F that sends n-forms on V into n-forms on U. Prove that F induces a linear map of vector spaces F : H nV H nU, for n = 0, 1. Moreover, prove if that F is a diffeomorphism, then F : H nV H nU is an isomorphism of vectors spaces. Problem 3. Prove that if U and V are connected, and H 1 (U V ) = 0, then U V is connected. Problem 4. Prove that if the open set U R2 can be written as an union U = U1 Un , where each U j is a c...