HOMEWORK 8 – 2019 Exercise 1. Let f : M → N be a map between two oriented compact manifolds of dimension n with fundamental classes [M] and [N], respectively. We say that f has degree d if f∗([M]) = d[N]. Prove that for every oriented compact manifold M of dimension n there is a map f : M → Sn of degree 1. (Hint: Find a geometric prescription and use the definition of the fundamental class.) Exercise 2. Use cup product and Z2 coefficients to show that RP3 is not homotopy equivalent to RP2 ∨ S3 . 1