Consider the irreducible complex polynomial p E C[r, y] given by p(x, y) = y + 1. Let := p=1(0) C C2 and let > be a compactification of E, as in Lecture 21. 3.1. Prove that is a smooth 2-manifold. It suffices to prove that 0 is a regular value of p. 3.2. For the map : EC, (x,y) x, determine the branch points ; of and the indicies np(x). Justify your answers. 3.3. Using the Riemann-Hurwitz Formula, determine x(E), the Euler characteristic of . 3.4. Determinen, (oo), the index at infinity, and so determine x() and hence the topological type of E. Justify your answers.