Please solve Exercise 8.

Please only use definitions, propositions, theorems given in the book ''Differential Topology'' by Guillemin and Pollack !

And if you refer to a proposition, theorem or exercise in the book please refer to them by chapter and section!

Also please explain each step, even if it seems trivial to you!

Please don't answer this question by using theory that is NOT covered in the book by Guillemin and Pollack!!!! The Propositions and Theorems in the book of Guillemin and Pollack are NOT numbered, so if you end up referring to Propositions by numbers, you are using the WRONG book!

Extension Theorem. Let It be a compact, connected, oriented k + 1 di- mensional manifold with boundary, and let f: dw - S* be a smooth map. Prove that f extends to a globally defined map F: W - S*, with OF = f, if and only if the degree of f is zero. In turn, the Extension Theorem follows from the special case, but to prove it you must first step aside and establish a lemma concerning the topo- logical triviality of Euclidean space. 146 CHAPTER 3 ORIENTED INTERSECTION THEORY 7. Let I be any compact manifold with boundary, and let f: dw - R*+1 be any smooth map whatsoever. Prove that f may be extended to all of W. 8. Prove the Extension Theorem.8. First use Exercise 7 to extend fto a map F: WV - R*+1. By the Trans- versality Extension Theorem, we may assume 0 to be a regular value of F. Use the Isotopy Lemma to place the finite point set F-1(0) in- side a subset U of Int (W) that is diffeomorphic to R*+1. Let B be a ball in U containing F-1(0), and show that OF: dB - R*+1 - (0) has winding number 0 with respect to the origin. (For this, note that F/| F| extends to the manifold W. = W - Int (B). But we already know that its degree on dil is zero. Now apply the corollary of the special case.)