Please solve Exercise 9.

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 W 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. 9. Conclude: The Hopf Degree Theorem. Two maps of a compact, connected, oriented k-manifold X into S* are homotopic if and only if they have the same degree