Please only solve Exercise 6.

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!

Special Case. Any smooth mapf: 3' - .5" having degree zero is homotopic to a constant map. 4. Check that the special case implies the following corollary. Corollary. Any smooth map f:S' . R3\" [0} having winding number zero with respect to the origin is homotopic to a constant. Prove the special case inductively. For I = I, you have already established Hopf's theorem, as Exercise 9, Section 3. So assume that the special case is true f0r I = k l, and extend it to 1' = k. The following exercise, which is the heart of the inductive argument, uses the corollary to pull maps away from the origin. 5- Letf: R\" v- R" be a smooth map with 0 as a regular value. Suppose that f"(0} is nite and that the number of preimage pointsinf"(0) is zero when counted with the usual orientation convention. Assuming the special case in dimension k l, prove that there exists a mapping g : R\" : R." {0] such that g =foutside a compact set. An obvious point worth mentioning before the next step is that becausef and g are equal outside a compact set, the homotomt ff+ {l Hg is con- stant outside a compact set. This fact noted, you need only devise a suitable method for reducing S\" to R" in order to complete the special mm. 6. Establish the special case in dimension k. 6. Pick distinct regular values a and b for f. Apply the Isotopy Lemma to S* - f-1(b) to find an open neighborhood U off-'(a) that is diffeo- morphic to R* and that satisfies the requirement b e f(U). Let a : R* U be a diffeomorphism, and choose another diffeomorphism p: S* - {b] - R* that maps a to 0. Now apply Exercise 5 to Bo fo a to find a map g: S* - S* - {b} that is homotopic to f. But since S* - {b] is diffeomorphic to R*, and thus contractible, g is homotopic to a constant