By K. H. Kamps

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Exercise 9. 5: (1) Using, for example, Exercise 2, show that the set of Morse functions on the compact manifold V is open for the C2 topology, and therefore also for the C∞ topology. (2) Let a ∈ V and let U be the open set of a chart centered at a. We let F denote the space of diﬀerentials of the functions deﬁned on U (or of exact 1-forms), seen as a subset of C∞ (U ; (Rn ) ) (for every x ∈ U , (df )x is a linear functional on Rn ). Note that F is not an open subset. Let α be a plateau function with support in U and value 1 on a compact neighborhood K of a.

Hence, if a and b are two (distinct) critical points and if the gradient that is used satisﬁes the Smale condition, then for M(a, b) or L(a, b) to be nonempty, we must have Ind(a) > Ind(b). In other words, the index decreases along the gradient lines. We will come back to these spaces at length in the next chapter. 4. All examples presented above satisfy the Smale condition,2 except for that of the height function on the torus. 10. Moreover, we have trajectories connecting two critical points of index 1, which, as we just saw, is forbidden.

We have two possibilities: • If this intersection contains two points, then the two are maxima, A1 ∪ Ai is diﬀeomorphic to S 1 and we are done. • If, on the contrary, it contains only one point, then A1 ∪Ai is diﬀeomorphic to [0, 1]. If A1 ∪ Ai = V , then we are done. And if this is not the case, then we continue adding Ai ’s until they run out. There exist other ways to prove this theorem, which may be simpler (see [55]). 1. c An Application, the Brouwer Fixed Point Theorem We will now use Sard’s theorem and our knowledge of manifolds of dimension 1 to prove Brouwer’s famous theorem (this proof comes from Milnor’s book [55]).