HW3 MA5311. Due February 15

Exercise. Let X \subset \mathbf{R}^k be a subset homeomorphic to the closed ball B_n \subset \mathbf{R}^n.  Show that if f: X \to X is continuous, then there exists x \in X such that f(x)=x.

Exercise. Let X be a one-dimensional compact manifold with boundary. Show that X is diffeomorphic to a finite union of segments and circles (You may use the appendix in Milnor’s lecture notes).

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