Exercises

1. Show that the sphere $S^2$ retracts onto one of its longitudes. If $X$ is the space obtained from $S^2$ by taking its union with a diameter, there is a surjective group homomorphism $\pi_1(X)\longrightarrow \mathbb{Z}$.
2. Prove that $A$ is a retract of $X$ if and only if every space $Y$, every continuous map $f:A\longrightarrow Y$ has a continuous extension ${\tilde f}:X\longrightarrow Y$.
3. Show that the fundamental group respects arbitrary products.
4. Construct a retraction from $\{(x, y) : x$    or $y$    is an integer$\}$ onto the boundary of $I^2$.
5. Show that every homeomorphism of $E^2$ onto itself must map the boundary to the boundary.
6. Given that there exists a functor $T$ from the category ${\bf Top}$ to the category ${\bf AbGr}$ such that $T(X)$ is the trivial group for every convex subset $X$ of a Euclidean space and $T(S^{n})$ is a non-trivial group, prove that $S^n$ is not a retract of the closed unit ball in $\mathbb{R}^{n+1}$.

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Lecture X - Brouwer's Theorem and its Applications.

in In this lecture we shall prove the Brouwer's fixed point theorem and deduce some of its consequences such as the Perron-Frobenius' theorem. The one dimensional Brouwer's theorem follows from the intermediate value property as is indicated in the exercises of lecture 3. We also include a proof of the fact that the spheres $S^n$ have trivial fundamental group when $n\geq 2$. This result has been included here to demonstrate why the fundamental group is insufficient to prove the Brouwer's fixed point theorem in dimension three or higher.

We begin by defining the fixed point property for a space. Here we require the fixed point property to hold for all continuous functions of the space into itself. Note that in analysis the spaces considered are somewhat special and so are the maps whose fixed point property are sought. A classic example of such a restricted fixed point theorem is the Banach's fixed point theorem.