Exercises:

1. Formulate and prove the Borsuk Ulam theorem for continuous maps from $S^1$ to the real line.
2. Use the Borsuk Ulam theorem to prove that a pair of homogeneous polynomials of odd degree in three real variables have a common non-trivial zero.
3. For the following three maps $f : S^1 \longrightarrow S^1$ compute the induced map $f_*:\pi_1(S^1, 1)\longrightarrow \pi_1(S^1, 1)$. All three maps preserve the base point $1$.

   (i)

   $f(z)=z^n$

   (ii)

   $f(z)=\bar{z}.$

   (iii)

   $f(z)= \frac{z^2-z+\frac{3}{2}}{\vert z^2-z+\frac{3}{2}\vert}.$ Hint: Is $(z^2-z)t + 3/2 = 0$ for any $z \in S^1$ and $0 \leq t \leq 1$?

4. Let $X$ be the union of the sphere $S^2$ and one of its diameters. Use exercise 1 of lecture 8 to determine a generator for $\pi_1(X, x_0)$, where $x_0$ is a point on the sphere.
5. Determine the generators of the group $\pi_1(S^1\times S^1, (1, 1))$. Determine the generators for the fundamental group of the space $X$ of example 11.3.
6. Compute $f_*:\pi_1(\mathbb{C} - \{0\}, 1)\longrightarrow \pi_1(\mathbb{C} - \{0\}, 1)$ for the function $f(z) = z^k$.

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Lecture XIV (Test - II)

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1. Suppose $X$ is a metric space and $A$ is a retract of $X$. Show that $A$ is closed in $X$. Is the space homeomorphic to the letter $Y$ a deformation retract of a space homeomorphic to $E^2$?
2. Show that if $X$ has the fixed point property and $A$ is a retract of $X$ then $A$ also has the fixed point property.
3. Find the degree of the following maps $f : S^1 \longrightarrow S^1$ given by:

   (i)

   $f(z) = \exp(z -$   Re$\;z)$. (ii) $f(z) = \overline{z}^2z^3$.

4. Show that $S^1$ is not homeomorphic to any subset of $\mathbb{R}$. Can $S^2$ be homeomorphic to a subset of $\mathbb{R}^2$?
5. Determine $\pi_1(\mathbb{R}P^2 - \{p \})$ where $p$ is any point of $\mathbb{R}P^2$.
6. For the map $f:S^1\longrightarrow S^1\times S^1$ given by $f(z) = (z^p, z^q)$, where $p$ and $q$ are positive integers, find the induced group homomorphism $f_*:\mathbb{Z} \longrightarrow \mathbb{Z}\times \mathbb{Z}$.

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Lecture XV - Covering Projections

in The theory of covering projections sets a common stage for the development of diverse branches of mathematics. In this course we develop the theory of covering projections only to the extent that is relevant for the computation of the fundamental group. It may be useful for the student to review the proof that $\pi_1(S^1) = \mathbb{Z}$. In fact one of the paradigms for a covering projection is the map

$$t \mapsto \exp(2\pi it)$$

wrapping the real line onto the circle.