Exercises:

1. Suppose that $G$ is a finite group acting freely on a Hausdorff space then the action is properly discontinuous and hence deduce that the group action in the example of the generalized Lens space is properly discontinuous.
2. Suppose that $p: {\tilde X}\longrightarrow X$ is a covering projection and ${\tilde X}$ is locally path connected and simply connected. Show that if $U$ is an evenly covered open set in $X$ and ${\tilde U}$ is a sheet lying above it then $\phi({\tilde U})\cap {\tilde U} = \emptyset$ for every $\phi \in$   Deck$({\tilde X}, X)$ and $\phi \neq$ id$_{\tilde X}$. Deduce that the group of deck transformations acts properly discontinuously on ${\tilde X}$. How does this relate to theorem 17.2?
3. Does the fundamental group of Klein's bottle have elements of finite order? Identify this group with a familiar group that we have already encountered in lecture 7. What is its abelianization?
4. Show that the torus is obtained as the orbit space of a group of homeomorphisms acting properly discontinuously on $\mathbb{R}^2$. Write out these homeomorphisms explicitly.
5. Show that the torus is a double cover of the Klein's bottle. Hence the fundamental group of the Klein's bottle must contain a subgroup of index two. Determine this subgroup.
6. Show that the cylinder is a two-sheeted cover of the Möbius band.
7. Suppose that $G$ is a topological group, $H$ is a discrete subgroup of $G$. Show that there exists a neighborhood $U$ of the identity such that $U = U^{-1}$, $U\cap H = \{1\}$ and that $\{hU/h\in H\}$ is a family of disjoint open sets. Deduce that the quotient map $\eta:G\longrightarrow G/H$ is a covering projection. Also show that $G/h$ is Hausdorff.

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Lecture XXI - Test - III

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1. Show that a homeomorphism of $E^2$ onto itself must preserve the boundary. That is it must map a boundary point to a boundary point.
2. Is it true that $\mathbb{R}P^3$ minus a point deformation retracts to a space homeomorphic to $\mathbb{R}P^2$?
3. Let $G$ be the infinite grid
   $$G = \{(x, y)\in \mathbb{R}^2\;/\; x \in \mathbb{Z}$$    or $$y \in \mathbb{Z}\}.$$
   Consider the covering map from $G$ onto the figure eight loop $(S^1\times\{1\})\cup (\{1\}\times S^1)$ given by
   $$p(x, y) = (\exp(2\pi i x),\; \exp (2\pi iy)).$$
   Determine the deck transformations of this covering. Is this a regular covering?
4. Given topological spaces $X$ and $Y$, a map $p:X\longrightarrow Y$ is said to be a local homeomorphism if each $x_0 \in X$ has a neighborhood $N_{x_0}$ such that the restriction map
   $$f\Big\vert _{N_{x_0}}:N_{x_0}\longrightarrow f(N_{x_0})$$
   is a homeomorphism. Show that a local homeomorphism which is a proper map is a covering projection.
5. Show that the map $f:\mathbb{C} - \{0, 1, -1\} \longrightarrow \mathbb{C} - \{\pm 2\}$ is a local homeomorphism. Is this map a covering projection? If so what is the group of deck transformations?

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Lecture XXII - Fundamental group of $SO(3, \mathbb R)$ and $SO(4, \mathbb R)$

For many applications, it is important to know The fundamental groups of the classical groups. We shall discuss in detail the orthogonal groups $SO(3, \mathbb{R})$ and $SO(4, \mathbb{R})$ since their underlying topological spaces are easily described. Indeed $SO(3, \mathbb{R})$ is the three dimensional real projective space and $SO(4, \mathbb{R})$, as a topological space, is the product of the three dimensional real projective space and the three dimensional sphere $S^3$. To unravel the structure of these spaces it is convenient to use quaternions. We shall assume some basic familiarity with quaternions (see [1]). We shall also use some basic facts from multi-variable calculus. The student who is unfamiliar with these parts of multi-variable calculus may omit these parts of the proof.