Exercises

1. Show that in a topological group, the connected component of the identity is a normal subgroup.
2. Show that the action of the group $SO(n, \mathbb{R})$ on the sphere $S^{n-1}$ given by matrix multiplication is transitive. You need to employ the Gram-Schmidt theorem to complete a given unit vector to an orthonormal basis.
3. Suppose a group $G$ acts transitively on a set $S$ and $x, y$ are a pair of points in $S$ and $y = gx$. Then the subgroups stab$\;x$ and stab$\;y$ are conjugates and $g^{-1}$(stab$\;y)g =$ stab$\;x$.

   (i)

   Show that the map $\;\overline{\phi}:G/$stab$\;x\longrightarrow S$ given by $\overline{\phi}(\overline{g}) = gx$ is well-defined, bijective and $\overline{\phi}\circ \eta = \phi$.

   (ii)

   Suppose that $S$ is a topological space, $G$ is a topological group and the action $G\times S\longrightarrow S$ is continuous. Show that the map $\overline{\phi}$ is continuous.

   (iii)

   Deduce that if $G$ is compact and $S$ is Hausdorff then $G/$stab$\;x$ and $S$ are homeomorphic.

4. Examine whether the map $\phi:SU(n)\times S^1\longrightarrow U(n)$ given by $\phi(A, z) = zA$ is a homeomorphism.
5. Show that the group of all unitary matrices $U(n)$ is compact and connected. Regarding $U(n-1)$ as a subgroup of $U(n)$ in a natural way, recognize the quotient space as a familiar space.
6. Show that the subgroups $SU(n)$ consisting of matrices in $U(n)$ with determinant one are connected for every $n$.
7. Suppose $G$ is a topological group and $H$ is a normal subgroup, prove that $G/H$ is Hausdorff if and only if $H$ is closed.

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Lecture VI (Test - I)

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1. Prove that the intervals $(a, b)$ and $[a, b)$ are non-homeomorphic subsets of $\mathbb{R}$. Prove that if $A$ and $B$ are homeomorphic subsets of $\mathbb{R}$, then $A$ is open in $\mathbb{R}$ if and only if $B$ is open in $\mathbb{R}$. Is an injective continuous map $f:\mathbb{R}\longrightarrow \mathbb{R}$ a homeomorphism onto its image?
2. Using Tietze's extension theorem or otherwise construct a continuous map from $\mathbb{R}$ into $\mathbb{R}$ such that the image of $\mathbb{Z}$ is not closed in $\mathbb{R}$.
3. If $K$ is a compact subset of a topological group $G$ and $C$ is a closed subset of $G$, is it true that $KC$ is closed in $G$? What if $K$ and $C$ are merely closed subsets of $G$?
4. Removing three points from $\mathbb{R}P^2$ we get an open set $G$ and a continuous map $f:G \longrightarrow \mathbb{R}P^2$ given by $f([x_1, x_2, x_3]) = [x_2x_3, x_3x_1, x_1x_2]$. Which three points need to be removed? Prove the continuity of $f$.
5. Let $C = \{({\bf v}_1, {\bf v}_2) \in S^2\times S^2\;/\; \langle {\bf v}_1, {\bf v}_2\rangle = 0\}$. Is $C$ connected? Is $C$ homeomorphic to $SO(3, \mathbb{R})$?
6. Prove that $\mathbb{R}P^1$ is homeomorphic to $S^1$.

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Lecture VII - Paths, homotopies and the fundamental group

in In this lecture we shall introduce the most basic object in algebraic topology, the fundamental group. For this purpose we shall define the notion of homotopy of paths in a topological space $X$ and show that this is an equivalence relation. We then fix a point $x_0 \in X$ in the topological space and look at the set of all equivalence classes of loops starting and ending at $x_0$. This set is then endowed with a binary operation that turns it into a group known as the fundamental group $\pi_1(X, x_0)$. Besides being the most basic object in algebraic topology, it is of paramount importance in low dimensional topology. A detailed study of this group will occupy the rest of part I of this course. However in this lecture we shall focus only on the most elementary results.

All spaces considered here are path connected Hausdorff spaces.