Exercises:

1. Check that the map $\phi$ constructed in the proof of theorem 11.3 is continuous and is indeed a homotopy. Work out the proof of theorem 11.5.
2. Show that the boundary $\partial M$ of the Möbius band $M$ is not a deformation retract of $M$ by taking a base point $x_0$ on the boundary and computing explicitly the group homomorphism
   $$i_*:\pi_1(\partial M, x_0)\longrightarrow \pi_1(M, x_0).$$

3. Show that the boundary of the Möbius band is not even a retract of the Möbius band.
4. Fill in the details on the continuity of the map $G$ in the example preceding corollary 11.9.
5. Show that the space $\mathbb{R}^3 - \{(x, y, z)/ x^2 + y^2 = 1,\; z = 0\}$ deformation retracts to a sphere with a diameter attached to it.
6. Let $X$ be the union of $S^2$ and one of its diameters $D$, $Y = S^2\vee S^1$ and $Z$ be the union of $S^2$ with a punctured half disc contained in a half with edge along $D$. Show that $X$ and $Y$ are both deformation retracts of $Z$ and so they have the same homotopy type.

in

Lectures XII - XIII The fundamental group of the circle.

in We have already stated the fact that the fundamental group of the circle is the group of integers and derived some important consequences form it. The importance of this result is attested by the fact that the Brouwer's fixed point theorem for a disc follows immediately from it. In this lecture will provide a detailed proof that $\pi_1(S^1, 1)=\mathbb{Z}$. Some of the ideas of the proof would appear again later in a general context of covering spaces. Though this result is a special one from the theory of covering spaces it is worthwhile looking at this important special case without reference to the general theory but rather as a preview to it. This topic will be covered in two lectures but the numbering will be as that of lecture 12. We begin with an algebraic lemma [14] (p. [//]).