Section 1.1

As a general convention, $t\in I$ will be the path variable and $s\in I$ will be the homotopy variable.

1.1.1 wrote:

Show that the composition of paths satisfies the following cancellation property: If $f_0\cdot g_0 \simeq f_1\cdot g_1$ and $g_0\simeq g_1$ , then $f_0\simeq f_1$ .

Solution

1.1.2 wrote:

Show that the change-of-basepoint homomorphism $\beta_h$ depends only on the homotopy class of $h$ .

Solution

1.1.3 wrote:

For a path connected space $X$ , show that $\pi_1(X)$ is abelian iff all basepoint-change homomorphisms depend only on the endpoints of the path $h$ .

Solution

1.1.4 wrote:

A subspace $X\subseteq \mathbb{R}^n$ is said to be star-shaped if there is a point $x_0\in X$ such that, for each $x\in X$ , the line segment from $x_0$ to $x$ lies in $X$ . Show that if a subspace $X\subseteq\mathbb{R}^n$ is locally star-shaped, in the sense that every point of $X$ has a star-shaped neighborhood in $X$ , then every path in $X$ is homotopic to a piecewise linear path, that is, a path consisting of a finite number of straight line segments traversed at constant speed. Show this applies in particular when $X$ is open or when $X$ is a union of finitely many closed convex sets.

Solution

1.1.5 wrote:

Show that for a space $X$ , the following three conditions are equivalent:

Every map $S^1\to X$ is homotopic to a constant map, with image a point.
Every map $S^1\to X$ extends to a map $D^2\to X$ .
$\pi_1(X,x_0)=0$ for all $x_0\in X$ .

Deduce that a space $X$ is simply connected iff all maps $S^1\to X$ are homotopic.

Solution

1.1.6 wrote:

We can regard $\pi_1(X,x_0)$ as the set of basepoint-preserving homotopy classes of maps $(S^1,s_0)\to (X,x_0)$ . Let $[S^1,X]$ be the set of homotopy classes of maps $S^1\to X$ with no conditions on basepoints. Thus there is a natural map $\Phi:\pi_1(X,x_0)\to [S^1,X]$ obtained by ignoring basepoints. Show that $\Phi$ is onto if $X$ is path connected, and that $\Phi([f])=\Phi([g])$ iff $[f]$ and $[g]$ are conjugate in $\pi_1(X,x_0)$ . Hence $\Phi$ induces a one-to-one correspondence between $[S^1,X]$ and the set of conjugacy classes in $\pi_1(X)$ , when $X$ is path connected.

Solution

1.1.7 wrote:

Define $f:S^1\times I\to S^1\times I$ by $f(\theta,s)=(\theta+2\pi s,s)$ , so $f$ restricts to the identity on the two boundary circles of $S^1\times I$ . Show that $f$ is homotopic to the identity by a homotopy that is stationary on one of the boundary circles, but not by any homotopy that is stationary on both.

Solution

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