## Warm-up Problems

• Let $A \subseteq X$ and $r : X \to A$ a retraction. Prove that $r _ { * } : \pi _ { 1 } ( X ) \to \pi _ { 1 } ( A )$ is a surjection.

• Let $B \subseteq Y$ and let $p : Y  B$ be a deformation retraction. Prove that $p _ { * } : \pi _ { 1 } ( Y ) \to$ $\pi _ { 1 } ( B )$ is an isomorphism.

1. (Jan 2002, B1) Show that if X is path-connected, locally path-connected space, with $\pi _ { 1 } ( X )$ finite, then every continuous map $f : X \to S ^ { 1 }$ is homotopic to a constant map.

2. (Jan 2002, B3) Let X be the 2-sphere, and $A \subseteq X$ the equitorial circle in X. Show that there is no retraction $r : X \to A$

3. (June 2014, A2)

(a) Let $q : A  B$ be continuous and $i : I \to I$ be the identity function. Define the function $q \times i : A \times I \to B \times I$ by $( q \times i ) ( a , t ) : = ( q ( a ) , i ( t ) )$ . Show that $q \times i$ is continuous.

(b) (You may assume the fact: If $q : A  B$ is a quotient map, then so is $q \times i . )$ Let X be a path-connected Hausdorff space. The cone CX of X is the quotient of the space $X \times I$ by the smallest equivalence relation such that $\left( p , 0 \right) \sim \left( q , 0 \right)$ for all $p , q \in X$ . Prove that there is a deformation retraction from CX to the singleton subspace $\{ [ ( p , 0 ) ] \}$

4. (May 2015, A4) Let $f : X \to Y$ be a continuous map, and suppose that there are continuous maps $g , h : Y \to X$ with $f \circ g \simeq \operatorname { I d } _ { Y }$ and $h \circ f \simeq \operatorname { I d } _ { X }$

(a) Show that $g \simeq h$

(b) Show, moreover, that f is a homotopy equivalence.

5. (Arizona Spring 2005) Let $f : S ^ { 1 } \vee S ^ { 1 }  T$ be a continuous map, where $T$ is the torus. Show that there is no continuous map $g : T  S ^ { 1 } \vee S ^ { 1 }$ such that $f \circ g$ is the identity map on $T$

6. (June 2004, A2) Let X be the unit sphere in $\mathbb { R } ^ { 3 }$ , and define an equivalence relation on X by

$$
( x , y , z ) \sim ( x ^ { \prime } , y ^ { \prime } , z ^ { \prime } ) \mathrm { ~ i f ~ a n d ~ o n l y ~ i f ~ } z = z ^ { \prime } .
$$

Let $Z = X / \sim$ be the quotient space under this equivalence relation, with the quotient topology. Show that Z is homeomorphic to the interval $[ - 1 , 1 ]$ . (Here $\mathbb { R } ^ { 3 }$ and $[ - 1 , 1 ]$ have their usual topologies, and X has the subspace topology.)

7. (June 2008, A3)

(a) Prove that a map $f : X \to Y$ between topological spaces is continuous if and only if $f ( { \overline { { A } } } ) \subseteq { \overline { { f ( A ) } } }$ for all $A \subseteq X$ . [Here A denotes the closure of A.]

(b) Prove that if f is continuous and $f ( { \overline { { A } } } )$ is closed for some $A \subseteq X$ then $f ( { \overline { { A } } } ) = { \overline { { f ( A ) } } }$

8. (June 2009, A3) Let $S ^ { 1 }$ and $D ^ { 2 }$ be the unit circle and closed unit disk in the (Euclidean) space $\mathbb { R } ^ { 2 }$ . Let ∼ be the smallest equivalence relation on the product space $S ^ { 1 } \times D ^ { 2 }$ satisfying $( x , y ) \sim ( x , ( 0 , 0 ) )$ for all $x \in S ^ { 1 }$ and $y \in D ^ { 2 }$ . Let $Z : = ( S ^ { 1 } \times D ^ { 2 } ) / \sim$ be the corresponding quotient space. Prove that Z is homeomorphic to $S ^ { 1 }$