# Topology Qual Workshop Day 6: Homotopy & Retractions

Warm-up Problems:

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

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

(1) (Michigan May ’10) Let $\mathbb { R } P ^ { 2 }$ and T denote, in this order, the real projective plane and the torus $S ^ { 1 } \times S ^ { 1 }$ . Prove that any map

$$
f : \mathbb { R } P ^ { 2 } \to T ,
$$

is homotopic to a constant map.

Variration: Let X be a path connected, locally path connected space with $\pi _ { 1 } ( X )$ finite. Show that every continuous map $f : X \to S ^ { 1 }$ is homotopic to a constant map.

(2) (Michigan Jan ’10) Let $D ^ { 2 }$ be a closed 2-disk, $S ^ { 1 }$ its boundary. Is the space $\ b X = \ b S ^ { 1 } \times \ b S ^ { 1 }$ a retract of the space $Y = D ^ { 2 } \times S ^ { 1 } ?$ (i.e. does there exist a continuous map $Y  X$ which is the identity on $X \subset Y ? )$

(3) (Arizona Winter ’05) Suppose $\phi : S ^ { 1 } \to S ^ { 1 }$ and $\psi : S ^ { 1 } \to S ^ { 1 }$ are continuous maps. Show that the compositions $\phi \circ \psi$ and $\psi \circ \phi$ are homotopic. (Hint: is this true for $f ( z ) = z ^ { n }$ where $n \in \mathbb { Z } ? )$

(4) (Arizona Fall ’05) Let $f : D ^ { 2 } \to D ^ { 2 }$ be a homeomorphism (where $D ^ { 2 }$ is the closed unit disk in $\mathbb { R } ^ { 2 } )$ Show that f must map boundary points $x \in \partial D ^ { 2 }$ to points $f ( x ) \in \partial D ^ { 2 }$ on the boundary.

(5) (Arizona Spring ’05) Let $f : S ^ { 1 } \vee S ^ { 1 }  T$ be a continuous map, where T is the torus. Show that there is no continuous map $g : T \to S ^ { 1 } \vee S ^ { 1 }$ such that $f \circ g$ is the identity map on $T .$

(6) Prove that $S ^ { n - 1 }$ is not a retract of $B ^ { n }$

Covering Space Problems (Suggested reading: Hatcher, page $7 7 - 7 8 )$

Let $G = \left. x | x ^ { 2 } \right.$ . Find the presentation complex, $X _ { G }$ , which is a familiar space, identify it as such. Then find the universal cover of $X _ { G } , \tilde { X } _ { G }$

Let $H = \left. x | x ^ { 6 } \right.$ . Find the presentation complex, $X _ { H }$ . Then find the universal cover of $X _ { H } , \tilde { X } _ { H }$