Warm-up Problems:

• Explicitly prove that R is that universal cover of $S ^ { 1 }$

• Prove or disprove: $S ^ { 2 } \vee S ^ { 2 }$ is a covering space of $S ^ { 2 }$

(1) (Wisconsin Jan ’98) Let $p : \tilde { X } \to X$ be a cover.
Suppose that $f , g : Y \to { \tilde { X } }$ are maps such that $p \circ f$ and $p \circ g$ are equal and assume that f and g agree at $y _ { 0 } \in Y$ . Show that if Y is connected, then $f = g .$

(2) (Jan ’06) Let $p : \tilde { X } \to X$ be a covering map, and $f : Y \to X$ be a continuous map.
Define ${ \tilde { Y } } = \left\{ ( y , { \tilde { x } } ) \in Y \times { \tilde { X } } : f ( y ) = p ( { \tilde { x } } ) \right\} \subseteq Y \times { \tilde { X } }$ , with the subspace topology inherited from $Y \times { \tilde { X } }$ and define $q : \tilde { Y }  Y \mathrm { ~ b y ~ } q ( y , \tilde { x } ) = y .$ . Show that q is also a covering map.

(3) (June ’05) Show, using covering spaces, that the fundamental group of the Klein bottle is not abelian.

(4) (Purdue Jan ’09) Let $p : E  B$ be a covering map with B connected.
Suppose that $p ^ { - 1 } ( b _ { 0 } )$ is finite for some $b _ { 0 } \in B$ . Prove that, for every $b \in B , p ^ { - 1 } ( b )$ has the same number of elements as $p ^ { - 1 } ( b _ { 0 } )$

(5) (Purdue Aug ’09) Let $p : E  B$ be a covering map.
Let Y be locally path-connected.
Let $g : Y  E$ be a function such that

$p \circ g$ is continuous

$g \circ \gamma$ is continuous for every path γ in Y .

Prove that g is continuous.

(6) (Purdue Jan ’08) Let $p : E  B$ be a covering map.
Suppose that points are closed in B. Let $A \subset E$ be compact.
Prove that for every $b \in B$ , the set $A \cap p ^ { - 1 } ( b )$ is finite.

(7) (Purdue Jan ’07) Let $p : E  B$ be a covering map.
Prove that p takes open sets to open sets.

(8) Let $G = \mathbb { Z } _ { 2 } * \mathbb { Z } _ { 2 } = \left. a , b | a ^ { 2 } = b ^ { 2 } = 1 \right.$ . Find the presentation complex $X _ { G }$ and the universal cover of $X _ { G } , \tilde { X } _ { G }$ . Find an index two subgroup of G and the corresponding covering space.

(9) (June ’07) Let $p : ( \tilde { X } , y _ { 0 } )  ( X , x _ { 0 } )$ be a covering space projection, $x _ { 0 } \in A \subseteq X .$ X, and $\iota : A  X$ the inclusion map.
Show that $q = p | _ { p ^ { - 1 } ( A ) } : \left( p ^ { - 1 } \left( A \right) , y _ { 0 } \right) \to \left( A , x _ { 0 } \right)$ is also a covering space, and ker $\iota _ { * } \subseteq \operatorname { i m } \left( q _ { * } \right) \subseteq \pi _ { 1 } \left( A , x _ { 0 } \right)$

(10) (June ’08) Let $p _ { i } : ( { \tilde { X } } _ { i } , { \tilde { x } } _ { i } )  ( X _ { i } , x _ { i } )$ be covering spaces for $i = 1 , 2$

(a) Show that the product space $\tilde { X } _ { 1 } \times \tilde { X } _ { 2 }$ together with the map $p : \tilde { X } _ { 1 } \times \tilde { X } _ { 2 }  X _ { 1 } \times X _ { 2 }$ defined by $p \left( y , z \right) : = \left( p _ { 1 } \left( y \right) , p _ { 2 } \left( z \right) \right)$ is also a covering space.

(b) Find the universal covering space of $S ^ { 1 } \times D ^ { 2 } \times S ^ { 1 }$

(11) (June ’09) Let $p : \tilde { X } \to X$ be a covering space.
Show that if X is Hausdorff, then $\tilde { X }$ is also Hausdorff.

(12) (May ’13) Let X be the space obtained from the 2-sphere $S ^ { 2 }$ by identifying the north and south poles (i.e. by identifying two diametrically opposite points).

(a) Show that X is homotopy equivalent to $S ^ { 1 } \vee S ^ { 2 }$

(b) Describe all connected covering spaces of $X .$