## Warm-up Problems

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

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

1. (Jan 2002, B4) Let $p : { \widetilde { X } } \to X$ be a covering map, with X locally path-connected, and $x _ { 0 } \in X$ . Show that for $y _ { 0 } , y _ { 1 } \in p ^ { - 1 } ( \{ x _ { 0 } \} )$ , there is a deck transformation of $\widetilde { X }$ taking y0 to $y _ { 1 }$ if and only if $p _ { * } ( \pi _ { 1 } ( \widetilde { X } , y _ { 0 } ) ) = p _ { * } ( \pi _ { 1 } ( \widetilde { X } , y _ { 1 } ) )$ .

2. (June 2005, B6) Show, using covering spaces, that the fundamental group of the Klein bottle is not abelian.

3. (Jan 2006, B7) Let $p : \widetilde { X } \to X$ be a covering map, and $f : Y \to X$ be a continuous map. Define ${ \widetilde { Y } } = \{ ( y , { \widetilde { x } } ) \in Y \times { \widetilde { X } } : f ( y ) = p ( { \widetilde { x } } ) \} \subseteq Y \times { \widetilde { X } }$ , with the subspace topology inherited from $Y \times { \widetilde { X } }$ , and define $q : \widetilde { Y }  Y$ by $q ( y , \widetilde { x } ) = y$ . Show that q is also a covering map.

4. (June 2007, B2) Let $p : ( \widetilde { X } , y _ { 0 } )  ( X , x _ { 0 } )$ be a covering space projection, $x _ { 0 } \in A \subseteq X$ and $\iota : A  X$ the inclusion map. Show that $q = p | _ { p ^ { - 1 } ( A ) } : ( p ^ { - 1 } ( A ) , y _ { 0 } ) \to ( A , x _ { 0 } )$ is also a covering space, and ker $( \iota _ { * } ) \subseteq \mathrm { i m } ( q _ { * } ) \subseteq \pi _ { 1 } ( A , x _ { 0 } )$

5. (June 2008, B6) Let $p _ { i } : ( \widetilde { X } _ { i } \widetilde { x } _ { i } )  ( X _ { i } , x _ { i } )$ be covering spaces for $i = 1 , 2$

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

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

6. (June 2009, B6) Let $p : { \widetilde { X } } \to X$ be a covering space. Show that if X is Hausdorff, then X is also Hausdorff.

7. (June 2009, B7) Use covering space theory to prove that any finitely presented group G has only finitely many subgroups of index 3.

8. (June 2010, B7) Use covering space theory to prove that if H is a subgroup of index 3 in a finitely presented group $G _ { i }$ then H is finitely presented. More generally, (June 2014, B7), show the result if H is only assumed to be of finite index.

9. (June 2014, B6) Let $p : \widetilde { X } \to X$ be a covering space with $\widetilde { X }$ path-connected. Let A be a subspace of X with inclusion map $i : A \hookrightarrow X$ , and suppose that for $a \in A$ the induced homomorphism $i _ { * } : \pi _ { 1 } ( A , a )  \pi _ { 1 } ( X , a )$ is a surjection. Show that if $r , s \in \widetilde { X }$ and $p ( r ) = p ( s ) = a$ , then there is a path from r to s in the subspace $p ^ { - 1 } ( A )$ of $\widetilde { X }$ .

10. (May 2015, B5) Suppose $X , X _ { 1 } , X _ { 2 }$ are path-connected spaces, $p : X _ { 1 } \to X$ and $q :$ $X _ { 2 }  X$ are (surjective) covering maps, and $f : X _ { 1 } \to X _ { 2 }$ is a continuous map with $q \circ f = p$ . Show that f is also a covering map.

11. (May 2015, B6) Show, using covering space theory, that a finitely presented group G has only finitely many distinct subgroups $H \leq G$ of index 4.