# Topology Qualifying Exam Workshop

May 2020

Worksheet 3B

Covering spaces: Covering map, Lifting theorems; covering space group action; universal covering, Cayley complex; Galois Correspondence Theorem, deck transformation, normal covering; applications to group theory.

Part 1 - 5-10 minutes

Warm-up:

1. Find the universal covers of these spaces:

(a) $X = S _ { 1 }$

(b) $Y = S _ { 1 } \vee S _ { 1 }$ (two circles glued together along a point).

2. What is a deck transformation?
   Describe the deck transformations from one of the universal covers above.

3. State the Galois Correspondence Theorem for a path-connected, locally path-connected, and semilocally simply connected space X.

## Part 2 - 1 hour and 40 minutes

Try these problems:

1. (June 2014) Use covering space theory to prove that if H is a finite index subgroup of a finitely presented group, then H is finitely presented.

2. (May 2018) Let $F _ { n }$ denote the free group on n letters.
   Use covering space theory to show that for all $n ,$ there exists a subgroup $H _ { n }$ of $F _ { 2 }$ such that $H _ { n } \cong F _ { n }$ . For your choices of $H _ { n }$ constructed, determine if each $H _ { n }$ is normal in $F _ { 2 }$

3. (January 2019) Suppose $p : { \tilde { X } }  X$ is a covering space and X is Hausdorff.
   Prove that $\tilde { X }$ is Hausdorff.

4. (May 2019) Let $p : { \tilde { X } }  X$ and $q : { \tilde { Y } }  Y$ be covering maps such that both $\tilde { X }$ and $\tilde { Y }$ are connected and simply connected.
   Show that for every map $f : X \to Y$ there is a map $\tilde { f } : \tilde { X } \to \tilde { Y }$ such that $f \circ p = q \circ { \tilde { f } }$

5. (January 2020) Suppose Y is Hausdorff, path-connected, locally path-connected, and semilocally simply-connected, and that its universal cover Y˜ is compact.
   Prove that $\pi _ { 1 } ( Y )$ is finite.

## Part 3 - 5-10 minutes

Wrap-up discussion with everyone:

1. Which of these problems did you find easy?
   Hard?

2. Which topics in the theme do you need to review more?