# Topology Qualifying Exam Workshop

May 2020

Worksheet 1

## Part 1 - 5-10 minutes

1. Discuss with your group: What topics on the qual do you feel confident about?
   What topics are you less confident about?
   (The syllabus can be found at https: // www.
   math.
   unl.
   edu/ graduate/ exams/ quals/ topology/ 871-872Qualifier_ Syllabus.
   pdf .)

2. Consider for yourself, and share with your group if you feel comfortable: Are you excited for the exam?
   Anxious?
   Worried?

## Part 2 - 1 hour and 40 minutes

This was the qual that I took.
This is not a reasonable amount of time to finish these problems, but look them all over, and pick and choose a few to try.

Do three problems from:

1. (May 2016) Given any topological space Z and subset $D \subseteq Z .$ , let $C l _ { Z } ( D )$ denote the closure of D in $Z .$ Show that if X and Y are topological spaces and $A \subseteq X , B \subseteq Y$ , then $C l _ { X \times Y } ( A \times B ) =$ $C l _ { X } ( A ) \times C l _ { Y } ( B )$

2. (May 2016) Let X be a connected space and $A , B \subseteq X$ be closed subsets of X with $X = A \cup B$ and A ∩ B a connected subset of X. Show that both A and B are connected.

3. (May 2016) Let X be the set of real numbers, let $\mathcal { T } _ { E }$ be the Euclidean topology on X, and let $\mathcal { T } _ { 0 }$ be the excluded point topology (that is, ${ \mathcal { T } } _ { 0 } = \{ U \subset X | 0 \not \in U \} \cup \{ X \} \}$ . For each of the following topological spaces, determine whether or not the space is compact:

(a) The set X with the topology $\mathcal { T } _ { E } \cap \mathcal { T } _ { 0 }$

(b) The set X with the topology generated by the subbasis $\mathcal { T } _ { E } \cup \mathcal { T } _ { 0 }$

4. (May 2016) Suppose that the space X has the fixed point property (that is, for any continuous function $f : X \to X$ there is a point $p \in X$ with $f ( p ) = p )$ . Suppose also that $A \subset X$ is a subspace admitting a retraction $r : X \to A$ . Show that A also has the fixed point property.

Do three problems from:

1. (May 2016) Let $X = S ^ { 1 } \times S ^ { 1 }$ , also thought of as the standard quotient of the unit square $[ 0 , 1 ] \times [ 0 , 1 ]$ and let $A = ( x , x ) : x \in S ^ { 1 } \}$ be the diagonal of X. Show that A is a retract of X, but not a deformation retract of X.

2. (May 2016) A group G is called residually finite if for every $g \in G$ with $g \neq 1$ , there is a finite group H and a (surjective) homomorphism $\varphi ; G \to H$ with $\varphi ( g ) \neq 1$ . Let G be a residually finite group and let X be the presentation complex for a presentation of $G ,$ with vertex $x _ { 0 }$ . Show that for any loop $\gamma : I  X$ at $x _ { 0 }$ with $1 \ne [ \gamma ] \in \pi _ { 1 } ( X , x _ { 0 } )$ , there is a finite-sheeted covering space $p : \tilde { X } \to X$ and a basepoint $\tilde { x } _ { 0 } \in p ^ { - 1 } ( \{ x _ { 0 } \} )$ such that γ does not lift to a loop at $\tilde { x } _ { 0 }$

3. (May 2016) Let $p : { \tilde { X } } \to X$ and $q : { \tilde { Y } }  Y$ be covering spaces of path-connected, locally path-connected spaces X and Y with X˜ and Y˜ locally path-connected and simply connected.
   Show that if X and Y are homeomorphic, then $\tilde { X }$ and $\tilde { Y }$ are homeomorphic.

4. (May 2016) Construct a ∆-complex structure, and use it to compute the simplicial homology groups, for the connected sum of two projective planes.

## Part 3 - 5-10 minutes

Wrap-up discussion with everyone:

1. Are these problems roughly what you expected?
   Harder?
   Easier?
   More technical?

2. What have you learned from these problems?
   Your answer doesn’t have to be strictly mathematical.