# Topology Qualifying Exam Workshop

May 2020

Worksheet 3A

Theme: Fundamental groups: Fundamental group, induced homomorphism; free group, group presentation, Tietze’s theorem, amalgamated product of groups, Seifert - van Kampen Theorem; cell complex, presentation complex, Classification of surfaces.

Part 1 - 5-10 minutes

Warm-up:

1. Create a cell complex for $S ^ { 2 }$ and a cell complex for $T = S ^ { 1 } \times S ^ { 1 }$

2. State the Seifert-van Kampen Theorem.

3. Briefly describe the classification of compact surfaces without boundary.

## Part 2 - 1 hour and 40 minutes

Try these problems:

1. (June 2012) Let X be the space obtained from the torus $T = S ^ { 1 } \times S ^ { 1 }$ by gluing in a disk $D ^ { 2 }$ along its boundary $\partial D ^ { 2 } = S ^ { 1 }$ using the map $\alpha : S ^ { 1 } \to T$ given by $z \mapsto ( z , ( 1 , 0 ) )$ for $z \in S ^ { 1 }$

(a) Find $\pi _ { 1 } ( X )$

(b) Give a CW complex for X.

2. (May 2017) If $H : X \times [ 0 , 1 ] \to X$ is a homotopy with $H _ { 0 } = H _ { 1 } =$ the identity map, show that the map $\gamma : I  X$ given by $\gamma ( t ) = H ( x _ { 0 } , t )$ is a loop in X representing an element $g = [ \gamma ] \in \pi _ { 1 } ( X , x _ { 0 } )$ which lies in the center of $\pi _ { 1 } ( X , x _ { 0 } )$ , i.e. gh = hg for all $h \in \pi _ { 1 } ( X , x _ { 0 } )$ .

3. (January 2019) Let $U = \{ ( x , y , 0 ) \in \mathbb { R } ^ { 3 } : x ^ { 2 } + y ^ { 2 } = 1 \}$ , and let $W = \mathbb { R } ^ { 3 } - U$

Let $V = \{ ( x , y , z ) : \mathbb { R } ^ { 3 } : x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4 \} \cup \{ ( 0 , 0 , z ) \in \mathbb { R } ^ { 3 } : - 2 \leq z \leq 2 \}$

(a) Describe a deformation retraction of W onto V .

(b) Use (a) and the Seifert-van Kampen Theorem to compute $\pi _ { 1 } ( W )$

4. (May 2019) Let A be the annulus $\{ r e ^ { i \theta } \in \mathbb { C } : 1 \leq r \leq 2 \}$ , and let B be the quotient space obtained from A by identifying each point $e ^ { i \theta }$ on the circle r = 1 with the point $2 e ^ { i ( \theta + \pi ) }$ on the circle $r = 2 .$ Determine $\pi _ { 1 } ( B )$

## 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?