# Topology Qualifying Exam Workshop

May 2020

Worksheet 5

## Part 1 - 2 hours

Try these problems.
On the exam you will be asked to complete three of the four questions in 3 hours.

1. (May 2013) Let $X _ { \alpha }$ be non-empty topological spaces and suppose that $X = \prod _ { \alpha } X _ { \alpha }$ is endowed with the product topology.

(a) Prove that each projection map $\pi _ { \alpha }$ is continuous and open.

(b) Prove that X is Hausdorff if and only if each space $X _ { \alpha }$ is Hausdorff.

## 2. (May 2013)

(a) Prove that every compact subspace of a Hausdorff space is closed.
Show by example that the Hausdorff hypothesis cannot be removed.

(b) Prove that every compact Hausdorff space is normal.

3. (May 2013) Define an equivalence relation ∼ on $\mathbb { R } ^ { 2 }$ by $( x _ { 1 } , y _ { 1 } ) \sim ( x _ { 2 } , y _ { 2 } )$ if and only if $x _ { 1 } ^ { 2 } + y _ { 1 } ^ { 2 } = x _ { 2 } ^ { 2 } + y _ { 2 } ^ { 2 }$

(a) Identify the quotient space $X = \mathbb { R } ^ { 2 } / \sim$ as a familiar space and prove that it is homeomorphic to this familiar space.

(b) Determine whether the natural map $p : \mathbb { R } ^ { 2 } \to X$ is a covering map.
Justify your answer.

4. (May 2013) A space is locally connected if for each point $x \in X$ and every neighborhood U of x, there is a connected neighborhood V of x contained in U.

(a) Prove that X is locally connected if and only if for every open set U of X, each connected component of U is open in X.

(b) Prove that if $p : X \to Y$ is a quotient map and X is locally connected, then Y is locally connected.

## Part 2 - 2 hours

Try these problems.
On the exam you will be asked to complete three of the four questions in 3 hours.

1. (May 2013) 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.

## 2. (May 2013)

(a) Explain in detail how the Seifert-van Kampen theorem may be used to calculate the fundamental group of a wedge sum $X \vee Y$ of two spaces under suitable assumptions on the spaces.
Clarify what assumptions on the spaces you are using and how you are using them.

(b) Describe the presentation complex $X _ { G }$ of the group $G = \langle a , b , c : a ^ { 2 } = 1 \rangle$ as a wedge sum of familiar spaces.
Explain carefully what results you are using.

3. (May 2013) Let Y be the standard 3-simplex $\Delta ^ { 3 }$ with a total ordering on its four vertices.
   Let X be the ∆-complex obtained from Y by identifying, for each $k \leq 3$ , all of its k-dimensional faces such that the identifications respect the vertex ordering.
   Thus X has a single k-simplex for each $k \leq 3$ . Compute the simplicial homology groups of the ∆-complex X.

4. (May 2013)

(a) Describe how to construct a cell structure on the 2-sphere $S ^ { 2 }$ consisting of one 0-cell, one 1-cell, and two 2-cells, and explain how to use this cell structure to calculate the simplicial homology groups of $S ^ { 2 }$

(b) Explain how a long exact sequence may be used to calculate all of the (singular) homology groups $H _ { i } ( S ^ { n } )$ of the n-sphere $S ^ { n }$ (and calculate these groups for all i and n).

## 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?