# Topology Qualifying Exam Workshop

May 2020

Worksheet 2

Theme: Continuous deformations: Retraction, deformation retraction, contractible, mapping cylinder, homotopic maps, homotopy type.

Also more homeomorphism invariants because those are heavily emphasized on the quals.

Part 1 - 5-10 minutes

Warm-up:

1. Determine whether the following spaces X admit a retraction and/or a deformation retraction onto the designated subspace Y .

(a) $X = [ 0 , 1 ] , Y = \{ 0 \} .$

(b) $X = [ 0 , 1 ] , Y = \{ 0 , 1 \}$

(c) $X = S ^ { 1 } , Y = \{ y \}$ , any singleton.

2. Construct an explicit deformation retraction from $\mathbb { R } ^ { n } \backslash \{ 0 \}$ onto $S ^ { n - 1 }$

3. Let $X = D ^ { 2 } , Y = \{ y \}$ a singleton, and $f : X \to Y$ be the constant function.
   Construct and describe the mapping cylinder.

## Part 2 - 1 hour and 40 minutes

Try these problems:

1. (May 2015) Let $f : X \to Y$ be a continuous map, and suppose that there are continuous maps $g , h : Y \to X$ with $f \circ g \simeq \operatorname { i d } _ { Y }$ and $h \circ f \simeq \operatorname { i d } _ { X }$

(a) Show that $g \simeq h .$

(b) Show, moreover, that f is a homotopy equivalence.

2. (May 2018) Show that if X and Y are topological spaces and $X \times Y$ , with the product topology, is contractible, then both X and Y are contactible.

3. (January 2019) Show that if $f , g : ( X , { \mathcal { T } } ) \to ( Y , { \mathcal { T } } ^ { \prime } )$ and $h , k : ( Y , T ^ { \prime } ) \to ( Z , T ^ { \prime \prime } )$ are continuous and $f \simeq g$ and $h \simeq k .$ , then $h \circ f \simeq k \circ g .$ . Show, however, that the converse $( h \circ f \simeq k \circ g$ implies $f \simeq g$ and $h \simeq k )$ need not be true.

4. (June 2010) Let $Z = X \cup Y .$ , for X and Y connected subspaces of Z with $X \cap Y = \emptyset$ Let $x _ { 0 } \in X$ and $y _ { 0 } \in Y .$ . Let ∼ be the equivalence relation generated by the equivalence $x _ { 0 } \sim y _ { 0 }$ . Show that the quotient space $Z / \sim$ is connected.

5. (June 2011) Suppose that A, B are disjoint, compact subspaces of the Hausdorff topological space X. Prove that there are open subsets $U , V$ of X such that $A \subset U , B \subset V .$ , and $U \cap V = \emptyset$

## 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?