## Topology Qualifying Exam

The Ph.D. qualifying exam committee tries to proofread the examinations as carefully as possible.
Nevertheless, the exam may contain misprints.
If you are convinced a problem has been stated incorrectly, mention this to the proctor and indicate your interpretation in your solution.
In such cases do not interpret the problem in such a way that it becomes trivial.

Directions: Do four problems in each section.
Budget your time.
Write your solution for each question on a separate page.

## Section I

1. Let A and B be closed subspaces of a topological space X with $X = A \cup B$ Suppose that $f : A  Y$ and $g : B  Y$ are continuous, and $f ( x ) = g ( x )$ for all $x \in A \cap B$

Prove that $h : X \to Y$ by

$$
h ( x ) = { \left\{ \begin{array} { l l } { f ( x ) } & { { \mathrm { ~ i f ~ } } \quad x \in A } \\ { g ( x ) } & { { \mathrm { ~ i f ~ } } \quad x \in B } \end{array} \right. }
$$

is continuous.
Is it necessary for both A and B to be closed?
Discuss.

2. Let $f : X \to Y$ by a quotient map.
   Let Y be connected and suppose that for each $y \in Y , f ^ { - 1 } ( y )$ is connected.
   Prove that X is connected.

3. Let I be a non empty index set, let $\{ X _ { \alpha } | \alpha \in I \}$ be a family of topological spaces, and let $A _ { \alpha } \subseteq X _ { \alpha }$ for each α.

(a) Show that if $A _ { \alpha }$ is closed in $X _ { \alpha }$ for each α, then $\prod A _ { \alpha }$ is closed in $\prod X _ { \alpha }$

(b) Show that $\overline { { \prod A _ { \alpha } } } = \prod \overline { { A _ { \alpha } } } .$

(c) Prove or disprove: If $A _ { \alpha }$ is open in $X _ { \alpha }$ for each α, then $\prod A _ { \alpha }$ is open in $\Pi X _ { \alpha } .$

4. Let D be the closed unit disk in the complex plane.
   Let ∼ be the equivalence relation on D defined by $z _ { 1 } \sim z _ { 2 }$ if and only if $z _ { 1 } ~ = ~ z _ { 2 }$ or $| z _ { 1 } | = | z _ { 2 } | < 1$ . Is the quotient topological space Hausdorff?
   (Prove your assertion.)

5. State the definition of compactness for topological spaces.
   Prove from your definition that the closed unit interval [0, 1] is compact.

## Section II

1. Define what it means for Y to be a strong deformation retract of $X ,$ where $Y \subseteq X$ are topological spaces.
   Prove that if $i : Y \to X$ is the inclusion map and $y \in Y$ , then the induced homomorphism $i _ { * } : \pi _ { 1 } ( Y , y )  \pi _ { 1 } ( X , y )$ is an isomorphism.

2. Prove by any method you know that:

(a) < is not homeomorphic to $\Re ^ { 2 }$

(b) $\Re ^ { 2 }$ is not homeomorphic to $\Re ^ { 3 }$

3. Let $X _ { 1 }$ and $X _ { 2 }$ be two copies of $S ^ { 2 }$ and let $N _ { 1 } , S _ { 1 }$ and $N _ { 2 } , S _ { 2 }$ be the north and south poles of $X _ { 1 }$ and $X _ { 2 } .$ , respectively.
   Define $X$ to be the quotient space obtained by identifying $N _ { 1 }$ with $N _ { 2 }$ and $S _ { 1 }$ with $S _ { 2 }$ . Compute the fundamental group of X by using the Seifert-van Kampen theorem.

4. (a) Define a covering space.

(b) State the main theorem about path lifting and covering spaces.

(c) Let $S ^ { 1 } \vee \Re P ^ { 2 }$ be the one point union of the circle and two dimensional real projective space, $i . e .$ . the quotient space obtained by taking the disjoint union of $S ^ { 2 }$ and $\Re P ^ { 2 }$ and then identifying a single point $x \in S ^ { 2 }$ with a single point in $y \in \Re P ^ { 2 }$ . Describe the universal cover of $S ^ { 1 } \vee \Re P ^ { 2 }$

(d) Describe the fundamental group of $S ^ { 1 } \vee \Re P ^ { 2 }$

5. Prove that $\Re ^ { 2 }$ cannot be retracted to $S ^ { 1 }$