# Wesleyan University Department of Mathematics

PhD Qualifying Exam, Written Part: Topology Profs. Hager and Hovey

August 1, 2005

This is a 3-hour exam; no books or notes or consultations are permitted. The exam has two parts. You must pass each part to pass the exam. "Pass" means "show us you know enough to start trying to write a thesis".

## Part One

What are the cardinalitis of these sets?P(N), {0,1}, [0, 1], (0, 1), $\{ f | f \colon  { \mathbb { N } } \to [ 0 , 1 ]$ is a function}. (Here N is the natural numbers and P denotes power set).

(2) Prove or disprove.

(a) If X is second countable and $Y \subseteq X ,$ then Y is second countable.

(b) If f : X → Y is continuous and onto, and X is second countable, then Y is second countable.

(c) The product of any set of second countable spaces is second countable.

→ (3) Why?

(a) Why is each of the following not metrizable? The Sorgenfrey line. The one-point compactification of an uncountable discrete space./The product of uncountably many copies of N (the countable discrete space)./

(b) Why is a compact Hausdorff space completely regular? Why is a completely regular Hausdorff space homeomorphic to a subspace of a compact Hausdorff space?

4 Give Hausdorff examples of the following:

(a) X,Y normal, X x Y not normal.

(b) X normal, Y  X, Y not normal.

-(c) Infinite compact X with no isolated points. The same with a dense set of isolated points instead.

(d) Infinite compact X with a point with no countable local base.

(e) X first countable but not second countable.

(f) X separable but not metrizable.

Consider the ech-Stone compactifiation β fcompletely regular Hausdorff X.

(a) State a set of characteristic properties of βX. (That is, "what is it?" not "how do we make it?")

() What are the cardialits of ββS(? β[,?

## Part Two

(6) Given a pointed space $( X , x _ { 0 } )$ , define the fundamental group $\pi _ { 1 } ( X , x _ { 0 } )$ and givea sketchof the pro that t is in fact a ru.

(7) Suppose we take an 8-sided polygon and identify the edges according to the scheme $a b ^ { - 1 } c d c ^ { - 1 } b d ^ { - 1 } a .$ That is, starting at one vertex and going around the edges clockwise, these are the labels you see, with $b ^ { - 1 }$ meaning that b is backwards. Find the fundamental group and the first homology group of this surface and identify it as one of the standard surfaces.

(8) Let U be an open cover of the connected space $X ,$ and let $a , b \in { \dot { X } }$ Show that there is a some sequence $U _ { 1 } , \dots , U _ { n }$ of open sets in U such that $a \in U _ { 1 } , b \in U _ { n } ,$ and $U _ { i } \cap U _ { i + 1 }$ is nonempty for $i = 1 , 2 , \ldots , n - 1$ (Hint: Let C be the set of all points b that can be connected to a by such a sequence. What can you say about C?)

(9) Give a space $X ,$ define the suspension of X, ΣX to be the space obtained by suspending X from two points. Technically, ∑X is the space obtained from the cylinder $X \times I$ by identifying (x, 1) with $( x ^ { \prime } , 1 )$ for all $x , x _ { \cdot } ^ { \prime } \in X$ and $( x , 0 )$ with $( x ^ { \prime } , 0 )$ . Calculate, with proof, the fundamental group $\frac { \operatorname { o f } \mathcal { Z } } { \pi } ,$ assuming X is path-connected.

(10) One of the most important results about topological groups is that the universal cover of a (well-behaved) topological group is another topological group. In this problem, you will prove part of this result. (a) Suppose we have a diagram

<!-- image-->

where p is a covering map and all spaces are path connected and locally path connected. Give necessary and sufficient conditions for there to exist a map $\tilde { f } \colon ( X , x _ { 0 } )  ( E , e _ { 0 } )$ such that $p { \widetilde { f } } = f .$ Assuming $\widetilde { f }$ does exist, how many choices are there for it?

(b) Now assume G is a path connected, locally path connected, topological group with identity $e ,$ multiplication $\mu \colon G \times G \to G ,$ and a universal cover p: $( H , e ^ { \prime } ) \to ( G , e )$ . Show that there is a map $\mu ^ { \prime } \colon H \times \dot { H }  H$ such that $p \circ \mu ^ { \prime } = \mu \circ ( p \times p )$

(c) Show that the multiplication $\mu ^ { \prime }$ defined above is associative. To complete the proof that H is a topological group, you would have to prove that H has an identity and a continuous inverse function, but we leave that as homiework for the interested student.

# Wesleyan University Department of Mathematics

PhD Qualifying Exam, Written Part: Topology Profs. Comfort and Hovey

August 30, 2004

This is a 3-hour exam; no books or notes or consultations are permitted. The exam has two parts; answer three questions from each part and then as many additional questions as time and ability permit. Partial credit may be awarded, but sparingly; therefore it is in your best interest to answer a number of questionsfully omitting some, rather than to answer all questions incompletely. Throughout this examination, the word "space" refers to a completely regular, Hausdorff space (i.e., a Tychonoff space).

## Part One

) Prove the following well-known theorem from calculus. If

$$
f \colon \ \{ a , b \}  \mathbb { R }
$$

is continuous, where $a < b \in \mathbb { R } ,$ then the image of f is a closed bounded interval.

) If $X _ { 0 }$ is the path-component o a space X containg the basoint $x _ { 0 } ,$ show that the inclusion $X _ { 0 }  X$ induces an isomorphism $\pi _ { 1 \cdot } ( X _ { 0 } , x _ { 0 } )  \pi _ { 1 } ( X , x _ { 0 } )$

3Compute the fundamental group, with proof,of the space obtained from two tori $S ^ { 1 } \times S ^ { 1 }$ by identifying a circle $S ^ { 1 } \times \{ x _ { 0 } \}$ in one torus with the corresponding circle $S ^ { 1 } \times \{ x _ { 0 } \}$ in the other torus.

Give the definitiona covering space.Prove that, p: $( E , e _ { 0 } ) \to ( X , x _ { 0 } )$ is a covering space, and $f \colon ( Y , y _ { 0 } )  ( X , x _ { 0 } )$ is a continuous map where Y is path-connected, then there exists at most one lift $\widetilde { f } \colon ( Y , y _ { 0 } )  ( E , e _ { 0 } )$ of f with $p { \widetilde { f } } = f .$

(5) (a) List all the compact connected surfaces, up to homeomorphism. (b) Consider the surface obtained from a hexagon (= 6-sided figure) by identifying the edges   
. in pairs, so that in going clockwise around the hexagon, we see the labels $a \dot { b } a c a ^ { - 1 } c ^ { - 1 }$ , where the inverse means the edge is traversed backwards. Which compact connected surface from your list is this, and how do you know?

(6) (a) Define carefully any three (standard, well-known) cardinal invariants.c which are associated with every topological space. [Note. Do not include on your list the simplest cardinal invariant: $c ( X ) = \backslash | X | =$ the cardinality of X.] (b) For each oe nvarnts you defed in (a), pove r isprove this statement: If X is space and F is a closed subspace of X, then $c ( F ) \leq c ( X )$ . (c) For each of the invariants c you defined in (a), prove or disprove this statement: If X is a metrizable space and F is a closed subspace of X, thenz $\because ( F ) \leq c ( X )$

Prove or disprove these two statements. a) The product of two Lindelof spaces is a Lindelofspace. b) The product of two metrizable Lindelof spaces is a Lindelof space.

(8Tobeginrecal these wodefnions.) A genealiCantr spaces aspace (homeomorphic to) {0,1}"; here, {0,1} is the 2-point discrete space, and κ is a cardinal number. (b) A space is zero-dimensional if it has a base of open-and-closed subsets. Now, outline a proof that a space is zero-dimensional if and only if it is homeomorphic to a subspace of a generalized Cantor space.

( For each ordinal ξ, let W() denote the set of ordinals less than  in its usual order topology. [Restated: $W ( \xi ) = \xi$ as sets, and the topology of W(ξ) is the topology for which $\{ A ( \eta ) ; \eta < \xi \} \cup \{ B ( \eta ) : \eta < \xi \}$ is a subbase. Here for $\eta < \xi , A ( \eta ) : = \{ \sigma : \eta < \sigma < \xi \}$ and $\mathcal { B } ( \eta ) : = \{ \sigma : \sigma < \eta \} ]$ (a) For which ordinals ξ is W(ξ) compact? Prove your assertion. (b) For which ordinals ξ is W(ξ) countably compact? Prove your assertion.

(10) Many familiar principles are equivalent to AC (The Axiom of Choice): Among the equivalents are these five: (Zorn's Lemma; Tukey's Lemma; the well-ordering principle; the Hausdorff maximality principle; König's Theorem. (a) State AC and any two of those five principles. (b) Choose any one of those five principles-let us call it P—and prove either that $A C \Rightarrow \cdot P$ or that $P \Rightarrow A C .$

# TOPOLOGY PRELIMINARY EXAM

# June 30, 2003

Directions. Do as much as you can; you are not expected to finish. The questions are weighted equally, though they may differ in difficulty. Do not spend too much time on one problem. You must be the judge of how much detail to provide.

1. Prove: For any set X , there is no surjection from X onto P(X) = the set of all subsets of X.

2.The fundamental group π1(X, x) is the set of pointed homotopy classses of pointed maps from $( S ^ { 1 } , p )$ to (X,x). What is the set of pointed homotopy classes of pointed maps from $( S ^ { 0 } , \dot { p } )$ to (X,x), where S° is the two-point set {-1, 1} and p = 1? $S ^ { 0 }$

3.Characterize each of the following, in terms of familiar properties. (Provide some justification for your answers.)

Subspaces of Hausdorff spaces.

Subspaces of compact spaces.compact" means only "each open cover has a finite subcover".)

Subspaces of compact Hausdorff spaces.

Ope bspacepct Hauso

Subspaces of path-connected spaces.

4(a) List all the compact connected surfaçes, up to homeomorphism.

(b) How do you know that no two of the surfaces you have listed are homeomorphic? (Some details, please)

The connected sum of a torus and a projective plane is a compact connected surface. To which element of your list is it homeomorphic, and how do you know?

Show that the number of points in a connecte, normal, T1 space Xis either ≤ 1 or $T _ { 1 }$

6.Describe the universal cover of the Klein bottle.

7. Let X be locally compact and Hausdorff.

Describ he-point ptifin $X ^ { \ast }$ of X.

Suppose also that X has a countable base. Show that X\*is metrizable.

8. Suppose that X is simply connected. Provide a proof of, or counterexample to, the following statement: Every map from X to the circle S1 is homotopic to a constant map. $S ^ { 1 }$

The Sorgenfry line S is the set of real numbers with $\{ [ a , b ) | a < b \}$ as base for the topology.

a)Explain why S is completely regular, separable, Lindelöf, not metrizable. Dli \$\ra }\$ u ,

Explain why $S \times S$ is separable, not normal, not Lindelöf.

10. Prove: If $f : X \to Y$ is continuous and onto, with X compact Hausdorff and Y Hausdorff, then there is a subspace F of X minimal among compact subspaces Z of X with $f ( Z ) = Y .$

## Wesleyan University

Doctoral Qualifying Topology Exam in Mathematics Profs. Adeboye and Collins, August 7, 2013

This is a 3-hour exam; no books or notes or consultations are permitted. The exam has two parts. Please choose at least two problems from each part and at least five questions total. Always explain your reasoning. Good luck!

## Part II

1. Let a and b be the generators of $\pi ( S ^ { 1 } \backslash S ^ { 1 } )$ . Draw a picture of a covering space of $S ^ { 1 } \vee S ^ { 1 }$ corresponding to the normal subgroup generated by $\hat { \alpha ^ { 2 } } , \hat { b ^ { 2 } }$ and ab.

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2. Consider a cylinder $S ^ { 1 } \times I .$ Identify the antipodal points of the boundary circle $S ^ { 2 } \times \{ 0 \}$ . Also identify the antipodal points of the boundary circle $S ^ { 1 } \times \{ 1 \}$ . Calculate the fundamental group of this surface.

3. Consider the space obtained from two tori $\mathcal { T } ^ { 2 } = \tilde { S } ^ { 1 } \times \dot { S } ^ { 1 }$ by identifying the circle $\dot { S } ^ { 1 } \times \{ ( 0 , 1 ) \}$ in one with $S ^ { 1 } \times \{ ( 0 , 1 ) \}$ in the other. Calculate the fundamental group of this space.

4. (a) Calculate the simplicial homology groups of $S ^ { 1 }$

(b) Calculate the simplicial homology groups of the torus using the $\triangle$ complex given below.

<!-- image-->

c) Let $ { T _ { 0 } } ^ { 2 }$ denote the the torus with the interior of a disk removed. Use the Mayer-Vietoris sequence for the torus, viewed as $ { T _ { 0 } } ^ { 2 }$ attached to a disk $D ^ { 2 }$ along a circle, to calculate the homology groups of $ { T _ { 0 } } ^ { 2 }$

<!-- image-->

Qualifying exam in Topology, June 2006

You have 3 hours. Do as many as you can. No one is expected to create a perfect score. You are to be the judge of how much detail to provide in responding to "why?", "explain", "prove", "show", "sketch", etc. Try to allocate time roughly as .5 to part A and .5 to part B.

## Part A

1. Let X be a space. The density character δX is defined as $\delta X =$ min $\{ m \mid X$ has a dense set of cardinal m}.

(a) Why does that "min" exist?

Let $C ( X ) = \{ f \in \mathbf { R } ^ { X } \mid f$ is continuous} and let m be infinite.

Prove: $\delta X \leq m$ implies $| C ( X ) | \leq 2 ^ { m }$

2. Let $\{ X _ { a } \mid a \in A \}$ (or just $\{ X _ { \mathfrak { a } } \} _ { A } )$ be a set of spaces. (A is arbitrary, not necessarily finite.)

(a) Define the topology of the topological (Tychonoff) product $\Pi \{ X _ { a } \mid a \in A \}$ (or just $\textstyle \prod _ { A } X _ { a } )$

(b) Here is the definition of topological sum $\Sigma \{ X _ { a } \mid a \in A \}$ (or just $\textstyle \sum _ { A } X _ { a } )$ ; we assume $\begin{array} { r } { a _ { 1 } \neq a _ { 2 } \Rightarrow X _ { a _ { 1 } } \cap X _ { a _ { 2 } } = \emptyset . \sum _ { A } X _ { a } } \end{array}$ is the set $\cup _ { A } { \bar { X } } _ { a } ; G \subseteq \cup _ { A } X _ { a }$ is "open" exactly when $G \cap X _ { a }$ is open in $X _ { a }$ $\forall a \in A$ Verify (quickly) that this gives us a topology.

3. A topological property is a class P of spaces for which $( X \in P ,$ Y homeomorphic to $X \Rightarrow Y \in P )$ . Consider these 6 topological properties: finite, countable, connected, locally connected, metrizable, normal. For each of these topological properties, answer the questions below.

(a) For exactly what $| A |$ is it always true that $\{ X _ { a } \} _ { A } \subseteq P \Rightarrow \Pi _ { A } X _ { a } \in$ P? Explain.

For exactly what |A| is it always true that $\{ X _ { \alpha } \} _ { A } \subseteq P \Rightarrow \sum _ { A } X _ { a } \in$ P? Explain.

4. Let C be a class of spaces. A space I is called injective in C if $\ b { I } \in \ b { C }$ and whenever $X , Y \in C , X \subseteq Y$ and $f : X \to I$ is continuous, then there exists continuous ${ \overline { { f } } } : Y \to I$ with ${ \overline { { f } } } \{ _ { X } = f$ Let $C = { \mathrm { a l l } }$ compact Hausdorff spaces. Explain why [0, 1] is injective in C.

5. For X a space, $c l o p ( { X } ) = \{ U \subseteq { X } | U$ is closed and open}. X is called zero-dimensional, zd, if clop(X) is a base. Recall that the weight of a space $Y , w Y ,$ is the minimum cardinal of a base.

Lt be the ratinals with theusual topology. Show that is $z d ,$ and $w \mathbf { Q } < | c l o p ( \mathbf { Q } ) |$

( Prove: if X is compact Hausdorff $z d ,$ then $w X = | c l o p ( X ) |$

6. Recall from algebra the definition of a commutative ring with identity, R, (or $( R , + , \cdot , 0 , 1 ) ) ,$ , and that $I \subseteq R$ is called an ideal if I is a subring and $R \cdot I \subseteq I .$

(a) Use Zorn's Lemma to show that each proper ideal is contained in a maximal ideal ("maximal" with respect to the partial order $^ { 6 6 } \subseteq$ of ideals").

Note that C(X) (see 1.) is a ring, with $( f + g ) ( x ) = f ( x ) + g ( x )$ 7 $( f g ) ( x ) = f ( \dot { x } ) \dot { g } ( x )$ and 0 and 1 the constant functions. For $f \in C ( X )$ $Z ( f ) \equiv \{ x \mid f ( x ) = 0 \}$

(b) Prove: for any X, $S \subseteq X , I ( S ) \equiv \{ f \in C ( X ) \mid Z ( f ) \supseteq { \tilde { \mathcal { X } } } \}$ is an ideal in C(X). Note that $I ( \overrightarrow { S } ) = I ( S )$ , and $S _ { 1 } \subseteq S _ { 2 } \Rightarrow ^ { \prime } I ( S _ { 1 } ) \supseteq$ I(S2).

(Prove: for compact X, the converse holds: if I is an ideal in $C ( X )$ , then $I = I ( S )$ for some closed S.

(d) What are the maximalideals in C(X) for compact X? Explain.

7.Prove: if X is locally compact, and X is dense in Hausdorf Y, then X is open in Y.

8.Recall the construction and characteristic properties of the Cech-Stone compactification βX, for X completely regular and Hausdorff. Recall also zd from 5. A simpler version of the argument constructing βX will prove: If X is zd Hausdorff, then there is compact zd Hausdorff zX containing X densely such that: if $f : X \to \{ 0 , 1 \}$ (discrete) is continuous, then there is unique ${ \overline { { f } } } : z X \to \{ 0 , 1 \} $ continuous with ${ \widehat { f } } | _ { X } = f$ Sketch the proof of this.

## Part B

1. Let $f , f ^ { \prime } : I \to X .$

) Define the term "path homotopy" as applied to $f , f ^ { \prime } .$

(b) Show that the relation of path homotopy among maps from I to X. is an equivalence relation.

) Let $X = \mathbf { R } ^ { 2 } - \{ ( 0 , 0 ) \}$ . Let $f : I \to X$ be

$$
f ( t ) = ( \cos ^ { 2 } ( 2 \pi t ) , \sin ^ { 2 } ( 2 \pi t ) )
$$

Sketch the graph of f and determine for which integers $k , \ f$ is path homotopic to $g _ { k } : I \to X$ where

$$
g _ { k } ( t ) = ( \cos ( 2 \pi k t ) , \sin ( 2 \pi k t ) )
$$

2. Let $g : X \to Y$ and $h : Y \to X$

' (a) Define the term "homotopy equivalence" as applied to g and h.

(b) Show that a map homotopic to a homotopy equivalence is a homotopy equivalence.

(c) Prove that the three spaces below are homotopy equivalent.

<!-- image-->

3. Let X be a topological space and $A \subset X$

(a) Define the term "deformation retract" as applied to A and X.

/(b) Construct an explicit deformation retraction of $\mathbf { R } ^ { n } - \{ 0 \}$ onto $S ^ { n - 1 }$

4. Let X be a topological space, and given $x \in X$ , let $c _ { x }$ denote the constant path $c _ { x } : I \to X$ carrying all of I to the point x. Let f be a path in X from $\scriptstyle { \mathcal { X } } _ { 0 }$ to $x _ { 1 }$ . Prove, by constructing a specific path homotopy between them, both algebraically and with a diagram, that

$$
[ f ] * [ e _ { x _ { 1 } } ] = [ f ] { \mathrm { ~ a n d ~ } } [ e _ { x _ { 0 } } ] * [ f ] = [ f ]
$$

5. Let $F : X \times I \to Y$ be a continuous map. Let $x _ { 0 } ~ \in ~ X$ and let $\alpha : I  Y$ by $\alpha ( t ) = F ( x _ { 0 } , t )$ . Prove that the following diagram of maps commutes.

$$
\pi _ { 1 } ( X , x _ { 0 } ) { \overset { F ( x , 1 ) * \quad \pi _ { 1 } ( Y , F ( x _ { 0 } , 1 ) ) } { \longleftrightarrow } }
$$

6Compute the fundamental group, with proof, of the spaces $X , Y$ below.

(a) Let X be the space which is two copies of the cylinder $S ^ { 1 } \times I$ identified at one point.

(b) Let Y be the space $B ^ { 2 }$ with 2 points on its boundary identified.

7.(a) Show that if a space X deformation retracts to a point $x \in X ,$ then for each neighborhood U of x in X, there exists a neighborhood $V \subset U$ of x such that the inclusion $V \hookrightarrow U$ is nullhomotopic.

(b) Let X be the subspace of $\scriptstyle \mathbf { R } ^ { 2 }$ consisting of the horizontal line segment $[ 0 , 1 ] \times \{ 0 \}$ together with all the vertical segments $\{ r \} \times$ $[ 0 , 1 - r ]$ for r a rational number in $[ 0 , 1 ]$ . Show that $X$ deformation retracts to any point in the segment $[ 0 , 1 ] \times \{ 0 \}$ , but not to any other point.

8. (a) State the Lebesgue number lemma.

b) Let $p : E  B$ be a covering map, and let $p ( e _ { 0 } ) = b _ { 0 }$ . Prove that any path $f : [ 0 , 1 ] \to B$ beginning at $b _ { 0 }$ has a unique lifting to a path $\tilde { f }$ in E beginning at $e _ { 0 }$

(c) Prove that the x-axis is a closed subspace of X whose subspace topology is the discrete topology.

(d) State the Tietze extension theorem, and use it to prove that X is not normal.

## (10) Distinguishing surfaces

Let $T _ { n }$ be the n-fold torus (sometimes known as the torus with n holes), where $n \geq 1$ . Prove that if p and q are distinct natural numbers, then $T _ { p }$ and $\textstyle { \mathcal { T } } _ { q }$ are not homeomorphic.

# TOPOLOGY PRELIMINARY EXAM June 30, 2003

s.  o Te eually, though they may dffrn difficulty.Donot spend tomuc time onone problem. You must be the judge of how much detail to provide.

.

2. The fundamental group $\pi _ { 1 } ( X , x )$ is the set of pointed homotopy classses of pointed maps from $( S ^ { 1 } , p )$ to $( X , x )$ . What is the set of pointed homotopy classes of pointed maps from $( \bar { S } ^ { 0 } , p )$ to $( X , x )$ , where $S ^ { 0 }$ is the two-point set $\{ - 1 , 1 \}$ and $p = 1 \mathord { ? }$

Characterize eac of the following, in terms f familiar propertis.rovide sme justification for your answers.)

Subspaces of Hausdorff spaces.

subcover".)

Subspaces of compact Hausdorff spaces.

Open subspaces of compact Hausdorff spaces.

Subspaces of path-connected spaces.

4(a) List all the compact connected surfaces, up to homeomorphism.

How do you know that no two of the surfaces you have listed are homeomorphi? (Some details, please)

The coned sum a torus nd a projectiv plan s a copact coe su To which element of your list is it homeomorphic, and how do you know?

hohat e $T _ { 1 }$ space X is either $\mathbf { \nabla } \cdot \leq 1 \ \mathrm { o r } \geq 2 ^ { \aleph _ { 0 } }$

Describe the universal cover of the Klein bottle.

7.Let X be locally compact and Hausdorff.

a)Describe the one-point compactification $X ^ { \ast }$ of X.

Suppos lso that Xhas a cntable baseSho ha $X ^ { \ast }$ is metrizable.

8Suppose that X is simply connected. Provide a proof of, or counterexample to, the following statement: Every map from X to the circle S1 is homotopic to a constant map. $S ^ { 1 }$

The Sorgenfry line S is the set of real numbers with $\{ [ a , b ) \mid a < b \}$ as base for the topology.

Explain why S is completely regular, separable, Lindelöf, not metrizable. 1

Explain why $s \times s$ is separable, not normal, not Lindelöf.

10. Prove: If $f \colon X \to Y$ is continuous and onto, with X compact Hausdorff and Y Hausdorff, then there is a subspace F of X minimal among compact subspaces Z of X with $\begin{array} { r } { f ( Z ) = Y . } \end{array}$