169169 vertical-align : middle;
170170 }
171171 /* Exam problem styling */
172- .problem {
173- margin-bottom : 1.5em ;
174- }
175172 .problem-number {
176173 display : block;
177174 margin-top : 1.5em ;
178175 margin-bottom : 0.5em ;
179176 }
180- .subproblem {
181- margin-left : 2em ;
182- margin-top : 0.5em ;
183- display : block;
184- }
185177 </ style >
186178</ head >
187179< body >
204196</ nav >
205197
206198< main >
207- < h1 id ="topology-general-exam-fall-2025 "> TOPOLOGY GENERAL EXAM FALL 2025</ h1 >
199+ < h1 class =" unnumbered " id ="topology-general-exam-fall-2025 "> TOPOLOGY GENERAL EXAM FALL 2025</ h1 >
208200< p > Instructions: This is a four hour exam. Your solutions should be
209201legible and clearly organized, written in complete sentences in good
210202mathematical style on your own paper. All work should be your own-no
211203outside sources are permitted-using methods and results from the first
212204year topology courses. Each problem is worth the same number of
213- points.</ p >
214-
215- < div class ="problem ">
216- < p > (1) Let
217- < math role ="math " aria-label ="X=S^{1} \times \mathbb{R} P^{2} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > X</ mi > < mo > =</ mo > < msup > < mi > S</ mi > < mn > 1</ mn > </ msup > < mo > ×</ mo > < mi > ℝ</ mi > < msup > < mi > P</ mi > < mn > 2</ mn > </ msup > </ mrow > < annotation encoding ="application/x-tex "> X=S^{1} \times \mathbb{R} P^{2}</ annotation > </ semantics > </ math > .< br /> < span class ="subproblem "> (a) Compute
218- < math role ="math " aria-label ="\pi_{1}(X) " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < msub > < mi > π</ mi > < mn > 1</ mn > </ msub > < mo stretchy ="false " form ="prefix "> (</ mo > < mi > X</ mi > < mo stretchy ="false " form ="postfix "> )</ mo > </ mrow > < annotation encoding ="application/x-tex "> \pi_{1}(X)</ annotation > </ semantics > </ math > .</ span > < br /> < span class ="subproblem "> (b) Describe the universal cover of
205+ points.< br /> < br /> < span class ="problem-number "> (1)</ span > Let
206+ < math role ="math " aria-label ="X=S^{1} \times \mathbb{R} P^{2} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > X</ mi > < mo > =</ mo > < msup > < mi > S</ mi > < mn > 1</ mn > </ msup > < mo > ×</ mo > < mi > ℝ</ mi > < msup > < mi > P</ mi > < mn > 2</ mn > </ msup > </ mrow > < annotation encoding ="application/x-tex "> X=S^{1} \times \mathbb{R} P^{2}</ annotation > </ semantics > </ math > .< br />
207+ (a) Compute
208+ < math role ="math " aria-label ="\pi_{1}(X) " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < msub > < mi > π</ mi > < mn > 1</ mn > </ msub > < mo stretchy ="false " form ="prefix "> (</ mo > < mi > X</ mi > < mo stretchy ="false " form ="postfix "> )</ mo > </ mrow > < annotation encoding ="application/x-tex "> \pi_{1}(X)</ annotation > </ semantics > </ math > .< br />
209+ (b) Describe the universal cover of
219210< math role ="math " aria-label ="X " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mi > X</ mi > < annotation encoding ="application/x-tex "> X</ annotation > </ semantics > </ math >
220211and explicitly describe each element in the group of deck
221- transformations of the universal cover.</ span > < br /> < br /> < span class ="problem-number "> (2)</ span > Let
212+ transformations of the universal cover.< br /> < br /> < span class ="problem-number "> (2)</ span > Let
222213< math role ="math " aria-label ="x_{1}, x_{2}, \ldots, x_{10} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < msub > < mi > x</ mi > < mn > 1</ mn > </ msub > < mo > ,</ mo > < msub > < mi > x</ mi > < mn > 2</ mn > </ msub > < mo > ,</ mo > < mi > …</ mi > < mo > ,</ mo > < msub > < mi > x</ mi > < mn > 10</ mn > </ msub > </ mrow > < annotation encoding ="application/x-tex "> x_{1}, x_{2}, \ldots, x_{10}</ annotation > </ semantics > </ math >
223214be 10 distinct points on the 2 -torus
224215< math role ="math " aria-label ="T^{2} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < msup > < mi > T</ mi > < mn > 2</ mn > </ msup > < annotation encoding ="application/x-tex "> T^{2}</ annotation > </ semantics > </ math > .
225216Let
226217< math role ="math " aria-label ="X " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mi > X</ mi > < annotation encoding ="application/x-tex "> X</ annotation > </ semantics > </ math >
227218be the quotient of
228219< math role ="math " aria-label ="T^{2} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < msup > < mi > T</ mi > < mn > 2</ mn > </ msup > < annotation encoding ="application/x-tex "> T^{2}</ annotation > </ semantics > </ math >
229- obtained by identifying all 10 points.< br /> < span class ="subproblem "> (a) Compute
230- < math role ="math " aria-label ="\pi_{1}(X) " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < msub > < mi > π</ mi > < mn > 1</ mn > </ msub > < mo stretchy ="false " form ="prefix "> (</ mo > < mi > X</ mi > < mo stretchy ="false " form ="postfix "> )</ mo > </ mrow > < annotation encoding ="application/x-tex "> \pi_{1}(X)</ annotation > </ semantics > </ math > .</ span > < br /> < span class ="subproblem "> (b) Compute
220+ obtained by identifying all 10 points.< br />
221+ (a) Compute
222+ < math role ="math " aria-label ="\pi_{1}(X) " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < msub > < mi > π</ mi > < mn > 1</ mn > </ msub > < mo stretchy ="false " form ="prefix "> (</ mo > < mi > X</ mi > < mo stretchy ="false " form ="postfix "> )</ mo > </ mrow > < annotation encoding ="application/x-tex "> \pi_{1}(X)</ annotation > </ semantics > </ math > .< br />
223+ (b) Compute
231224< math role ="math " aria-label ="H_{n}(X) " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < msub > < mi > H</ mi > < mi > n</ mi > </ msub > < mo stretchy ="false " form ="prefix "> (</ mo > < mi > X</ mi > < mo stretchy ="false " form ="postfix "> )</ mo > </ mrow > < annotation encoding ="application/x-tex "> H_{n}(X)</ annotation > </ semantics > </ math >
232225for all
233- < math role ="math " aria-label ="n \geq 0 " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > n</ mi > < mo > ≥</ mo > < mn > 0</ mn > </ mrow > < annotation encoding ="application/x-tex "> n \geq 0</ annotation > </ semantics > </ math > .</ span > < br /> < br /> < span class ="problem-number "> (3)</ span > (a) Let
226+ < math role ="math " aria-label ="n \geq 0 " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > n</ mi > < mo > ≥</ mo > < mn > 0</ mn > </ mrow > < annotation encoding ="application/x-tex "> n \geq 0</ annotation > </ semantics > </ math > .< br /> < br /> < span class ="problem-number "> (3)</ span > (a) Let
234227< math role ="math " aria-label ="f: S^{1} \rightarrow \mathbb{R} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > f</ mi > < mo > :</ mo > < msup > < mi > S</ mi > < mn > 1</ mn > </ msup > < mo > →</ mo > < mi > ℝ</ mi > </ mrow > < annotation encoding ="application/x-tex "> f: S^{1} \rightarrow \mathbb{R}</ annotation > </ semantics > </ math >
235228be a continuous map. Show that there exists some
236229< math role ="math " aria-label ="x \in S^{1} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > x</ mi > < mo > ∈</ mo > < msup > < mi > S</ mi > < mn > 1</ mn > </ msup > </ mrow > < annotation encoding ="application/x-tex "> x \in S^{1}</ annotation > </ semantics > </ math >
237230such that
238- < math role ="math " aria-label ="f(x)=f(-x) " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > f</ mi > < mo stretchy ="false " form ="prefix "> (</ mo > < mi > x</ mi > < mo stretchy ="false " form ="postfix "> )</ mo > < mo > =</ mo > < mi > f</ mi > < mo stretchy ="false " form ="prefix "> (</ mo > < mi > −</ mi > < mi > x</ mi > < mo stretchy ="false " form ="postfix "> )</ mo > </ mrow > < annotation encoding ="application/x-tex "> f(x)=f(-x)</ annotation > </ semantics > </ math > .< br /> < span class ="subproblem "> (b) Show that any continuous map
231+ < math role ="math " aria-label ="f(x)=f(-x) " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > f</ mi > < mo stretchy ="false " form ="prefix "> (</ mo > < mi > x</ mi > < mo stretchy ="false " form ="postfix "> )</ mo > < mo > =</ mo > < mi > f</ mi > < mo stretchy ="false " form ="prefix "> (</ mo > < mi > −</ mi > < mi > x</ mi > < mo stretchy ="false " form ="postfix "> )</ mo > </ mrow > < annotation encoding ="application/x-tex "> f(x)=f(-x)</ annotation > </ semantics > </ math > .< br />
232+ (b) Show that any continuous map
239233< math role ="math " aria-label ="g: \mathbb{R} P^{2} \times \mathbb{R} P^{2} \rightarrow T^{2} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > g</ mi > < mo > :</ mo > < mi > ℝ</ mi > < msup > < mi > P</ mi > < mn > 2</ mn > </ msup > < mo > ×</ mo > < mi > ℝ</ mi > < msup > < mi > P</ mi > < mn > 2</ mn > </ msup > < mo > →</ mo > < msup > < mi > T</ mi > < mn > 2</ mn > </ msup > </ mrow > < annotation encoding ="application/x-tex "> g: \mathbb{R} P^{2} \times \mathbb{R} P^{2} \rightarrow T^{2}</ annotation > </ semantics > </ math >
240- is nullhomotopic.</ span > < br /> < br /> < span class ="problem-number "> (4)</ span > Let (
234+ is nullhomotopic.< br /> < br /> < span class ="problem-number "> (4)</ span > Let (
241235< math role ="math " aria-label ="C, d " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > C</ mi > < mo > ,</ mo > < mi > d</ mi > </ mrow > < annotation encoding ="application/x-tex "> C, d</ annotation > </ semantics > </ math >
242236) and (
243237< math role ="math " aria-label ="C^{\prime}, d^{\prime} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < msup > < mi > C</ mi > < mi > ′</ mi > </ msup > < mo > ,</ mo > < msup > < mi > d</ mi > < mi > ′</ mi > </ msup > </ mrow > < annotation encoding ="application/x-tex "> C^{\prime}, d^{\prime}</ annotation > </ semantics > </ math >
@@ -263,13 +257,14 @@ <h1 id="topology-general-exam-fall-2025">TOPOLOGY GENERAL EXAM FALL 2025</h1>
263257< math role ="math " aria-label ="f_{c} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < msub > < mi > f</ mi > < mi > c</ mi > </ msub > < annotation encoding ="application/x-tex "> f_{c}</ annotation > </ semantics > </ math >
264258transverse to the surface
265259< math role ="math " aria-label ="S " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mi > S</ mi > < annotation encoding ="application/x-tex "> S</ annotation > </ semantics > </ math >
266- ?< br /> < span class ="subproblem "> (b) For what values of
260+ ?< br />
261+ (b) For what values of
267262< math role ="math " aria-label ="c \in \mathbb{R} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > c</ mi > < mo > ∈</ mo > < mi > ℝ</ mi > </ mrow > < annotation encoding ="application/x-tex "> c \in \mathbb{R}</ annotation > </ semantics > </ math >
268263is the intersection of the image of
269264< math role ="math " aria-label ="f_{c} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < msub > < mi > f</ mi > < mi > c</ mi > </ msub > < annotation encoding ="application/x-tex "> f_{c}</ annotation > </ semantics > </ math >
270265with
271266< math role ="math " aria-label ="S " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mi > S</ mi > < annotation encoding ="application/x-tex "> S</ annotation > </ semantics > </ math >
272- a manifold?</ span > < br /> < br /> < span class ="problem-number "> (6)</ span > Consider the 3 -form</ p >
267+ a manifold?< br /> < br /> < span class ="problem-number "> (6)</ span > Consider the 3 -form</ p >
273268< p > < math role ="math " aria-label ="\theta=e^{w}(x d y \wedge d z+y d z \wedge d x+z d x \wedge d y) " display ="block " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > θ</ mi > < mo > =</ mo > < msup > < mi > e</ mi > < mi > w</ mi > </ msup > < mo stretchy ="false " form ="prefix "> (</ mo > < mi > x</ mi > < mi > d</ mi > < mi > y</ mi > < mo > ∧</ mo > < mi > d</ mi > < mi > z</ mi > < mo > +</ mo > < mi > y</ mi > < mi > d</ mi > < mi > z</ mi > < mo > ∧</ mo > < mi > d</ mi > < mi > x</ mi > < mo > +</ mo > < mi > z</ mi > < mi > d</ mi > < mi > x</ mi > < mo > ∧</ mo > < mi > d</ mi > < mi > y</ mi > < mo stretchy ="false " form ="postfix "> )</ mo > </ mrow > < annotation encoding ="application/x-tex "> \theta=e^{w}(x d y \wedge d z+y d z \wedge d x+z d x \wedge d y)</ annotation > </ semantics > </ math > </ p >
274269< p > on
275270< math role ="math " aria-label ="\mathbb{R}^{4} " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < msup > < mi > ℝ</ mi > < mn > 4</ mn > </ msup > < annotation encoding ="application/x-tex "> \mathbb{R}^{4}</ annotation > </ semantics > </ math >
@@ -282,7 +277,7 @@ <h1 id="topology-general-exam-fall-2025">TOPOLOGY GENERAL EXAM FALL 2025</h1>
282277with the orientation induced as being part of the boundary of the 4
283278-ball with the standard orientation. Compute</ p >
284279< p > < math role ="math " aria-label ="\int_{S} \mathrm{~d} \theta . " display ="block " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < msub > < mo > ∫</ mo > < mi > S</ mi > </ msub > < mrow > < mspace width ="0.222em "> </ mspace > < mi mathvariant ="normal "> d</ mi > </ mrow > < mi > θ</ mi > < mi > .</ mi > </ mrow > < annotation encoding ="application/x-tex "> \int_{S} \mathrm{~d} \theta .</ annotation > </ semantics > </ math > </ p >
285- < p > (7) Let
280+ < p > < br /> < br /> < span class =" problem-number " > (7)</ span > Let
286281< math role ="math " aria-label ="p: M \rightarrow N " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mrow > < mi > p</ mi > < mo > :</ mo > < mi > M</ mi > < mo > →</ mo > < mi > N</ mi > </ mrow > < annotation encoding ="application/x-tex "> p: M \rightarrow N</ annotation > </ semantics > </ math >
287282be a smooth covering map between closed manifolds of some dimension
288283< math role ="math " aria-label ="n " display ="inline " xmlns ="http://www.w3.org/1998/Math/MathML "> < semantics > < mi > n</ mi > < annotation encoding ="application/x-tex "> n</ annotation > </ semantics > </ math > .
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