Topology Final
Math 131 – Harvard University – Fall 2013
Due Tuesday, 10 December 2013, 12:00 pm
Hand in your completed final to the staff in room 325 Science Center.
Name:
Aim for concise, clear answers. Refer only to Munkres, your class notes and
the course notes online. Do not collaborate; all work must be done on your
own. Do not consult any internet resources, except the course web page.
Part I. Do any 7 of the following 8 problems. Write the answer to each
problem on a separate sheet, and attach it to Part II.
1. Let fn: [0,1] →Rbe a sequence of polynomials such that lim fn(x) =
1 if x= 1/2 and lim fn(x) = 0 otherwise. Prove that
sup
n
sup
x∈[0,1]
|f0
n(x)|=∞.
Give an example of such a sequence.
2. Show that the continuous functions C([0,1]) are dense in the space
R[0,1] with the product topology. Given an example of a space Xsuch
that C(X) is not dense in RX.
3. Let E , B and Xbe reasonable, connected spaces, let p: (E, e)→(B , b)
be a covering map, and let f: (X, x)→(B, b) a continuous map. We
say all lifts exist if for every e0∈Ewith p(e0) = b, there exists a
map F: (X, x)→(E, e0) satisfying p◦F=f. Give a necessary and
sufficient condition, in terms of fundamental groups, such that all lifts
exist.
4. Let X∼
=R2/Z2be the torus, with basepoint x. Let f: (X, x)→(X, x)
be continuous map, and suppose f∗gives the identity map on π1(X, x).
Prove that fis homotopic to the identity map on X.
5. Prove there is no retraction of the M¨obius band onto its boundary.
6. Let [f]∈π1(B , b) be represented by a path f: [0,1] →Bthat begins
and ends at b. Let g: [0,1] →[0,1] be a homeomorphism. Prove that
[f◦g]=[f] or [f]−1in π1(B, b).
7. Compute the fundamental group of X=RP2∨RP2. (Here RP2is the
real projective plane.) Draw a picture of the universal cover of X.
8. Let N⊂F2be a normal subgroup such that F2/N ∼
=Z. Prove that
Ncannot be finitely generated.
