Download 2008 Final Exam Answers

Final Examination
Maths 7510
Mon., 8 Dec., 2008
Read each question carefully. Organize your answers clearly. Write your answers using
complete sentences starting on a right hand page of the blue book.
Problem 1.
(1) Prove that (0, 1] is not a compact subspace of R. Ans: (1/n, 1] is
a cover that does not have any finite refinement (or part (2)).
(2) Prove that a compact subspace of a Hausdorff space is closed. Ans: (Thm 26.3)
Let A be the compact subset. For each point x not in A and each a ∈ A choose
disjoint open sets, a ∈ Ua . By compactness there is a finite refinement and the
intersection of the corresponding open sets abour x yields an open set about x not
meeting A. Hence the complement is open. Q.E.D.
Problem 2. Let p : X → Y be a continuous function between topological spaces.
(1) Show that if there is a continuous map f : Y → X such that p ◦ f is the identity on
Y then p is a quotient map. Ans: (Homework: Sect. 22: 2) If p−1 (U ) is open,
then f −1 (p−1 (U )) = U is open by continuity.
(2) Show that a retraction is a quotient map.
Ans: Apply the first part to the
inclusion composed with the retraction.
Problem 3. Consider the cartesian product X × Y of two topological spaces
(1) Describe a base for the topology for the cartesian product. Ans: The base is the
set of products of an open set in X with an open set in Y .
(2) Prove that the cartesian product is Hausdorff if both X and Y are Hausdorff.
Ans: (Homework: Sect. 17: 11) Let (xi , yi ) be the coordinates of two points. One
of the coordinates must differ, say xi . Let Ui be disjoint open sets containing xi .
Then Ui × Y are dijoint open sets containing the original points.
(3) Prove X is Hausdorff if and only the diagonal ∆ = {(x, x)} is closed in X × X.
Ans: See midterm answers.
Problem 4.
(1) Show that a contractible space is path connected. Ans: The image
of a point under the homotopy from the identity to a constant map is a path to
the image of the constant map. Composing any two gives a path between the two
points.
(2) If X is contractible and Y is path connected show the homotopy classes of maps,
[X, Y ], has a single element. Ans: (Homework: Sect. 52: 3) The contraction
homotopy implies any map from X to Y is a homotopic to a constant map into Y .
By path connectivity, any two constant maps into Y are homotopic.
Problem 5. Let p : E → B be a covering map with B connected.
(1) Show that the cardinality of p−1 (b) is constant on B. Ans: (Homework: Sect.
54: 3) Note first that the cardinality of the fibre is constant on the evenly covered
open set about any point b. Next show the set {b : #(p−1 (b)) = #(p−1 (b0 ))} is also
closed by using a evenly covered open set about any limit point. As B is connected
this set is all of B.
1
2
(2) Show that p : S 1 → S 1 with p(z) = z 2 is a covering. Ans: Any interval with angle
measure less than π has image with angle measure less than 2π hence injective.
Cover the circle with 3 such intervals, which are then evenly covered.
(3) Determine the cardinality of p−1 (1) for this covering. Ans: The solutions of
z 2 = 1 are ±1 hence the cover is 2-fold.
Problem 6. Let p : E → B be a covering map.
(1) State the path lifting theorem. Ans: Given a continuous path: γ : [0, 1] → B
with γ(0) = b0 . Then is a unique path γ̂ : [0, 1] → E with γ̂(0) = e0 with p ◦ γ̂ = γ.
(2) Prove the uniqueness of path lifting. Ans: Show that the set of points where two
lifting agree is connected.
(3) If B is simply connected and E is path connected show that p is a homeomorphism.
Ans: If π1 (B, b0 ) is finite, the number of points in the fibre is the number of
elements in the fundamental group. The map p is then an injective and surjective
maps of sets. As p is a local homeomorphism, the set inverse is continuous, hence
p is a homeomorphism.
Problem 7. Let M be the Moebius band, [0, 1] × [−1, 1]/(0, t) ∼ (1, −t).
(1) State the Seifert-van-Kampen Theorem. Ans: If {Aα } is a finite covering of pathconnected open sets each containing the base points and the intersection of any two
is path connected, then the free product of {π1 (Aα , x0 )} surjects onto π1 (ΣX). If
the triple intersections are path connected then the kernel is the normal subgroup
generated by ιαβ (ω)ιβα (ω)−1 .
(2) Using the decomposition of the real projective plane as the union of a M and a twodisc identified along their boundary circles, compute its fundamental group. Ans:
M deformation retracts to the central circle hence has infinite cyclic fundamental
group. Note that the map from the circle boundary to the central circle is the twofold covering map. The fundamental group of the disc is trivial. By the first part
of S-v-K, the fund. group of the projective plane has a surjection from the infinite
cyclic group. Let U1 be RP 2 with the central circle of M removed and U2 be RP 2
with the center of disk removed. (Both deform to M and D2 respectively). Next
observe that ι12 is the degree two map. ι21 is the trivial homomorphism. Hence the
normal subgroup is generated by even multiples of the infinite cyclic (free product)
and the fundamental group is cyclic of order two.
Problem 8.
(1) Determine the fundamental group of the join of two cones by their
cone points. Ans: This space is contractible to the cone point.
(2) Use Seifert-van-Kampen to determine the fundamental group of a suspension of a
path connected space. Ans: Let Ai be the suspension with one of the suspension
points removed. The intersection retracts to X and is path connected, so S-v-K
applies. Each Ai deforms to the cone on X hence has trivial fundamental group.
Hence, π1 (ΣX)is trivial since it is the surjective image of the free product of two
trivial groups (which is trivial).