MATH5340 SEMESTER 2, 2023
ASSIGNMENT 2
1. Instructions
Your solutions to Questions 1–4 below should be:
• typeset using LATEX;
• submitted as a pdf file on Canvas, by 11:59pm on Friday 27 October (unless you
have an extension);
• written up independently, although you may discuss the questions with other stu-
dents.
You may use without proof any results from Weeks 5–10 lectures and/or the corresponding
sections of Hatcher; please give a precise reference for these results e.g. “by Proposition 2.9”
or “by a Lemma in Week 6 Lecture 2”.
For any commutative diagrams, I recommend the package tikz-cd. For any other
diagrams, unless you want to spend a lot of time on them, I suggest you draw by hand
and take a photo to include using the graphicx package (or draw by hand on a tablet and
include this file, if you have a suitable device).
This assignment counts for 25% of your final grade. It will be marked out of 50, with 10
marks for each of Questions 1–4 below for mathematical correctness, and the remaining 10
marks (Question 5) for overall quality of exposition. For Question 5, what I’m looking for
includes:
• appropriate use of LATEX e.g. using math mode, displaying math for readability,
cross-referencing, use of sections, use of enumerate or itemize environments;
• writing style e.g. writing in sentences, clear explanations of logical dependencies,
well-structured answers, accuracy, appropriate level of formality, explaining new
notation before it is used, proof-reading;
• notational choices e.g. consistency with lectures, sensible choices;
• appropriate referencing; and
• helpful figures.
If you are in doubt as to how much detail is needed in your solutions, please just ask me.
The basic guidelines are that you shouldn’t say anything that is incorrect e.g. claiming
that two spaces are homeomorphic when they’re not, and you should include enough detail
that I am convinced that you understand the situation.
1
2 ASSIGNMENT 2
2. Questions
(1) Let X, Y and Z be topological spaces which are path-connected and locally path-
connected (as these terms are defined in Hatcher). Consider the following commu-
tative diagram of topological spaces and continuous maps.
X
Y
Z
q
p
r
(a) Prove that if p : X → Y and r : Y → Z are covering spaces, then q : X → Z
is also a covering space.
(b) Prove that if q : X → Z and r : Y → Z are covering spaces, then p : X → Y
is also a covering space.
(c) Now assume that p : X → Y , q : X → Z and r : Y → Z are all covering spaces.
Choose x0 ∈ X and let y0 = p(x0) and z0 = q(x0). Write 1 for the basepoint
of S2 and let f : S2 → Z be a continuous map such that f(1) = z0. Prove
that there exists a unique continuous map g : S2 → Y such that r ◦ g = f and
g(1) = y0, and that g factors through X.
(2) The Baumslag–Solitar group BS(2, 3) is given by the presentation:
BS(2, 3) = ⟨a, t | ta2t−1a−3⟩.
Let X be the 2-dimensional cell complex with π1(X) ∼= BS(2, 3) which corresponds
to this presentation, as constructed in Corollary 1.28 of Hatcher.
(a) Prove that X is a K(BS(2, 3), 1) space.
(b) Prove that every nontrivial element g of BS(2, 3) can be expressed uniquely in
the form
g = an0tε1an1tε1 · · · tεk−1anktεkank+1
where k ≥ 0, for 0 ≤ j ≤ k + 1 we have nj ∈ Z and εj ∈ {±1}, if k = 0 then
n0 ̸= 0, and this expression has no subwords of the form ta2ℓt−1 or t−1a3ℓt for
ℓ ∈ Z.
(c) (i) Verify that there is a homomorphism φ : BS(2, 3) → BS(2, 3) induced
by a 7→ a2 and t 7→ t.
(ii) Show that φ is surjective.
(iii) Show that φ is not injective, by considering φ([a, tat−1]).
(d) Deduce that there is a normal covering space q : Y → X with group of deck
transformations isomorphic to BS(2, 3), such that Y is not contractible.
MATH5340 SEMESTER 2, 2023 3
(3) Let K be the Klein bottle.
(a) Compute the simplicial homology of K, regarding it as a ∆-complex as on
page 102 of Hatcher.
(b) Explain why the simplicial homology of K is independent of the choice of a
∆-complex structure on K.
(4) Let X be a topological space. Recall that the suspension of X is the quotient
SX = (X × [−1, 1])/ ∼
where the equivalence relation ∼ identifies X × {1} to a point and X × {−1} to
another point.
(a) Use the quotient map q : X × [−1, 1]→ SX to construct homomorphisms
φn : Cn(X)→ Cn+1(SX) for all n ≥ 0
such that the following (infinite) diagram commutes:
· · · Cn(X) Cn−1(X) · · · C0(X) 0
· · · Cn+1(SX) Cn(SX) · · · C1(SX) C0(SX)
δ δ
φn
δ
φn−1
δ 0
φ0 0
δ δ δ δ δ
(b) Using part (a) or otherwise, prove that H˜n(X) ∼= H˜n+1(SX) for all n ≥ 0.
(c) Deduce that H˜n(S
n) ∼= Z, for all n ≥ 0.