Prof. Rahul Pandharipande
ETH Zürich
Algebraic Topology I
HS 2014
Problem set 1
CW complexes, homotopy equivalence,
mapping cones
1. Let X be a contractible space and let r : X → A be a retraction. Show
that also A is contractible.
2. Let f, g : X → S n be maps such that f (x) 6= −g(x) for all x ∈ X.
Show that f is homotopic to g.
3. Consider the subsets of R3
A = {(x, y, z) ∈ R3 |x2 + y 2 + z 2 = 1}
B = {(x, y, z) ∈ R3 |x = y = 0, −1 ≤ z ≤ 1}
C = {(x, y, z) ∈ R3 |x2 + y 2 ≤ 1, z = 0}.
Consider moreover the spaces X = A ∪ B and Y = A ∪ C.
(a) Draw a picture of the spaces.
(b) Define a CW structure on X and Y .
(c) Use [Hatcher, Proposition 0.17] to prove that X ' S 2 ∨ S 1 and
Y ' S 2 ∨ S 2.
4. Recall that the Klein bottle K can be obtained from a square identifying the sides as shown in the first figure. We also consider the subspace
X ⊂ R3 formed by a Klein bottle intersecting itself in a circle, as shown
in the second figure.
1
>
∨
∨
∨
∨
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(a) Draw in the square the curve γ that corresponds to the self intersection of X.
(b) Define a CW structure on the Klein bottle and on X
(c) Prove that X is homotopy equivalent to S 1 ∨ S 1 ∨ S 2 .
5. Recall that if f : X → Y is a continuous map, the mapping cone of f ,
Cf is the quotient space
Cf = (X × I t Y )/{(x1 , 0) ∼ (x2 , 0), (x, 1) ∼ f (x)}.
Consider the map f : S 1 → S 1 , e2πit 7→ e4πit (that can also be expressed
as z 7→ z 2 ). Prove that Cf is homeomorphic to RP 2 .
Extra exercise (to be discussed in the exercise class)
(a) Show that if a space X deformation retracts to a point x ∈ X, then for
each neighborhood U of x in X there exists a neighborhood V ⊂ U of
x such that the inclusion map V ,→ U is nullhomotopic.
(b) Let X be the subspace of R2 consisting of the horizontal segment [0, 1]×
{0} together with all the vertical segments {r} × [0, 1 − r] for every
rational number r ∈ [0, 1]. Show that X deformation retracts to any
point in the segment [0, 1] × {0} but not to any other point. Hint:
Exercise 2.
(c*) Let Y be the subspace of R2 that is the union of an infinite number
of copies of X arranged as in the picture below. Show that Y is contractible, but doesn’t deformation retract onto any point. Hint: show
that the map that moves each point at unit speed along the segment
it belongs to towards the right is continuous.
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