problem sheets
4. Covering spaces
Basic theory
1. Show that neither the torus nor RP2can cover the other.
2. The closed orientable surfaces Σgof genus gare drawn below. Construct a 5-sheeted cover
Σ11 →Σ3and a 3-sheeted cover Σ4→Σ2, and find their groups of deck transformations.
Σ2Σg
3. Show that the torus is a double (2-sheeted) cover of the Klein bottle. Can it be a triple cover?
4. Let Gbe a (path-connected, locally path-connected) topological group with multiplication and
inversion maps m, i. Using existence and uniqueness of liftings, show that any path-connected cover
˜
Gof Ghas a unique (modulo choice of basepoint living over the identity element of G) topological
group structure such that ˜
G→Gis a homomorphism.
5. Consider the group Znacting on Rnby integer translations, so that the quotient is the n-torus
Tn. Use this covering space to prove that the homotopy groups πi(Tn) vanish when i≥2.
Constructing covers
6. Consider the CW-complex Θ made by taking two vertices and attaching three edges between
them so as to form the shape of the letter theta. Draw the the universal cover of Θ and justify
your answer.
7. For each of the following pictures of a cover of S1∨S1, write down generators for their corre-
sponding subgroups in the free group ha, bi.
b
a
a
b
a
b
b
b
a
b
a
Conversely, for each of the following subgroups of < a, b >, draw the corresponding cover:
ha2, b2,(ab)2i ha2, b2, aba−1, bab−1i ha, b3i h{ambna−mb−n:∀m, n}i h{a2mb2na−2mb−2n:∀m, n}i.
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