problem sheets
3. Van Kampen’s theorem
1. Recall from sheet 2 that the torus Tcan be made from a solid hexagon with sides glued in pairs
according to the labelled arrows. Use van Kampen’s theorem to compute a presentation of the
fundamental group, and show that it is isomorphic to Z2.
2. Each of the following pictures represents a solid polygon with its boundaries identified (not
necessarily in pairs) according to the arrows shown. Describe the fundamental group in each case.
a
a
b
b
b b
a
a
aa
a
3. Let Xbe a torus minus a hole. Explain why Xis homotopy-equivalent to S1∨S1. Use the
fundamental group to show that there does not exist a retraction r:X→∂X .
4. Each figure depicts a space made by gluing bands (copies of I×I) to disc(s). For each one,
compute its fundamental group, and the fundamental group of the space made by gluing a disc to
each of its boundary circles.
5. Take a solid cube and identify its faces in pairs, as shown below. (The top and sides are
identified using translation, but the left and right faces get rotated as well as translated.) Compute
the fundamental group of this space.
Q
Q
P
P
R
R
6. Compute the fundamental group of the space obtained by identifying the faces of a solid prism
as shown below.
Q
Q
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