problem sheets
2. The fundamental group and group actions
The fundamental group of the circle
1. Prove that S2minus two points is homotopy-equivalent to S1, and that S3minus two points is
homotopy-equivalent to S2. Show that S2and S3are not homeomorphic.
2. By choosing suitable explicit identifications of the fundamental group of S1with Z, identify the
induced maps f∗of the following maps f:S1→S1, as maps Z→Z:
(a). The map z7→ zn;
(b). The antipodal map;
(c). The map eiθ 7→ e2πi sin θ.
3. A map f:S1→C− {0}is given by the formula z7→ 8z4+4z3+ 2z2+z−1. What is the winding
number of this map about the origin, and why?
4. Identify the fundamental group of the M¨obius strip M, based at the point [(1,1
2)]. What class
does the boundary curve of Mrepresent in this group?
5. Recall that any path γ:x→yin a space Xinduces an isomorphism γ#:π1(X, x)→π1(X, y).
If Xis the torus S1×S1and γis a loop based at some point x0, identify the induced map γ#.
Group actions
6. Define the torus Tas the quotient of R2by the action of the integral translation group Z2.
(a) Show that the space X– a solid hexagon with sides glued in pairs according to the labelled
arrows shown below – is homeomorphic to T. (You don’t have to write down explicit formulae; a
clear description of the homeomorphism is enough.)
a
a
b
bc
c
(b) Let x0∈Tbe the image of 0 ∈R2. Show that there is an order 6 homeomorphism f:T→T
which fixes x0. With respect to the standard identification π1(T , x0)∼
=Z2, describe the induced
map f∗:π1(T, x0)→π1(T , x0) as a 2 ×2 integer matrix, and comment on its determinant.
7. The Klein bottle Kis formed from a square by the following identifications.
a
bb
a
(a). Construct a covering map from R2to K, and use it to show that π1(K) is isomorphic to a
group whose elements are pairs (m, n) of integers, with group operation given by (m, n)∗(p, q) =
(m+ (−1)np, n +q).
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