MAT1301 Assignment 1 Due date: Friday, Jan 29, 2010 1. Let X, Y ...

MAT1301 Assignment 1
Due date: Friday, Jan 29, 2010
1. Let X, Y, U, V be the subsets of R 2 pictured below. Let f : X → Y be
the fold map (i.e. the map whose restriction to each circle is the identity).
Let g : U → V be the map which in polar coordinates doubles the
angle while leaving the radius fixed (i.e. re iθ ↦→ re 2iθ as complex numbers).
Determine whether f and g are covering projections. Compute
the fundamental group of the space V .
(Note: the spaces include only the solid lines shown, not the regions
they surround.)
2. Let p : R → S 1 be given by p(t) = e it . Discuss whether every map
f : RP 2 → S 1 can be lifted to a map ˜ f : RP 2 → R such that f = p◦ ˜ f.
3. Describe the fundamental groups and the universal covers of the following
spaces:
(a) The torus S 1 × S 1
(b) The punctured torus X = T 2 \ {p}
(c) The real projective plane RP 2
(d) The punctured projective plane RP 2 − {p}
4. Find the fundamental groups of the following spaces.
(a) S 2 with two points removed
(b) Klein bottle with one point removed
(c) Torus with two points removed
5. Let X ⊂ R 3 be the union of S 2 with a chord joining the north and
south poles. Describe the universal covering space of X and compute
π1(X).
6. Show that the torus is a 2-fold covering of the Klein bottle. Compute
the fundamental group of the Klein bottle.
7. Compute the fundamental group of the ’dunce cap’ space obtained by
taking a triangle (including the interior) and identifying the boundary
sides.
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8. Read the algorithm in Hatcher (p. 55 no. 22) describing how to compute
the fundamental group of a knot complement when the knot is
described by the Wirtinger presentation. Apply the method to compute
the fundamental group of the complement of a trefoil knot. You
do not have to answer Hatcher p. 55 no. 22.
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