Topology, MM8002/SF2721, Spring 2017. Exercise set 13
Let {x1 , x2 , ..., xn } be a set. A word in {x1 , x2 , ..., xn } is a sequence x✏i11 x✏i22 ...x✏ikk ...,
where ✏l 2 Z. We can define an equivalence relation on the words in {x1 , x2 , ..., xn },
generated by
✏
✏
✏
✏
k+1
k+1
k+2
k+3
x✏i11 x✏i22 ...x✏ikk xik+1
xik+2
xik+3
... ⇠ x✏i11 x✏i22 ...x✏ikk xik+1
+✏k+2 ✏k+3
xik+3 ...,
when xik+1 = xik+2 and
✏
✏
k+2
k+2
x✏i11 x✏i22 ...x✏ikk x0ik+1 xik+2
... ⇠ x✏i11 x✏i22 ...x✏ikk xik+2
....
Denote by F ({x1 , x2 , ..., xn }) the free group generated by {x1 , x2 , ..., xn }, i.e.
words x✏i11 x✏i22 ...x✏ikk , with group operation the concatenation of words and neutral
element the empty word, modulo the equivalence relation above.
Exercise 1. Show using the universal covering that ⇡1 (S 1 _ S 1 , ⇤) ⇠
= F ({x1 , x2 }).
(This is in fact true for arbitrary wedges of spheres.)
Exercise 2. Describe the coverings of S 1 _ S 1 corresponding to the normal subgroups generated by x21 , x22 and (x1 x2 )4 .
Exercise 3. Show using the classification of coverings that subgroups of (finite)
free groups are free.
Exercise 4. Consider the quotient space X from the sphere S 2 by identifying the
north pole with the south pole. Calculate the fundamental group of X. What
happens if you identify n points on the sphere?
Exercise 5. Exercise 12-8. in Lee.
Other interesting Exercises. Exercises 11-20, 12-3 and 12-12 in Lee.
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