### Exercises

1. Describe geometrically the image of the space of triples of colinear
points under the homeomorphism of the space of
unordered triples of points in the plane considered up to dilations and
translations onto the 3-sphere.

2. Consider the space of unordered pairs of distinct
points on a circle. Find its fundamental group. Does it contain a
natural deformation retract? Identify the topological type of the space.

3. Find the fundamental group of the space of (non-degenerate)
triangles on the plane. Does this space contain a nice deformation
retract with a natural retraction?

4. Formulate the Gram-Schmidt orthogonalization theorem from Linear
Algebra as a theorem about deformation retractions.

5. Find a presentation of the commutator subgroup of the braid group
*B*_{3}
by generators and relations.

6. What is the quotient of *Bn* by its commutator
subgroup.

7. How to decide if a braid of *n* strings presented by a picture
belongs to the commutator subgroup of *B*_{n}?

8. Find a presentation of the pure braid group *P*_{n}
by generators and relations.

9. Prove that the center of *B*_{n} is contained in
*P*_{n} for *n>2*.

10. Prove that the restriction of the forgetting the last string
projection *P*_{n} -> P_{n-1} to the center of
*P*_{n} is injective.

11. Prove that the center of *B*_{n} is infinite cyclic
group generated by the Dehn twist along the circle surrounding all the
strings.

12. Prove that the quotient of the 3 string braid group by the center
is a free product of two non-trivial finite groups. Which ones?

13. Construct a topological space of 5 points with non-abelian
fundamental group. Find its universal covering space.

14. Are the digital circles made of 4 and 8 points homotopy equivalent to
each other? What about digital circles made of 4 and 6 points?