### 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 B3 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 Bn?

8. Find a presentation of the pure braid group Pn by generators and relations.

9. Prove that the center of Bn is contained in Pn for n>2.

10. Prove that the restriction of the forgetting the last string projection Pn -> Pn-1 to the center of Pn is injective.

11. Prove that the center of Bn 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?