By Calegari D.

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21. Let ∼ be an equivalence relation on X, and let π : X → X/ ∼ be the quotient map to the space of equivalence classes. The quotient topology on X/ ∼ is the topology whose open sets are those U ⊂ X/ ∼ such that π −1 (U ) is open in X. Thus, X/ ∼ has as many open sets as it is allowed subject to the condition that π is continuous. 22. A homeomorphism from X to Y is a continuous map which is invertible and has a continuous inverse. The category of topological manifolds is denoted TOP and has as objects topological manifolds and as morphisms all continuous maps.

46. A pair of pants is the topological surface obtained from a disk by removing two subdisks — that is, a disk with two holes. A pair of pants can also be thought of as a sphere minus three subdisks. The Euler characteristic of a pair of pants is −1. Since the Euler characteristic of its boundary is 0, a surface obtained from glueing n pairs of pants has Euler characteristic −n. So Σg can be decomposed (in many different ways) into 2g − 2 pairs of pants. 47. Show that the number of decompositions of Σg into pairs of pants, up to combinatorial equivalence, is equal to the number of graphs with 2g − 2 vertices with 3 edges at every vertex.

The plane R2 is a metric space with d((x1 , y1 ), (x2 , y2 )) = |x1 − x2 | + |y1 − y2 | This metric is known as the Manhattan metric. Can you see why? 9. An isometry of a metric space X is a 1–1 and onto transformation of X to itself which preserves distances between points. The set of isometries of a space X is a group Isom(X), where multiplication in the group is composition of symmetries, and e is the trivial symmetry which fixes every x in X. This is an example of a group of the form Aut(X) where the relevant structure on X is that of the category of metric spaces MET whose objects are metric spaces and whose morphisms are isometries.

### Classical geometry (lecture notes) by Calegari D.

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