## Classical Geometry - Lecture Notes by Calegari D. By Calegari D.

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Let Σ be a closed surface. The mapping class group of Σ, denoted MC(Σ), is defined to be the quotient group MC(Σ) = Homeo(Σ)/Homeo0 (Σ) where Homeo0 (Σ) denotes the normal subgroup of self–homeomorphisms of Σ which are homotopic to the identity. 55. Show Homeo0 (Σ) is a normal subgroup of Homeo(Σ). Notice that Homeo(Σ) acts on MH(Σ) by ψ : (f, Σ ) → (ψ ◦ f, Σ ) Moreover, if ψ ∈ Homeo0 (Σ), then ψ(f, Σ ) ∼ (f, Σ ) 36 DANNY CALEGARI with respect to the equivalence relation defined on representatives.

10. Isometries are frequently too restrictive for many circumstances; a typical metric space of study might admit no non–trivial isometries at all. We can enrich the structure by allowing as morphisms those maps which, though they don’t literally preserve distances between points, at least don’t increase distances between points by too much. Such a map is called a Lipschitz map, and metric spaces with these as morphisms define a category LIP which is in many ways a much more interesting object than MET.

1x m = m for any m with t(m) = x m1x = m for any m with s(m) = x The composition of two morphisms is another morphism, and this composition is associative; composition can be expressed as a function c : M × M → M. That is, c satisfies c(m, c(n, r)) = c(c(m, n), r) for any m, n, r ∈ M Sometimes a category is written as a 5–tuple (O, M, s, t, c), but in practice it is frequently sufficient to specify the objects and the morphisms. A category is something like a class in an object–oriented programming language like C++; one defines at the same time the data types (the objects in the category) and the admissible functions which operate on them (the morphisms).