# 02/18/2020: A Primer on Mapping Class Groups (PMS-49) (Princeton Mathematical) (Benson Farb, Dan Margalit)
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Last Annotation: 02/18/2020
## Highlights
- It is defined to be the group of isotopy classes of orientationpreserving diffeomorphisms of S \(that restrict to the identity on ∂S if ∂S = ∅\): Mod\(S\) = Diﬀ + \(S, ∂S\)/ Diﬀ 0 \(S, ∂S\)\. (Benson Farb, Dan Margalit 18)
- Here Diﬀ 0 \(S, ∂S\) is the subgroup of Diﬀ + \(S, ∂S\) consisting of elements that are isotopic to the identity\. (Benson Farb, Dan Margalit 18)
- M\(S\) = Teich\(S\)/ Mod\(S\) is the moduli space of Riemann surfaces homeomorphic to S (Benson Farb, Dan Margalit 18)
- Teich\(S\) = HypMet\(S\)/ Diﬀ 0 \(S\)\. The space Teich\(S\) is a metric space homeomorphic to an open ball\. The group Diﬀ + \(S\) acts on HypMet\(S\) by pullback\. This action descends to an action of Mod\(S\) on Teich\(S (Benson Farb, Dan Margalit 18)
- mathematics (Benson Farb, Dan Margalit 18)
- Mod\(S\) encodes most of the topological features of M\(S\)\. (Benson Farb, Dan Margalit 18)
- any closed curve α on a flat torus is homotopic to a geodesic: one simply lifts α to R2 and performs a straight-line homotopy\. Note that the corresponding geodesic is not unique\. (Benson Farb, Dan Margalit 41)
- Proposition 1\.10 Let α and β be two essential simple closed curves in a surface S\. Then α is isotopic to β if and only if α is homotopic to β\. (Benson Farb, Dan Margalit 51)
- Given an isotopy between two simple closed curves in S, it will often be useful to promote this to an isotopy of S, which we call an ambient isotopy of S\. (Benson Farb, Dan Margalit 51)
- in order to prove a topological statement about an arbitrary nonseparating simple closed curve, we can prove it for any specific simple closed curve\. (Benson Farb, Dan Margalit 53)
- a homotopy of homeomorphisms can be improved to an isotopy; a homeomorphism of a surface can be promoted to a diffeomorphism; and Homeo0 \(S\) is contractible, so in particular any isotopy from the identity homeomorphism to itself is homotopic to the constant isotopy\. (Benson Farb, Dan Margalit 58)
- T HEOREM 1\.12 Let S be any compact surface and let f and g be homotopic homeomorphisms of S\. Then f and g are isotopic unless they are one of the two examples described above \(on S = D 2 and S = A\)\. In particular, if f and g are orientation-preserving, then they are isotopic\. (Benson Farb, Dan Margalit 58)
- It turns out that these two examples are the only examples of homotopic homeomorphisms that are not isotopic (Benson Farb, Dan Margalit 58)
- T HEOREM 1\.13 Let S be a compact surface\. Then every homeomorphism of S is isotopic to a diffeomorphism of S\. (Benson Farb, Dan Margalit 59)
- It is a general fact that any homeomorphism of a smooth manifold can be approximated arbitrarily well by a smooth map\. By taking a close enough approximation, the resulting smooth map is homotopic to the original homeomorphism\. However, this general fact, which is easy to prove, is much weaker than Theorem 1\.13 because the resulting smooth map might not be smoothly invertible; indeed, it might not be invertible at all\. (Benson Farb, Dan Margalit 59)
- T HEOREM 1\.14 Let S be a compact surface, possibly minus a finite number of points from the interior\. Assume that S is not homeomorphic to S 2 , R2 , D 2 , T 2 , the closed annulus, the once-punctured disk, or the oncepunctured plane\. Then the space Homeo0 \(S\) is contractible\. (Benson Farb, Dan Margalit 60)
- It answers the fundamental question: how can one prove that a homeomorphism is or is not homotopically trivial? Equivalently, how can one decide when two homeomorphisms are homotopic or not? (Benson Farb, Dan Margalit 61)
- As a general rule, the term “mapping class group” refers to the group of homotopy classes of homeomorphisms of a surface, (Benson Farb, Dan Margalit 62)
- that there are four surfaces for which homotopy is not the same as isotopy: the disk D 2 , the annulus A, the once-punctured sphere S0,1 , and the twice-punctured sphere S0,2 (Benson Farb, Dan Margalit 63)