By Mahowald M., Priddy S. (eds.)

ISBN-10: 0821851020

ISBN-13: 9780821851029

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**Extra info for Algebraic Topology**

**Example text**

S and fix the punctures individually. 4 Computing the Cohomology of Mapping Class Groups between r* r and 39 T^'. Consider the following maps $ r > 2, * : r>2, n: r > 2, which we define as follows. For Vf, sew a pair of pants (a copy of the surface So 3 ) to the regular surface E* r along two components of its boundary. For $ , sew a pair of pants to £ ' r along one component of its boundary. Finally, for T], sew two components of 5E* together. The maps $, $ and n induce maps of the mapping class group, that is to say, these are extensions of the identity on £°3 f° r ^ anc ^ *• Let $*, $*, and n*, respectively be the maps they induce on homology .

The proof that Tsg is generated by Dehn twists on a finite number of curves is due to Dehn [7]. Humphries later determined the minimal number of generators necessary [8]. McCool [9] gave an indirect proof that Fg is finitely presented, but the first explicit presentation was provided by Hatcher and 32 3. 5 References Thurston [10]. The later was somewhat simplified by Wajnryb [11] whose presentation uses as generators: Ti = rCi, 1 < i < 2g + 1, and has I I + 3 relations when g > 2. What are these generators?

The following exposition and those to come serve as this foundation. Moduli space can be approached in several distinct ways. In particular, when the genus g is greater than one, it is the space of isometry classes of hyperbolic metrics on a surface; it is also the space of conformal equivalence classes of Riemann surfaces, and the space of algebraic curves of genus g, up to isomorphism. These definitions reveal an elaborate interplay between hyperbolic geometry, complex analysis, algebraic topology and physics.

### Algebraic Topology by Mahowald M., Priddy S. (eds.)

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