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A Primer on Mapping Class Groups (PMS-49) by Dan Margalit, Benson Farb

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Chapter Five

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Presentations and Low-dimensional Homology

Having found a finite set of generators for the mapping class group, we now begin to focus on relations. Indeed, one of our main goals in this chapter is to give a finite presentation for Mod(S). In doing so, we will see some beautiful topological ideas, as well as some useful techniques from geometric group theory.

The relations in a group G are intimately related to the first and second homology groups of G. Recall that the homology groups of G are defined to be the homology groups of any K(G, 1)-space. The first and second homology groups have direct, group-theoretic interpretations. For ...

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