Homology of Linear Groups pp 33-64 | Cite as

# Stability

Chapter

## Abstract

Suppose we have a sequence
is an isomorphism for

*G*_{1}⊂*G*_{2}⊂ ··· ⊂*G*_{ n }⊂ ··· of groups. A natural question to ask is the following: for a fixed*k*is there an integer*n*(*k*) such that the map$$
H_k (G_i ) \to H_k (G_{i + 1} )
$$

*i*≥*n*(*k*)? The answer is certainly no if for example*G*_{ n }is the free abelian group of rank*n*, but there are many examples for which stability does happen. For example, there are stability results for the sequence of symmetric groups [**89**] and also for the mapping class groups of orientable surfaces [**54**] (*g*= genus,*r*= number of boundary components);*H*^{ k }stabilizes at*n*= 2*k*in the first case and at*g*= 3*k*in the second.## Keywords

Abelian Group Spectral Sequence Simplicial Complex Local Ring Mapping Class Group
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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