Homological stability

In mathematics, homological stability is any of a number of theorems asserting that the group homology of a series of groups formula_1 is stable, i.e.,
is independent of "n" when "n" is large enough (depending on "i"). The smallest "n" such that the maps formula_3 is an isomorphism is referred to as the "stable range".
The concept of homological stability was pioneered by Daniel Quillen whose proof technique has been adapted in various situations.
Examples.
Examples of such groups include the following:
Applications.
In some cases, the homology of the group
can be computed by other means or is related to other data. For example, the Barratt–Priddy theorem relates the homology of the infinite symmetric group agrees with mapping spaces of spheres. This can also be stated as a relation between the plus construction of formula_5 and the sphere spectrum. In a similar vein, the homology of formula_6 is related, via the +-construction, to the algebraic K-theory of "R". 