In mathematics, homological stability is any of a number of theorems asserting that the

group homology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology lo ...

of a series of groups

G_1 \subset G_2 \subset \cdots

is stable, i.e., :

H_i(G_n)

is independent of ''n'' when ''n'' is large enough (depending on ''i''). The smallest ''n'' such that the maps

H_i(G_n) \to H_i(G_)

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 :

G_\infty = \bigcup_n G_n

can be computed by other means or is related to other data. For example, the

Barratt–Priddy theorem In homotopy theory, a branch of mathematics, the Barratt–Priddy theorem (also referred to as Barratt–Priddy–Quillen theorem) expresses a connection between the homology of the symmetric groups and mapping spaces of spheres. The theorem (nam ...

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 In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups. Explicitly, if X is a based connected CW complex and P is a perfect normal subgroup of \pi_ ...

\operatorname_\infty

and the

sphere spectrum In stable homotopy theory, a branch of mathematics, the sphere spectrum ''S'' is the monoidal unit in the category of spectra. It is the suspension spectrum of ''S''0, i.e., a set of two points. Explicitly, the ''n''th space in the sphere spectrum ...

. In a similar vein, the homology of

\operatorname_\infty(R)

is related, via the +-construction, to the algebraic K-theory of ''R''.

References

{{Reflist Algebraic topology Algebraic K-theory