Homological stability explained

In mathematics, homological stability is any of a number of theorems asserting that the group homology of a series of groups

G _{1} \subset G _{2} \subset … 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} _{n+1} ) 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.^[1]

Examples

Examples of such groups include the following:

group	name
symmetric group S _{n}	Nakaoka stability^[2]
braid group B _{n}	^[3]
general linear group \operatorname{GL} _{n(R)} for (certain) rings R	^[4]
mapping class group of surfaces (n is the genus of the surface)	Harer stability^[5]
automorphism group of free groups, \operatorname{Aut}(F _{n)}	^[6]

Applications

In some cases, the homology of the group

G _{infty} = c up _{n} G _{n} 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 \operatorname{BS} _{infty} and the sphere spectrum. In a similar vein, the homology of \operatorname{GL} _{infty(R)} is related, via the +-construction, to the algebraic K-theory of R.

Notes and References

1. Book: Quillen, D.. Daniel Quillen. Finite generation of the groups K_i of rings of algebraic integers.. Algebraic K-theory, I: Higher K-theories. 179–198. Lecture Notes in Math.. 341. Springer. 1973.
2. Nakaoka, Minoru. Homology of the infinite symmetric group. Ann. Math. . 2. 73. 1961. 229 - 257. 10.2307/1970333.
3. Arnol’d. V.I.. Vladimir Arnold. The cohomology ring of the colored braid group. Mathematical Notes. 5. 2. 1969. 138–140. 10.1007/bf01098313.
4. Van der Kallen, W.. Homology stability for linear groups. Invent. Math.. 60. 1980. 269–295. 10.1007/bf01390018.
5. Harer, J. L.. Stability of the homology of the mapping class groups of orientable surfaces. Annals of Mathematics. 121. 1985. 215–249. 10.2307/1971172.
6. Hatcher. Allen. Allen Hatcher. Vogtmann. Karen. Karen Vogtmann. Cerf theory for graphs. J. London Math. Soc. . Series 2. 58. 3. 1998. 633–655. 10.1112/s0024610798006644.