“Homological stability”的意思、由来-开放百科全书

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

is stable, i.e.,

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

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

Applications

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

and the sphere spectrum. In a similar vein, the homology of

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

References

1. ^{{cite book|author=Quillen, D.|chapter=Finite generation of the groups K_i of rings of algebraic integers.|title=Algebraic K-theory, I: Higher K-theories|pages=179–198|series=Lecture Notes in Math.|volume=341|publisher=Springer|year=1973}}
2. ^{{cite journal|author =Nakaoka, Minoru|title =Homology of the infinite symmetric group|journal =Ann. Math. |series=2|volume=73|year=1961|pages=229–257|doi=10.2307/1970333}}
3. ^{{cite journal|author=Arnol’d, V.I.|title=The cohomology ring of the colored braid group|journal=Mathematical Notes|volume=5|issue=2|year=1969|pages=138–140|doi=10.1007/bf01098313}}
4. ^{{citation|author=Suslin, A. A.|title=Stability in algebraic K-theory. Algebraic K-theory, Part I (Oberwolfach, 1980)|pages=304–333|series=Lecture Notes in Math., 966|publisher=Springer|year=1982}}
5. ^{{cite journal|author=Van der Kallen, W.|title=Homology stability for linear groups|journal=Invent. Math.|volume=60|year=1980|pages=269–295|url=http://www.math.cinvestav.mx/sites/default/files/kallen.pdf|doi=10.1007/bf01390018}}
6. ^{{cite journal|author=Harer, J. L.|title=Stability of the homology of the mapping class groups of orientable surfaces|journal=Annals of Mathematics|volume=121|year=1985|pages=215–249|doi=10.2307/1971172}}
7. ^{{cite journal|author1=Hatcher, Allen|author2=Vogtmann, Karen|title=Cerf theory for graphs|journal=J. London Math. Soc. |series=Series 2|volume=58|issue=3|year=1998|pages=633–655|doi=10.1112/s0024610798006644}}