“Bockstein homomorphism”的意思、由来-开放百科全书

词条

Bockstein homomorphism

释义

See also
References

In homological algebra, the Bockstein homomorphism, introduced by {{harvs|txt|authorlink=Meyer Bockstein|last=Bockstein|first=Meyer |year1=1942|year2=1943|year3=1958}}, is a connecting homomorphism associated with a short exact sequence

of abelian groups, when they are introduced as coefficients into a chain complex C, and which appears in the homology groups as a homomorphism reducing degree by one,

To be more precise, C should be a complex of free, or at least torsion-free, abelian groups, and the homology is of the complexes formed by tensor product with C (some flat module condition should enter). The construction of β is by the usual argument (snake lemma).

A similar construction applies to cohomology groups, this time increasing degree by one. Thus we have

The Bockstein homomorphism associated to the coefficient sequence

is used as one of the generators of the Steenrod algebra. This Bockstein homomorphism has the following two properties:

if ,

;

in other words, it is a superderivation acting on the cohomology mod p of a space.

References

{{Citation | last1=Bockstein | first1=Meyer |authorlink=Meyer Bockstein| title=Universal systems of ∇-homology rings | mr=0008701 | year=1942 | journal=C. R. (Doklady) Acad. Sci. URSS (N.S.) | volume=37 | pages=243–245}}
{{Citation | last1=Bockstein | first1=Meyer |authorlink=Meyer Bockstein| title=A complete system of fields of coefficients for the ∇-homological dimension | mr=0009115 | year=1943 | journal=C. R. (Doklady) Acad. Sci. URSS (N.S.) | volume=38 | pages=187–189}}
{{citation

{{citation

{{citation|mr=0666554|last= Spanier|first= Edwin H.|author-link=Edwin Spanier| title= Algebraic topology. Corrected reprint |publisher=Springer-Verlag|publication-place= New York-Berlin|year= 1981|pages= xvi+528| isbn= 0-387-90646-0}}

2 : Algebraic topology|Homological algebra

随便看

开放百科全书收录14589846条英语、德语、日语等多语种百科知识，基本涵盖了大多数领域的百科知识，是一部内容自由、开放的电子版国际百科全书。

See also

References