Misplaced Pages

Bockstein homomorphism

Article snapshot taken from Wikipedia with creative commons attribution-sharealike license. Give it a read and then ask your questions in the chat. We can research this topic together.
Homological map

In homological algebra, the Bockstein homomorphism, introduced by Meyer Bockstein (1942, 1943, 1958), is a connecting homomorphism associated with a short exact sequence

0 → P → Q → R → 0 {\displaystyle 0\to P\to Q\to R\to 0}

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,

β : H i ( C , R ) → H i − 1 ( C , P ) . {\displaystyle \beta \colon H_{i}(C,R)\to H_{i-1}(C,P).}

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

β : H i ( C , R ) → H i + 1 ( C , P ) . {\displaystyle \beta \colon H^{i}(C,R)\to H^{i+1}(C,P).}

The Bockstein homomorphism β {\displaystyle \beta } associated to the coefficient sequence

0 → Z / p Z → Z / p 2 Z → Z / p Z → 0 {\displaystyle 0\to \mathbb {Z} /p\mathbb {Z} \to \mathbb {Z} /p^{2}\mathbb {Z} \to \mathbb {Z} /p\mathbb {Z} \to 0}

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

β β = 0 {\displaystyle \beta \beta =0} ,
β ( a ∪ b ) = β ( a ) ∪ b + ( − 1 ) dim ⁡ a a ∪ β ( b ) {\displaystyle \beta (a\cup b)=\beta (a)\cup b+(-1)^{\dim a}a\cup \beta (b)} ;

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

See also

References

Categories: