Bockstein series
In algebraic topology , a branch of mathematics , the Bockstein sequence is an aid for comparing cohomology groups with different coefficients; it is named after Meir Bockstein .
construction
Homology
Be
a short exact sequence of Abelian groups and a topological space. From the short exact sequence of chain complexes
one obtains a long exact sequence of homology groups by means of the snake lemma
- ,
the so-called Bockstein sequence or Bockstein sequence. The connecting homomorphism is called Bockstein homomorphism .
Cohomology
also provides a short exact sequence of coquette complexes
and again with the snake lemma, a long exact sequence of cohomology groups
- ,
which is also referred to as the Bockstein sequence or Bockstein sequence and the connecting homomorphism as the Bockstein homomorphism .
Examples
- The short exact sequence gives the Bockstein homomorphisms
- and .
- The Bockstein homomorphism associated with the short exact sequence
- is important for the construction of the Steenrod algebra .
- The Bockstein homomorphisms associated with the short exact sequences and
- and
- are important in the construction of secondary characteristic classes and in Deligne cohomology .
literature
- Bockstein, M. (1942). Universal systems of ∇-homology rings. 《CR (Doklady) Acad. Sci. URSS (NS)》 37: 243-245. MR0008701.
- Bockstein, M. (1943). A complete system of fields of coefficients for the ∇-homological dimension. 《CR (Doklady) Acad. Sci. URSS (NS)》 38: 187-189. MR0009115.
- Bockstein, M. (1958). Sur la formule des coefficients universels pour les groupes d'homologie. 《Comptes Rendus de l'Académie des Sciences. Série I. Mathématique》 247: 396-398. MR0103918.