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

${\displaystyle 0\rightarrow N\rightarrow G\rightarrow H\rightarrow 0}$

a short exact sequence of Abelian groups and a topological space. From the short exact sequence of chain complexes ${\displaystyle X}$

${\displaystyle 0\rightarrow C_{*}(X;N)\rightarrow C_{*}(X;G)\rightarrow C_{*}(X;H)\rightarrow 0}$

one obtains a long exact sequence of homology groups by means of the snake lemma

${\displaystyle \ldots \rightarrow H_{*+1}(X;H)\rightarrow H_{*}(X;N)\rightarrow H_{*}(X;G)\rightarrow H_{*}(X;H)\rightarrow H_{*-1}(X;N)\rightarrow \ldots }$,

the so-called Bockstein sequence or Bockstein sequence. The connecting homomorphism is called Bockstein homomorphism . ${\displaystyle H_{*}(X;H)\rightarrow H_{*-1}(X;N)}$

Cohomology

${\displaystyle 0\rightarrow N\rightarrow G\rightarrow H\rightarrow 0}$also provides a short exact sequence of coquette complexes

${\displaystyle 0\rightarrow C^{*}(X;N)\rightarrow C^{*}(X;G)\rightarrow C^{*}(X;H)\rightarrow 0}$

and again with the snake lemma, a long exact sequence of cohomology groups

${\displaystyle \ldots \rightarrow H^{*-1}(X;H)\rightarrow H^{*}(X;N)\rightarrow H^{*}(X;G)\rightarrow H^{*}(X;H)\rightarrow H^{*+1}(X;N)\rightarrow \ldots }$,

which is also referred to as the Bockstein sequence or Bockstein sequence and the connecting homomorphism as the Bockstein homomorphism . ${\displaystyle H^{*}(X;H)\rightarrow H^{*+1}(X;N)}$

Examples

• The short exact sequence gives the Bockstein homomorphisms ${\displaystyle 0\rightarrow \mathbb {Z} \rightarrow \mathbb {Z} \rightarrow \mathbb {Z} /2\mathbb {Z} \rightarrow 0}$

${\displaystyle H_{i}(X;\mathbb {Z} /2\mathbb {Z} )\rightarrow H_{i-1}(X;\mathbb {Z} )}$and . ${\displaystyle H^{i}(X;\mathbb {Z} /2\mathbb {Z} )\rightarrow H^{i+1}(X;\mathbb {Z} )}$

• The Bockstein homomorphism associated with the short exact sequence ${\displaystyle 0\rightarrow \mathbb {Z} /n\mathbb {Z} \rightarrow \mathbb {Z} /n^{2}\mathbb {Z} \rightarrow \mathbb {Z} /n\mathbb {Z} \rightarrow 0}$

${\displaystyle H^{i}(X;\mathbb {Z} /n\mathbb {Z} )\rightarrow H^{i+1}(X;\mathbb {Z} /n\mathbb {Z} )}$

is important for the construction of the Steenrod algebra .

• The Bockstein homomorphisms associated with the short exact sequences and ${\displaystyle 0\rightarrow \mathbb {Z} \rightarrow \mathbb {R} \rightarrow \mathbb {R} /\mathbb {Z} \rightarrow 0}$ ${\displaystyle 0\rightarrow \mathbb {Z} \rightarrow \mathbb {C} \rightarrow \mathbb {C} /\mathbb {Z} \rightarrow 0}$

${\displaystyle H^{i}(X;\mathbb {R} /\mathbb {Z} )\rightarrow H^{i+1}(X;\mathbb {Z} )}$ and ${\displaystyle H^{i}(X;\mathbb {C} /\mathbb {Z} )\rightarrow H^{i+1}(X;\mathbb {Z} )}$

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.