Bockstein homomorphism

Bockstein homomorphism

In mathematics, the Bockstein homomorphism in homological algebra is a connecting homomorphism associated with a short exact sequence

:0 → "P" → "Q" → "R" → 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").

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").

This is important as a source of cohomology operations (see Steenrod algebra). For coefficients in a finite cyclic group of order "n" as "R", the mapping β can be combined with reduction modulo "n"; and then iterated.

History

The name is for the Soviet topologist from Moscow, Meer Feliksovich Bokshtein (Bokstein), with Bockstein being a French transliteration. Little known in the West, he was born October 4, 1913, and died May 2, 1990.

References

* citation
last= Bockstein
first= Meyer
title= Sur la formule des coefficients universels pour les groupes d'homologie
journal=Comptes Rendus de l'académie des Sciences. Série I. Mathématique
volume= 247
year= 1958
pages= 396–398
url=
doi=
id= MR|0103918
* citation
first= Allen
last= Hatcher
author-link= Allen Hatcher
title= Algebraic Topology
url= http://www.math.cornell.edu/%7Ehatcher/AT/ATpage.html
year= 2002
publisher= Cambridge University Press
isbn= 978-0-521-79540-1
id= MR|1867354 .
*citation|id=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


Wikimedia Foundation. 2010.

Игры ⚽ Поможем написать реферат

Look at other dictionaries:

  • Meyer Bockstein — or Meer Feliksovich Bokshtein or Bokstein (October 4, 1913, to May 2, 1990) was a topologist from Moscow who introduced the Bockstein homomorphism. The Bockstein spectral sequence is named after him. References Meyer Bockstein at the Mathematics… …   Wikipedia

  • Spin structure — In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical …   Wikipedia

  • Steenrod algebra — In algebraic topology, a branch of mathematics, the Steenrod algebra is a structure occurring in the theory of cohomology operations. It is an object of great importance, most especially to homotopy theorists. More precisely, for a given prime… …   Wikipedia

  • Stiefel–Whitney class — In mathematics, the Stiefel–Whitney class arises as a type of characteristic class associated to real vector bundles E ightarrow X. It is denoted by w ( E ), taking values in H^*(X; /2), the cohomology groups with mod 2 coefficients. The… …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

  • Differential graded algebra — In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded algebra with an added chain complex structure that respects the algebra structure. Contents 1 Definition 2 Examples of DGAs 3 Other facts about …   Wikipedia

  • Trigenus — In low dimensional topology, the trigenus is an invariant consisting of a triplet:(g 1,g 2,g 3) assigned to closed 3 manifolds. The definition is by minimizing the genera of three orientable handle bodies mdash; with no intersection between their …   Wikipedia

  • Spectral sequence — In the area of mathematics known as homological algebra, especially in algebraic topology and group cohomology, a spectral sequence is a means of computing homology groups by taking successive approximations. Spectral sequences are a… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”