In the mathematical discipline of
topology, the Brown–Gitler spectrum is a
spectrum whose
cohomology is a certain
cyclic module over the
Steenrod algebra In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology.
For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, c ...
.
Brown–Gitler spectra are defined by the
isomorphism:
:
History
The concept was introduced by mathematicians
Edgar H. Brown and
Samuel Gitler in a 1973 paper.
In topology, Brown–Gitler spectrum is related to the concepts of the
Segal conjecture
Segal's Burnside ring conjecture, or, more briefly, the Segal conjecture, is a theorem in homotopy theory, a branch of mathematics. The theorem relates the Burnside ring of a finite group ''G'' to the stable cohomotopy of the classifying space ''B ...
(proven in 1984) and the
Burnside ring
In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century. The algebraic r ...
.
Applications
Brown–Gitler spectra have had many important applications in
homotopy theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...
.
References
External links
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{{DEFAULTSORT:Brown-Gitler spectrum
Topology