In algebraic topology, a G-spectrum is a
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
with an action of a (finite) group.
Let ''X'' be a spectrum with an action of a finite group ''G''. The important notion is that of the homotopy fixed point set
. There is always
:
a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition,
is the
mapping spectrum .)
Example:
acts on the complex ''K''-theory ''KU'' by taking the
conjugate bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces.
Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...
of a
complex vector bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces.
Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...
. Then
, the real ''K''-theory.
The cofiber of
is called the
Tate spectrum
Tate is an institution that houses, in a network of four art galleries, the United Kingdom's national collection of British art, and international modern and contemporary art. It is not a government institution, but its main sponsor is the U ...
of ''X''.
''G''-Galois extension in the sense of Rognes
This notion is due to J. Rognes . Let ''A'' be an
E∞-ring with an action of a finite group ''G'' and ''B'' = ''A''
''hG'' its invariant subring. Then ''B'' → ''A'' (the map of ''B''-algebras in E
∞-sense) is said to be a ''G-Galois extension'' if the natural map
:
(which generalizes
in the classical setup) is an equivalence. The extension is faithful if the
Bousfield class In algebraic topology, the Bousfield class of, say, a spectrum ''X'' is the set of all (say) spectra ''Y'' whose smash product with ''X'' is zero: X \otimes Y = 0. Two objects are Bousfield equivalent if their Bousfield classes are the same.
The no ...
es of ''A'', ''B'' over ''B'' are equivalent.
Example: ''KO'' → ''KU'' is a ℤ./2-Galois extension.
See also
*
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 ...
References
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External links
*
Algebraic topology
Homotopy theory
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