Atiyah-Hirzebruch spectral sequence
Encyclopedia
In mathematics
, the Atiyah–Hirzebruch spectral sequence is a spectral sequence
for calculating generalized cohomology, introduced by in the special case of K-theory
. For a CW complex
X, it relates the generalized cohomology groups
with 'ordinary' cohomology groups H j with coefficients in the generalized cohomology of a point. More precisely, the E2 term of the spectral sequence is Hj(X,hj(point)), and the spectral sequence converges conditionally to hi+j(X).
Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence
, and reduces to it in the case where h=H. It can be derived from an exact couple that gives the E1 page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with h.
In detail, assume X to be the total space of a Serre fibration with fibre F and base space B. The filtration of B by its n-skeletons
gives rise to a filtration of X. There is a corresponding spectral sequence
with E2 term
and abutting to
This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre F is a point.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Atiyah–Hirzebruch spectral sequence is a spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
for calculating generalized cohomology, introduced by in the special case of K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
. For a CW complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
X, it relates the generalized cohomology groups
- hi(X)
with 'ordinary' cohomology groups H j with coefficients in the generalized cohomology of a point. More precisely, the E2 term of the spectral sequence is Hj(X,hj(point)), and the spectral sequence converges conditionally to hi+j(X).
Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence
Serre spectral sequence
In mathematics, the Serre spectral sequence is an important tool in algebraic topology...
, and reduces to it in the case where h=H. It can be derived from an exact couple that gives the E1 page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with h.
In detail, assume X to be the total space of a Serre fibration with fibre F and base space B. The filtration of B by its n-skeletons
N-skeleton
In mathematics, particularly in algebraic topology, the n-skeleton of a topological space X presented as a simplicial complex refers to the subspace Xn that is the union of the simplices of X of dimensions m ≤ n...
gives rise to a filtration of X. There is a corresponding spectral sequence
Spectral sequence
In homological algebra and algebraic topology, a spectral sequence is a means of computing homology groups by taking successive approximations...
with E2 term
- Hp(B;h q(F))
and abutting to
- hp + q(X).
This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre F is a point.