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Serre spectral sequence

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In ordinary cohomology

The Serre spectral sequence or Leray-Serre spectral sequence is a spectral sequence for computation of ordinary cohomology (ordinary homology) of topological spaces in a Serre-fiber sequence of topological spaces.

Given a homotopy fiber sequence

F E p X \array{ F &\longrightarrow& E \\ && \downarrow^{\mathrlap{p}} \\ && X }

over a simply connected space XX, then the corresponding cohomology Serre spectral sequence looks like

E 2 p,q=H p(X,H q(F))H p+q(E). E_2^{p,q}= H^p(X, H^q(F)) \Rightarrow H^{p+q}(E) \,.

In generalized cohomology

The generalization of this from ordinary cohomology to generalized (Eilenberg-Steenrod) cohomology is the Atiyah-Hirzebruch spectral sequence, see there for details.

In relative cohomology

There are two kinds of relative Serre spectral sequences.

For FEXF \to E \to X as above and AXA \hookrightarrow X a subspace, the induced restriction of the fibration

F F p 1(A) E p A X \array{ F & \simeq & F \\ \downarrow && \downarrow \\ p^{-1}(A) &\longrightarrow& E \\ \downarrow && \downarrow^{\mathrlap{p}} \\ A &\hookrightarrow& X }

induces a spectral sequence in relative cohomology of the base space of the form

E 2 p,q=H p(X,A;H q(F))H (E,p 1(A)). E_2^{p,q} = H^p(X,A; H^q(F)) \;\Rightarrow\; H^\bullet(E, p^{-1}(A)) \,.

(e.g. Davis 91, theorem 9.33)

Conversely, for

F F E E p X X \array{ F' & \hookrightarrow & F \\ \downarrow && \downarrow \\ E' &\hookrightarrow& E \\ \downarrow && \downarrow^{\mathrlap{p}} \\ X &\hookrightarrow& X }

a sub-fibration over the same base, then this induces a spectral sequence for relative cohomology of the the total space in terms of ordinary cohomology with coefficients in the relative cohomology of the fibers:

E 2 p,q=H p(X;H q(F,F))H (E,E). E^{p,q}_2 = H^p(X; H^q(F,F')) \;\Rightarrow\; H^\bullet(E,E') \,.

(e.g. Kochmann 96, theorem 2.6.3, Davis 91, theorem 9.34)

In equivariant cohomology

There is also a generalization to equivariant cohomology: for cohomology with coefficients in a Mackey functor withRO(G)-grading for representation spheres S VS^V, then for EXE \to X an FF-fibration of topological G-spaces and for AA any GG-Mackey functor, the equivariant Serre spectral sequence looks like (Kronholm 10, theorem 3.1):

E 2 p,q=H p(X,H V+q(F,A))H V+p+q(E,A), E_2^{p,q} = H^p(X, H^{V+q}(F,A)) \,\Rightarrow\, H^{V+p+q}(E,A) \,,

where on the left in the E 2E_2-page we have ordinary cohomology with coefficients in the genuine equivariant cohomology groups of the fiber.

Details

For details on the plain Serre spectral sequence see at Atiyah-Hirzebruch spectral sequence and take E=HRE = H R to be ordinary cohomology.

Consequences

References

General

The original article is

  • Jean-Pierre Serre, Homologie singuliére des espaces fibrés Applications, Ann. of Math. 54 (1951),

Textbook accounts include

Lecture notes etc. includes

  • Greg Friedman, Some extremely brief notes on the Leray spectral sequence (pdf)

Discussion in homotopy type theory includes

In equivariant cohomology

In equivariant cohomology, for Bredon cohomology:

  • Ieke Moerdijk, J.-A. Svensson, The Equivariant Serre Spectral Sequence, Proceedings of the American Mathematical Society Vol. 118, No. 1 (May, 1993), pp. 263-278 (JSTOR)

and for genuine equivariant cohomology, i.e. for RO(G)-graded cohomology with coefficients in a Mackey functor:

  • William Kronholm, The RO(G)RO(G)-graded Serre spectral sequence, Homology Homotopy Appl. Volume 12, Number 1 (2010), 75-92. (pdf, Euclid)

See also

Revised on June 7, 2016 07:06:31 by Urs Schreiber (131.220.184.222)