Serre spectral sequence



Algebraic topology

Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories




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. Kochman 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.


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




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

and implementation in Lean is in

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

Last revised on January 25, 2021 at 10:32:59. See the history of this page for a list of all contributions to it.