algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
(also nonabelian homological algebra)
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
over a simply connected space $X$, then the corresponding cohomology Serre spectral sequence looks like
The generalization of this from ordinary cohomology to generalized (Eilenberg-Steenrod) cohomology is the Atiyah-Hirzebruch spectral sequence, see there for details.
There are two kinds of relative Serre spectral sequences.
For $F \to E \to X$ as above and $A \hookrightarrow X$ a subspace, the induced restriction of the fibration
induces a spectral sequence in relative cohomology of the base space of the form
(e.g. Davis 91, theorem 9.33)
Conversely, for
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.g. Kochman 96, theorem 2.6.3, Davis 91, theorem 9.34)
There is also a generalization to equivariant cohomology: for cohomology with coefficients in a Mackey functor withRO(G)-grading for representation spheres $S^V$, then for $E \to X$ an $F$-fibration of topological G-spaces and for $A$ any $G$-Mackey functor, the equivariant Serre spectral sequence looks like (Kronholm 10, theorem 3.1):
where on the left in the $E_2$-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 $E = H R$ to be ordinary cohomology.
The original article is
Textbook accounts include
Alan Hatcher, Spectral sequences in algebraic topology I: The Serre spectral sequence (pdf)
Stanley Kochman, section 2.2. of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
Davis, Lecture notes in algebraic topology, 1991
Lecture notes etc. includes
Discussion in homotopy type theory includes
and implementation in Lean is in
In equivariant cohomology, for Bredon cohomology:
and for genuine equivariant cohomology, i.e. for RO(G)-graded cohomology with coefficients in a Mackey functor:
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.