Serre spectral sequence
The Serre spectral sequence or Leray-Serre spectral sequence is a spectral sequence for computation of singular homology of topological spaces in a Serre-fiber sequence of topological spaces.
Given a homotopy fiber sequence
over a simply connected space , 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 is also a generalization to equivariant cohomology: for cohomology with coefficients in a Mackey functor withRO(G)-grading for representation spheres , then for an -fibration of topological G-spaces and for any -Mackey functor, the equivariant Serre spectral sequence looks like (Kronholm 10, theorem 3.1):
where on the left in the -page we have ordinary cohomology with coefficients in the genuine equivariant cohomology groups of the fiber.
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
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 -graded Serre spectral sequence, Homology Homotopy Appl. Volume 12, Number 1 (2010), 75-92. (pdf, Euclid)
Revised on May 10, 2016 06:50:15
by Urs Schreiber