Bousfield–Kan spectral sequence


Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories




The classical and motivating examples of a homotopy spectral sequence is the Bousfield-Kan spectral sequence which computes homotopy groups of a topological space/simplicial set/homotopy type realized as the totalization of a cosimplicial homotopy type. It may be regarded as the unstable analog of the Adams spectral sequence, which computes homotopy groups of certain spectra. The corresponding spectral sequence for homology groups of the totalization of a cosimplicial homotopy type is the Eilenberg-Moore spectral sequence.


The Bousfield-Kan spectral sequence was introduced and originally studied in

  • Aldridge Bousfield, Daniel Kan, The homotopy spectral sequence of a space with coefficients in a ring. Topology, 11, pp. 79–106, 1972.

  • Aldridge Bousfield, Daniel Kan, A second quadrant homotopy spectral sequence, Transactions of the American Mathematical Society Vol. 177, Mar., 1973

  • Aldridge Bousfield, Daniel Kan, Pairings and products in the homotopy spectral sequence. Transactions of the American Mathematical Society, 177, pp. 319–343, 1973.

  • Aldridge Bousfield, Homotopy Spectral Sequences and Obstructions, Isr. J. Math. 66 (1989), 54-104.

An early textbook account of this work is in

Lecture notes include

Discussion of computations using effective homology includes

  • Romero Ibanez, Effective homology and spectral sequences, 2007 (pdf)

Discussion in homotopy type theory is in

Revised on March 1, 2016 13:29:24 by Urs Schreiber (