cobar型のEilenberg-Mooreスペクトル系列

余単体的空間の homology spectral sequence の代表としては, cobar型の Eilenberg-Moore spectral sequence が挙げられる。その構成には様々な方法があるが, 代表的なのは以下のものだろう。

収束性については, 色々難しい問題がある。Jeanneret と Osse の [JO99] では, Larry Smith の構成に基づいて議論されている。 Tilman Bauer の [Bau] は, cosimplicial space の homology spectral sequence として一般的に議論されている点で, Dwyer の結果の延長にあたるものと言えるだろう。

References

[Bau]

Tilman Bauer. Convergence of the Eilenberg-Moore spectral sequence for generalized cohomology theories. arXiv: 0803.3798.

[Dwy74]

W. G. Dwyer. “Strong convergence of the Eilenberg-Moore spectral sequence”. In: Topology 13 (1974), pp. 255–265.

[Dwy75]

William G. Dwyer. “Exotic convergence of the Eilenberg-Moore spectral sequence”. In: Illinois J. Math. 19.4 (1975), pp. 607–617.

[EM64]

Samuel Eilenberg and J. C. Moore. “Homological algebra and fibrations”. In: Colloque de Topologie (Brussels, 1964). Librairie Universitaire, Louvain, 1964, pp. 81–90.

[EM66]

Samuel Eilenberg and John C. Moore. “Homology and fibrations. I. Coalgebras, cotensor product and its derived functors”. In: Comment. Math. Helv. 40 (1966), pp. 199–236.

[JO99]

A. Jeanneret and A. Osse. “The Eilenberg-Moore spectral sequence in \(K\)-theory”. In: Topology 38.5 (1999), pp. 1049–1073. url: http://dx.doi.org/10.1016/S0040-9383(98)00046-9.

[Rec70]

David L. Rector. “Steenrod operations in the Eilenberg-Moore spectral sequence”. In: Comment. Math. Helv. 45 (1970), pp. 540–552.

[Smi70a]

Larry Smith. Lectures on the Eilenberg-Moore spectral sequence. Lecture Notes in Mathematics, Vol. 134. Berlin: Springer-Verlag, 1970, pp. vii+142.

[Smi70b]

Larry Smith. “On the Künneth theorem. I. The Eilenberg-Moore spectral sequence”. In: Math. Z. 116 (1970), pp. 94–140.

[Tam02]

Dai Tamaki. “The fiber of iterated Freudenthal suspension and Morava \(K\)-theory of \(\Omega ^{k}S^{2l+1}\)”. In: Recent progress in homotopy theory (Baltimore, MD, 2000). Vol. 293. Contemp. Math. Providence, RI: Amer. Math. Soc., 2002, pp. 299–329.

[Tam12]

Dai Tamaki. “The Salvetti complex and the little cubes”. In: J. Eur. Math. Soc. (JEMS) 14.3 (2012), pp. 801–840. arXiv: math/0602085. url: http://dx.doi.org/10.4171/JEMS/319.

[Tam94]

Dai Tamaki. “A dual Rothenberg-Steenrod spectral sequence”. In: Topology 33.4 (1994), pp. 631–662. url: http://dx.doi.org/10.1016/0040-9383(94)90002-7.