The homology of \(\operatorname{MSpin}\). (English) Zbl 0554.55003
Let \(A\) denote the mod 2 Steenrod algebra, and let \(A_1\) be the dual of the subalgebra of \(A\) generated by \(Sq^1\) and \(Sq^2\). The authors determine the structure of \(H_*(\operatorname{MSpin};\mathbb{Z}/2)\) as an \(A\)-comodule algebra. The simultaneous determination of these two structures leads, via the Adams spectral sequence, to interesting new information on the product structure of the bordism ring \(\Omega_*^{\operatorname{Spin}}\) extending the result of D. W. Anderson, E. H. Brown jun. and F. P. Petersen [Ann. Math. (2) 89, 271–298 (1967; Zbl 0156.21605)]. The authors’ method is to use the identification of \(H_*(\operatorname{MSpin};\mathbb{Z}/2)\) with \(A\square_{A_1}N\) [the second author, Proc. Am. Math. Soc. 87, 355–356 (1983; Zbl 0517.55012)]. The bulk of this paper is devoted to the analysis of the structure of \(N\).
Reviewer: S. O. Kochmann
MSC:
55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |
55T15 | Adams spectral sequences |
57R90 | Other types of cobordism |
55S10 | Steenrod algebra |
55P42 | Stable homotopy theory, spectra |
References:
[1] | Davis, Proc. Symposium on Algebraic Topology in Honor of Jose Adem. Contemporary Mathematics 12 pp 91– (1982) · Zbl 0505.55009 · doi:10.1090/conm/012/676320 |
[2] | DOI: 10.2307/2042276 · Zbl 0436.57014 · doi:10.2307/2042276 |
[3] | DOI: 10.2307/1970690 · Zbl 0156.21605 · doi:10.2307/1970690 |
[4] | Adams, Math. Proc. Cambridge Philos. Soc. 80 pp 475– (1976) |
[5] | Stong, Notes on Cobordism Theory. (1968) |
[6] | Davis, Proc. Conf., Current Trends in Algebraic Topology 2 pp 297– (1982) |
[7] | DOI: 10.1112/jlms/s2-25.3.467 · Zbl 0458.55003 · doi:10.1112/jlms/s2-25.3.467 |
[8] | Pengelley, Proc.Conf., Current Trends in Algebraic Topology 2 pp 5110– (1982) |
[9] | DOI: 10.2307/2374085 · Zbl 0508.55007 · doi:10.2307/2374085 |
[10] | DOI: 10.2307/1970615 · Zbl 0163.28202 · doi:10.2307/1970615 |
[11] | DOI: 10.2307/2043716 · Zbl 0517.55012 · doi:10.2307/2043716 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.