The loop cohomology of a space with the polynomial cohomology algebra. (English) Zbl 1380.55007
Let \(X\) (resp. \(\Omega X\)) be a simply connected topological space (resp. the loop space of \(X\)). The aim of this paper is to calculate the algebra \(H^{*}(\Omega X; \; \mathbb{Z}/2\mathbb{Z})\) when the algebra \(H^{*}(X; \; \mathbb{Z}/2\mathbb{Z})\) is polynomial, by means of the Steenrod operation \(Sq_{1}\) on \(H^{*}(X; \; \mathbb{Z}/2\mathbb{Z})\) (compare with [A. Borel, Ann. Math. (2) 57, 115–207 (1953; Zbl 0052.40001)]).
The author gives also a criterion for \(H^{*}(\Omega X; \; \mathbb{Z}/2\mathbb{Z})\) to be an exterior algebra. The method used is based on the filtered Hirsch model for \(X\) (see: [S. Saneblidze, Trans. A. Razmadze Math. Inst. 170, No. 1, 114–136 (2016; Zbl 1385.55011)]).
The author gives also a criterion for \(H^{*}(\Omega X; \; \mathbb{Z}/2\mathbb{Z})\) to be an exterior algebra. The method used is based on the filtered Hirsch model for \(X\) (see: [S. Saneblidze, Trans. A. Razmadze Math. Inst. 170, No. 1, 114–136 (2016; Zbl 1385.55011)]).
Reviewer: Saïd Zarati (Tunis)
MSC:
| 55P35 | Loop spaces |
| 55N07 | Steenrod-Sitnikov homologies |
| 55R20 | Spectral sequences and homology of fiber spaces in algebraic topology |
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