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**On the cohomology of quotients of moment-angle manifolds.**
*(English.
Russian original)*
Zbl 1342.57019

Russ. Math. Surv. 70, No. 4, 779-781 (2015); translation from Usp. Mat. Nauk 70, No. 4, 209-210 (2015).

From the text: We describe the cohomology of the quotient \({\mathcal Z}_{{\mathcal K}}/H\) of a moment-angle complex \({\mathcal Z}_{{\mathcal K}}\) by a freely acting subtorus \(H\subset T^m\). We establish a ring isomorphism between \(H^*({\mathcal Z}_{{\mathcal K}}/H,R)\) and an appropriate Tor-algebra of the face ring \(R[{\mathcal K}]\), with coefficients in an arbitrary commutative ring \(R\) with unit.

This result was stated by V. Buchstaber and the author [Torus actions and their applications in topology and combinatorics. University Lecture Series. 24. Providence, RI: American Mathematical Society (AMS). viii, 144 p. (2002; Zbl 1012.52021)], §7.37 for a field \(R\), but the argument was not sufficiently detailed in the case of non-trivial \(H\) and finite characteristic. We prove the collapse of the corresponding Eilenberg-Moore spectral sequence using the extended functoriality of Tor with respect to ‘strongly homotopy commutative’ maps in the category DASH by H. J. Munkholm [J. pure appl. Algebra 5, 1–50 (1974; Zbl 0294.55011)]. Our collapse result does not follow from the general results by V. K. A. M. Gugenheim and J. P. May [Mem. Am. Math. Soc. 142, 94 p. (1974; Zbl 0292.55019)] and Munkholm [loc. cit.].

This result was stated by V. Buchstaber and the author [Torus actions and their applications in topology and combinatorics. University Lecture Series. 24. Providence, RI: American Mathematical Society (AMS). viii, 144 p. (2002; Zbl 1012.52021)], §7.37 for a field \(R\), but the argument was not sufficiently detailed in the case of non-trivial \(H\) and finite characteristic. We prove the collapse of the corresponding Eilenberg-Moore spectral sequence using the extended functoriality of Tor with respect to ‘strongly homotopy commutative’ maps in the category DASH by H. J. Munkholm [J. pure appl. Algebra 5, 1–50 (1974; Zbl 0294.55011)]. Our collapse result does not follow from the general results by V. K. A. M. Gugenheim and J. P. May [Mem. Am. Math. Soc. 142, 94 p. (1974; Zbl 0292.55019)] and Munkholm [loc. cit.].

### MSC:

57R19 | Algebraic topology on manifolds and differential topology |

57T35 | Applications of Eilenberg-Moore spectral sequences |

55T20 | Eilenberg-Moore spectral sequences |

### References:

[1] | F. Bosio and L. Meersseman 2006 Acta Math.197 1 53–127 · Zbl 1157.14313 · doi:10.1007/s11511-006-0008-2 |

[2] | V. M. Buchstaber and T. E. Panov 2002 Torus actions and their applications in topology and combinatorics Amer. Math. Soc., Providence, RI viii+144 pp. · doi:10.1090/ulect/024 |

[3] | V. M. Buchstaber and T. E. Panov 2015 Toric topology Amer. Math. Soc., Providence, RI |

[4] | V. Gugenheim and J. P. May 1974 On the theory and applications of differential torsion products Mem. Amer. Math. Soc. Amer. Math. Soc., Providence, RI 142 |

[5] | H. Ishida 2013 Complex manifolds with maximal torus actions 1302.0633 |

[6] | H. J. Munkholm 1974 J. Pure Appl. Algebra5 1–50 · Zbl 0294.55011 · doi:10.1016/0022-4049(74)90002-4 |

[7] | T. Panov and Yu. Ustinovsky 2012 Mosc. Math. J.12 1 149–172 |

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