Abstract
We compute the equivariant cohomology \(H^*_{T_I}(\mathcal Z_{\mathcal K})\) of moment–angle complexes \(\mathcal Z_{\mathcal K}\) with respect to the action of coordinate subtori \(T_I \subset T^m\). We give a criterion for \(\mathcal Z_{\mathcal K}\) to be equivariantly formal, and obtain specifications for the cases of flag complexes and graphs.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
S. A. Abramyan and T. E. Panov, “Higher Whitehead products in moment–angle complexes and substitution of simplicial complexes,” Proc. Steklov Inst. Math. 305, 1–21 (2019) [transl. from Tr. Mat. Inst. Steklova 305, 7–28 (2019)].
I. V. Baskakov, V. M. Buchstaber, and T. E. Panov, “Cellular cochain algebras and torus actions,” Russ. Math. Surv. 59 (3), 562–563 (2004) [transl. from Usp. Mat. Nauk 59 (3), 159–160 (2004)].
V. M. Buchstaber and T. E. Panov, Toric Topology (Am. Math. Soc., Providence, RI, 2015), Math. Surv. Monogr. 204.
S. Eilenberg and J. C. Moore, “Homology and fibrations. I: Coalgebras, cotensor product and its derived functors,” Comment. Math. Helv. 40, 199–236 (1966).
M. Franz, “The integral cohomology of toric manifolds,” Proc. Steklov Inst. Math. 252, 53–62 (2006).
M. Franz, “Dga models for moment–angle complexes,” arXiv: 2006.01571 [math.AT].
J. Grbić and S. Theriault, “Homotopy theory in toric topology,” Russ. Math. Surv. 71 (2), 185–251 (2016) [transl. from Usp. Mat. Nauk 71 (2), 3–80 (2016)].
S. Luo, T. Matsumura, and W. F. Moore, “Moment angle complexes and big Cohen–Macaulayness,” Algebr. Geom. Topol. 14 (1), 379–406 (2014).
S. Mac Lane, Homology (Springer, Berlin, 1963), Grundl. Math. Wiss. 114.
J. McCleary, A User’s Guide to Spectral Sequences, 2nd ed. (Cambridge Univ. Press, Cambridge, 2001), Cambridge Stud. Adv. Math. 58.
D. Notbohm and N. Ray, “On Davis–Januszkiewicz homotopy types. I: Formality and rationalisation,” Algebr. Geom. Topol. 5 (1), 31–51 (2005).
T. Panov and S. Theriault, “The homotopy theory of polyhedral products associated with flag complexes,” Compos. Math. 155 (1), 206–228 (2019).
T. E. Panov and Ya. A. Veryovkin, “Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups,” Sb. Math. 207 (11), 1582–1600 (2016) [transl. from Mat. Sb. 207 (11), 105–126 (2016)].
Funding
The work of the first author (Sections 1–3) was supported by the Russian Science Foundation under grant no. 20-11-19998, https://rscf.ru/project/20-11-19998/, and performed at Steklov Mathematical Institute of Russian Academy of Sciences. The work of the second author (Section 4) was performed at the Steklov International Mathematical Center and supported by the Ministry of Science and Higher Education of the Russian Federation (agreement no. 075-15-2022-265).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2022, Vol. 317, pp. 157–167 https://doi.org/10.4213/tm4282.
Rights and permissions
About this article
Cite this article
Panov, T.E., Zeinikesheva, I.K. Equivariant Cohomology of Moment–Angle Complexes with Respect to Coordinate Subtori. Proc. Steklov Inst. Math. 317, 141–150 (2022). https://doi.org/10.1134/S0081543822020079
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543822020079