Abstract
We compute the rational Borel equivariant cohomology ring of a cohomogeneity-one action of a compact Lie group.
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Notes
That is, \(H^*_{K^-}\times _{H^*_{H}} H^*_{K^+}< H^*_{K^-}\times H^*_{K^+}\) is the subring of pairs \((x_-,x_+)\) such that \(\rho ^*_{-}(x_-) = \rho ^*_{+}(x_+)\).
To make this account self-contained, the proof of normality is thus. The transitive action of \(K_0\) on \(K_0{/}H_1\) induces a map \({\lambda }:K_0 \longrightarrow {{\,\mathrm{Homeo}\,}}K_0{/}H_1\) whose image, which acts effectively by definition, can only be \(S^1\) itself if \(K_0{/}H_1 \approx S^1\) and \(\mathrm {SO}(3)\) if \(K_0{/}H_1 \approx P^3\) [4, Thm. 1.1]. As \(\ker \lambda \) stabilizes all points, it is in particular contained in \(H_1\). The stabilizer of the coset \(1H_1\in K_0{/}H_1\) under the effective action of \({{\,\mathrm{im }\,}}\lambda \) is \(\lambda (H_1) \cong H_1{/}\ker \lambda \), which must be finite since \({{\,\mathrm{im }\,}}\lambda \) is of rank one, so \(\ker \lambda \) is of finite index in \(H_1\); particularly, its identity component must be \(H_0\). Since \(\ker \lambda \) is normal in \(K_0\) by definition, so also must be \(H_0\).
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Acknowledgements
The authors would like to thank the referee for careful proofreading, for suggesting a reference, and for making an important correction to their statement of Theorem 3.2. The first author would like to thank Omar Antolín Camarena for helpful conversations and the National Center for Theoretical Sciences (Taiwan) for its hospitality during a phase of this work.
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Carlson, J.D., Goertsches, O., He, C. et al. The equivariant cohomology ring of a cohomogeneity-one action. Geom Dedicata 203, 205–223 (2019). https://doi.org/10.1007/s10711-019-00434-4
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DOI: https://doi.org/10.1007/s10711-019-00434-4