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TORUS ACTIONS OF COMPLEXITY ONE IN NON-GENERAL POSITION

Ayzenberg, Anton 大阪大学 DOI:10.18910/84953

2021.10

概要

Let the compact torus Tn−1 act on a smooth compact manifold X2n effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space X2n/Tn−1 if the action is cohomologically equivariantly formal (which essentially means that Hodd(X2n; Z) = 0)? It happens that homology of the orbit space can be arbitrary in degrees 3 and higher. For any finite simplicial complex L we construct an equivariantly formal manifold X2n such that X2n/Tn−1 is homotopy equivalent to Σ3L. The constructed manifold X2n is the total space of a projective line bundle over the permutohedral variety hence the action on X2n is Hamiltonian and cohomologically equivariantly formal. We introduce the notion of an action in j-general position and prove that, for any simplicial complex M, there exists an equivariantly formal action of complexity one in j-general position such that its orbit space is homotopy equivalent to Σ j+2 M.

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参考文献

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