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Article

Torus actions of complexity one in non-general position

Let the compact torus Tn1 act on a smooth compact manifold X2n e ec-
tively with nonempty nite set of xed points. We pose the question: what can be said
about the orbit space X2n{Tn1 if the action is cohomologically equivariantly formal
(which essentially means that HoddpX2n;Zq  0)? It happens that homology of the orbit
space can be arbitrary in degrees 3 and higher. For any nite simplicial complex L we
construct an equivariantly formal manifold X2n such that X2n{Tn1 is homotopy equiv-
alent to 3L. 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 cohomolog-
ically 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 j􀀀2M.