Let a torus $T$ act smoothly on a compact smooth manifold $M$. If the rational equivariant cohomology $H^*_T(M)$ is a free $H^*_T(pt)$-module, then according to the Chang-Skjelbred Lemma, it can be determined by the $1$-skeleton consisting of the $T$-fixed points and $1$-dimensional $T$-orbits of $M$. When $M$ is an even-dimensional, orientable manifold with 2-dimensional 1-skeleton, Goresky, Kottwitz and MacPherson gave a graphic description of the equivariant cohomology. In this paper, first we revisit the even-dimensional GKM theory and introduce a notion of GKM covering, then we consider the case when $M$ is an odd-dimensional, possibly non-orientable manifold with $3$-dimensional $1$-skeleton, and give a graphic description of its equivariant cohomology.
An early draft of this paper was part of the author's PhD thesis at Northeastern University, Boston. The author thanks Shlomo Sternberg for teaching him symplectic geometry. The author thanks Victor Guillemin and Jonathan Weitsman for guidance. The author thanks Catalin Zara for many useful discussions and carefully reading the manuscript, and thanks Volker Puppe, Oliver Goertsches and Liviu Mare for helpful suggestions. The author also thanks the anonymous referees for their excellent comments. The author thanks the Ling-Ma graduate research fund at Northeastern University and the China Postdoctoral Science Foundation (Grant No.\,2018T110083) for the generous support.
"Localization of certain odd-dimensional manifolds with torus actions." Osaka J. Math. 58 (3) 609 - 635, July 2021.