[Paper Review] Localization of certain odd-dimensional manifolds with torus actions

Chen He|Osaka City University (Osaka City University)|Aug 15, 2016

Advanced Combinatorial Mathematics19 references3 citations

TL;DR

This paper extends GKM theory to odd-dimensional manifolds with torus actions by introducing a GKM-type theorem for 3-dimensional 1-skeleta, providing a graphic description of equivariant cohomology using GKM graphs. It establishes a ring isomorphism between the equivariant cohomology of certain odd-dimensional manifolds and a polynomial ring modulo relations, generalizing even-dimensional GKM theory to non-orientable, odd-dimensional settings without requiring orientability or symmetry of the cohomogeneity-one action.

ABSTRACT

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.

Motivation & Objective

• To generalize GKM theory from even-dimensional to odd-dimensional manifolds with torus actions.
• To develop a graphic description of equivariant cohomology for odd-dimensional, possibly non-orientable manifolds with 3-dimensional 1-skeleta.
• To introduce the notion of a GKM covering in the even-dimensional case to support the odd-dimensional generalization.
• To provide a ring isomorphism for equivariant cohomology in terms of polynomial rings and congruence relations, valid without assuming orientability or symmetric cohomogeneity-one structure.

Proposed method

• Revisit even-dimensional GKM theory and define a GKM covering to extend the framework to odd dimensions.
• Define the GKM condition for odd-dimensional manifolds with 3-dimensional 1-skeleta, based on isotropy weights and orbit structure.
• Construct GKM graphs from the 1-skeleton, encoding fixed points and 1-dimensional orbits with edge labels corresponding to isotropy weights.
• Use the Chang-Skjelbred Lemma to realize equivariant cohomology as a subring of the cohomology of fixed points, subject to congruence relations on edges.
• Establish a ring isomorphism between the equivariant cohomology of the odd-dimensional manifold and a quotient of a polynomial ring, using the GKM graph data.
• Apply the method to examples including products of even and odd GKM manifolds, contact and cosymplectic manifolds, and odd-dimensional Grassmannians.

Experimental results

Research questions

• RQ1 How can GKM theory be extended to odd-dimensional manifolds with torus actions, especially when the 1-skeleton is 3-dimensional?
• RQ2 What conditions ensure that the equivariant cohomology of an odd-dimensional manifold is determined by its 1-skeleton, even if the manifold is non-orientable?
• RQ3 Can a graphic description of equivariant cohomology be constructed for odd-dimensional manifolds using GKM graphs and congruence relations?
• RQ4 How does the GKM covering construction in even dimensions support the odd-dimensional generalization?
• RQ5 What is the structure of the equivariant cohomology ring for odd-dimensional cohomogeneity-one manifolds without assuming $K_-=K_+$ or orientability?

Key findings

• The equivariant cohomology of an odd-dimensional, possibly non-orientable manifold with a 3-dimensional 1-skeleton is isomorphic to a subring of the product of equivariant cohomologies of fixed points, subject to congruence relations on edges of the GKM graph.
• For the 7-dimensional manifold $N_G^7$, the equivariant cohomology ring is isomorphic to $H^*_{T^3}(b{C}P^2)[r]/r^2$, matching the structure $H^*_{T^3}(b{C}P^2) imes H^*(S^3)$.
• The GKM graph of $N_G^7$ consists of three inner $oxplus$ vertices representing $S^2 imes S^1$ and three outer $oxplus$ vertices representing $S^3$, with edge labels corresponding to differences of weights $eta_i = eta_j - eta_k$.
• The congruence relations for the cohomology classes are encoded in a system of six modular conditions on the components $f_i, g_i$, with $g_i$ divisible by the corresponding edge label.
• The method generalizes beyond orientable or symmetric cohomogeneity-one manifolds, as demonstrated by the construction of $M o G imes_{K_0} S^{n_0+1}$ without requiring $K_- = K_+$.
• The framework does not require orientability or equivariant formality in the general case, distinguishing it from prior cohomogeneity-one results that assume orientability.