Equivariant cohomology epimorphisms and face ring quotients for Hamiltonian and complexity one GKM$_4$ manifolds

Equivariant Cohomology Epimorphisms and Face Ring Quotients for Hamiltonian and Complexity One GKM$_4$ Manifolds

Introduction and Motivation

The paper investigates the formal algebraic structure of equivariant cohomology for manifolds with torus actions, focusing on the interplay between geometric and combinatorial properties captured by GKM theory. The primary concern is the surjectivity and module freeness of restriction maps in equivariant cohomology, and their combinatorial avatars for abstract GKM graphs. Specifically, the authors consider manifolds with GKM$_j$-type torus actions (where the isotropy weights at fixed points exhibit $j$-general position), explore extensions of GKM graphs, and detail the relationship between the cohomology rings of different group actions, emphasizing the complexity one Hamiltonian and GKM$_4$ settings.

Summary of Main Results

Two main results are established:

1. Surjectivity and Module Structure of Graph Cohomology (\textbf{Theorem M.1}): For any GKM$_3$ action of a torus $K$ on a manifold $M$, and any extension of its GKM graph to a $T$-labeled abstract GKM graph (with $K\subset T$), the restriction map in equivariant graph cohomology $$p_*: H^*_T(\Gamma_T; \Bbbk) \to H^*_K(\Gamma_K; \Bbbk)$$ is surjective, and $_j$0 is a free $_j$1-module. The kernel of $_j$2 coincides with $_j$3, where $_j$4 denotes the epimorphism $_j$5 corresponding to the torus inclusion.
2. Face Ring Presentation in the Complexity One GKM$_j$6 Case (\textbf{Theorem M.2}): The equivariant cohomology ring of Hamiltonian and complexity one GKM$_j$7 manifolds admits an explicit presentation as a quotient of the face ring $_j$8 by an ideal $_j$9 generated in degree two. The ideal $j$0 is directly linked to the kernel of the restriction map and the structure of the extended torus action.

Technical Overview

GKM Theory and Graph Extensions

The formalism relies on the GKM theorem, which provides a combinatorial model for the equivariant cohomology of certain torus actions on manifolds via labeled graphs, capturing the fixed-point data and isotropy representations. The paper defines abstract (unsigned and signed) GKM graphs and introduces extensions in the sense of graph labelings corresponding to larger tori.

Surjectivity fails in general for GKM$j$1 graphs but holds for GKM$j$2, as demonstrated via counterexamples and diagram-chasing arguments built on ABFP (Atiyah–Bredon–Franz–Puppe) sequences. The proof of Theorem M.1 uses exactness properties of these sequences and leverages syzygy-theoretic results to infer module freeness and kernel structure.

Face Ring Quotients and Cohomology Computation

The authors establish a bridge from equivariant graph cohomology to face rings of simplicial posets associated with the GKM graph. In the GKM$j$3 Hamiltonian and complexity one cases, extensions to torus graphs allow for explicit face ring presentations. The key algebraic insight is that the cohomology ring is isomorphic to a quotient of the face ring $j$4 by an ideal $j$5 generated by degree two relations, with the ideal arising from the kernel of the epimorphism in the extended GKM graph.

A critical technical step is proving that every unsigned torus graph admits a lift to a $j$6-graph, ensuring that face ring models are available for the entire class under study. This is achieved using characteristic functions on face posets and congruence relations among weights.

Counterexamples and Generalization

The failure of surjectivity and freeness for GKM$j$7 is illustrated with the flag manifold $j$8 and its GKM graph extension, where the restriction map is not surjective and the module is not free. This underscores the necessity of the higher independence hypothesis ($j$9) for the main results.

Numerical Results and Contradictory Claims

The paper establishes the strong claim that for GKM$_4$0 actions and their GKM graph extensions, the restriction map in equivariant graph cohomology is always surjective, and the graph cohomology module is free. This sharply contradicts previously assumed generalizations to GKM$_4$1, exposing flaws in arguments not respecting the independence conditions.

Additionally, the explicit quotient structure of the cohomology ring in Theorem M.2, with generators and relations dictated by the face ring and its degree two ideal, gives a computational handle on the ring structure, amenable to explicit combinatorial evaluation for Hamiltonian and complexity one GKM$_4$2 manifolds.

Implications and Future Directions

Practically, these results facilitate the computation of equivariant cohomology rings for a broad class of manifolds with torus actions, extending the algebraic toolkit with face ring models and surjectivity criteria. Theoretically, they clarify module structure and syzygy properties, deepening the connection between topology, combinatorics, and commutative algebra in equivariant settings.

Future research could focus on:

• Extending these presentations to more general GKM manifolds, including higher complexity or less restrictive independence, possibly involving new combinatorial objects.
• Analyzing syzygy orders and their topological implications for other classes of group actions.
• Investigating non-Hamiltonian actions and settings with non-connected stabilizers, exploring how the face ring and ABFP sequence structures adapt.
• Algorithmic implementations for explicit ring computations, exploiting the face ring and ABFP models.

In the broader AI context, these algebraic structures and combinatorial models serve as probes for symmetry, orbit space decomposition, and knowledge representation, with potential for application in equivariant architectures and geometric learning methodologies.

Conclusion

The paper provides a rigorous algebraic foundation for equivariant cohomology of GKM$_4$3 manifolds under Hamiltonian and complexity one actions, establishing surjectivity and module freeness for GKM$_4$4 graph extensions and delivering an explicit face ring quotient description. Through detailed analysis, counterexamples, and structural theorems, the authors clarify the constraints and power of GKM theory, opening avenues for both theoretical investigation and practical computation in algebraic topology and equivariant geometry (2604.13629).