PSD - Dr Amaranta Membrillo Solis

Toric spaces from finite posets 

Supervisor: Dr Amaranta Membrillo Solis

Project description:

Toric topology studies spaces constructed from combinatorial data that support torus actions, revealing deep relationships between topology, combinatorics, and geometry. Classical examples include moment-angle complexes and quasitoric manifolds, where the underlying combinatorics arises from simplicial complexes or convex polytopes. Recent generalisations have broadened this framework to include constructions based on finite posets, allowing combinatorial data that go beyond simplicial or polytopal settings. This extension enables the definition of toric-type spaces indexed by finite posets, providing a unified approach that encompasses smooth, cornered, and orbifold-like structures. 

This project will investigate the topology of toric spaces derived from finite posets, building upon and connecting the theories of Panov–Lü, Yu, and Kishimoto–Levi. It will explore how the combinatorial and isotropy data encoded in a poset determine the global topological and cohomological properties of the resulting spaces, and under what conditions these admit natural orbifold interpretations. The methodology will combine techniques from homotopy theory and  geometric topology, aiming to generalise the rich algebraic and geometric behaviour of classical toric spaces to the poset setting. 

Beyond its theoretical contributions, the project seeks to provide new computational and conceptual tools applicable to stratified and symmetry-driven spaces in areas such as configuration-space topology, data geometry, and mathematical physics. 

Objectives: 

• Define a functorial construction assigning to each finite poset P, possibly with isotropy data, a toric space Z(P), generalising moment-angle and quasitoric constructions. 

• Construct an explicit algebraic model for cohomology, extending the Buchstaber–Panov face ring to a poset ring, and prove its isomorphism (or spectral-sequence convergence) with the cohomology ring of Z(P). 

• Develop cochain-level and homotopy-colimit models describing Z(P) as a homotopy colimit of diagrammatic data, and analyse how combinatorial operations on P affect the resulting cohomology. 

• Construct and study explicit examples from low-dimensional posets to illustrate cohomology models, ring structures, and potential orbifold phenomena. 

References:

• Davis, M. & Januszkiewicz, T. (1991). Convex polytopes, Coxeter orbifolds and torus actions. Duke Math. J., 62(2). 

• Buchstaber, V. M., & Panov, T. E. Toric Topology. American Mathematical Society, Providence, RI; University Lecture Series, Vol. 73, 2015. 

• Yu, L. A generalization of moment-angle manifolds with noncontractible orbit spaces. Algebraic & Geometric Topology, Vol. 24 (2024), No. 1, 

• Lü, Z. and Panov, T. “Moment-angle complexes from simplicial posets.” Open Mathematics, vol. 9, no. 4 (2011), pp. 715–730. 

• Kishimoto, D. & Levi, R. (2022). Polyhedral products over finite posets. Kyoto J. Math., 62(3), 615–654. 

Funding Notes:

Further information: 
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Entry requirements 
Fees and funding

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