Toric Topology and Polyhedral Products

This book explores toric topology, polyhedral products and related mathematics from a wide range of perspectives, collectively giving an overview of the potential of the areas while contributing original research to drive the subject forward in interesting new directions. Contributions to this volume were written in connection to the thematic program Toric Topology and Polyhedral Products held at the Fields Institute from January-June 2020. 16 original conributions were inspired or influenced by the program.

Toric Topology arose as a subject in its own right about twenty-five years ago. It sits at the intersection of commutative algebra, topology, combinatorics, algebraic geometry, and symplectic and convex geometry. Polyhedral products are a functorial generalization of a construction that is at the centre of Toric Topology. They are of independent interest and unify several constructions that arise in a diverse range of areas, such as geometric group theory, homotopy theory, algebraic combinatorics and subspace arrangements.

High calibre contributions play a role in answering important questions and generating new research Explains the reach of toric topology and polyhedral products and their usefulness in diverse areas of mathematics Interdisciplinary content is useful for providing a constructive and productive overview of an exciting mathematical area

Auteur
Anthony Bahri is Professor of Mathematics at Rider University. He obtained his D. Phil. in 1980 from the University of Oxford and held postdoctoral positions at Purdue University and at Rutgers University. His research area in algebraic topology includes bordism theory, homotopy theory, polyhedral products, toric spaces and toric varieties, mainly from the topological point of view.

Lisa Jeffrey is Professor of Mathematics at University of Toronto. She obtained her D.Phil. in 1992 at University of Oxford (under the supervision of Michael Atiyah) and then held postdoctoral positions at IAS and Cambridge University. She held a junior faculty position at Princeton University (1993-5) followed by a tenure-track position at McGill University (1995-8) before moving to her present position in 1998. Her research area is symplectic geometry and mathematical physics.

Taras Panov is Professor of Mathematics at Moscow State University. He obtained his PhD in 1999 at Moscow State University and then held postdoctoral positions at the University of Manchester and Osaka City University. His research area is cobordism theory, toric topology, geometry and topology of manifolds, and homotopy theory of polyhedral products.

Don Stanley received his PhD from the University of Toronto in 1997. After postdoctoral positions in Europe and Canada he moved to the University of Regina where he is now a professor in the Department of Mathematics and Statistics. His thesis was on ring spectra and he subsequently worked on Lusternik-Schnirelmann category, rational homotopy theory and classifications problems in derived and abelian categories. These days his interests have shifted towards topological data analysis and using polyhedral products and other techniques to study which graded algebras are the cohomology of spaces.

Stephen Theriault is a Professor of Mathematics at the University of Southampton. Heearned a PhD at the University of Toronto in 1997. After having postdoctoral positions at MIT, the University of Illinois at Chicago and the University of Virginia, he held a position at the University of Aberdeen before moving to Southampton. His research area is homotopy theory, and he has done work on the homotopy theory of spheres and Moore spaces, Lie groups and gauge groups, manifolds and polyhedral products.

Contenu

Preface.- Connected sums of sphere products and minimally non-Golod complexes.- Toric manifolds over 3-polytopes.- Symmetric products and a Cartan-type formula for polyhedral products.- Multiparameter persistent homology via generalized Morse theory.- Compact torus action on the complex Grassmann manifolds.- On the enumeration of Fano Bott manifolds.- Dga models for moment-angle complexes.- Duality in toric topology.- Bundles over connected sums.- The SO(4) Verlinde formula using real polarizations.- GKM graph locally modelled by TnxS1-action on TCn and its graph equivariant cohomology.- On the *genera of moment-angle manifolds associated to dual-neighborly polytopes: combinatorial formulas and sequences.- Homeomorphic model for the polyhedral smash product of disks and spheres.- Invariance of polarization induced by symplectomorphisms.- Polyhedral products for wheel graphs and their generalizations.- On the cohomology ring of real moment-angle complexes.