[Paper Review] Lefschetz Properties and Basic Constructions on Simplicial Spheres

Eric Babson, Eran Nevo|ArXiv.org|Feb 7, 2008

Advanced Combinatorial Mathematics12 references4 citations

TL;DR

This paper establishes that the strong-Lefschetz property (SL) is preserved under three fundamental constructions on simplicial homology spheres: join, connected sum, and stellar subdivision. The key contribution is proving that if a homology sphere is SL over a field of characteristic zero (or any field for connected sum), then so are its join, connected sum, and stellar subdivisions—providing critical algebraic tools toward proving the g-conjecture for piecewise-linear spheres.

ABSTRACT

The well known $g$-conjecture for homology spheres follows from the stronger conjecture that the face ring over the reals of a homology sphere, modulo a linear system of parameters, admits the strong-Lefschetz property. We prove that the strong-Lefschetz property is preserved under the following constructions on homology spheres: join, connected sum, and stellar subdivisions. The last construction is a step towards proving the $g$-conjecture for piecewise-linear spheres.

Motivation & Objective

• To prove that the strong-Lefschetz property is preserved under basic topological constructions on simplicial homology spheres.
• To support the g-conjecture for PL spheres by showing that SL is preserved under constructions that generate all PL spheres.
• To provide algebraic evidence for the g-conjecture via the strong-Lefschetz property in Stanley-Reisner rings modulo linear systems of parameters.
• To explore the behavior of the SL property under stellar subdivisions, a key step toward proving the g-conjecture for piecewise-linear spheres.
• To establish that the SL property is preserved under join and connected sum, even over arbitrary fields, when the input complexes are SL.

Proposed method

• Uses the Stanley-Reisner ring construction and reduction modulo a linear system of parameters (l.s.o.p.) to analyze the algebraic structure of simplicial spheres.
• Applies Mayer-Vietoris and Euler characteristic arguments to show that connected sums of homology spheres are again homology spheres with symmetric h-vectors.
• Employs exact sequences of A-modules to relate the homology and cohomology of the original complex, its connected sum, and the link of a face.
• Uses commutative algebra techniques, particularly the behavior of multiplication maps by a generic element ω, to verify the strong-Lefschetz property via isomorphisms in graded components.
• Leverages the fact that the set of Lefschetz elements is Zariski open and nonempty, so their intersection is nonempty, to construct a common Lefschetz element for the connected sum.
• Relies on the algebraic shifting framework and properties of exterior and symmetric shifting to interpret the results in terms of combinatorial shifting.

Experimental results

Research questions

• RQ1 Does the strong-Lefschetz property persist under the join of two simplicial homology spheres over a field of characteristic zero?
• RQ2 Is the strong-Lefschetz property preserved under the connected sum of two homology spheres, even over arbitrary fields?
• RQ3 Does stellar subdivision of a homology sphere preserve the strong-Lefschetz property, particularly when the link of the subdivided face is SL?
• RQ4 Can the strong-Lefschetz property be used to prove the g-conjecture for piecewise-linear spheres via a sequence of algebraic constructions?
• RQ5 Is there a connection between the SL property of a complex and that of its barycentric subdivision?

Key findings

• The strong-Lefschetz property is preserved under the join of two homology spheres over a field of characteristic zero, provided both are SL.
• The connected sum of two SL homology spheres of the same dimension is SL over any field, not just characteristic zero.
• Stellar subdivision preserves the strong-Lefschetz property over ℝ if both the original complex and the link of the face are SL.
• The barycentric subdivision of an SL homology sphere is itself SL, as it is a special case of stellar subdivision.
• The SL property is preserved under the closure of a class of complexes under join, connected sum, and stellar subdivisions, provided the class is closed under taking links.
• The results imply that any PL-sphere obtained via a sequence of stellar subdivisions from a simplex inherits the SL property, assuming the link condition is maintained at each step.