[Paper Review] Diffeomorphic moment-angle manifolds with different Betti numbers

Frédéric Bosio|arXiv (Cornell University)|Oct 13, 2014

Advanced Combinatorial Mathematics2 references4 citations

TL;DR

This paper constructs two 47-dimensional simple polytopes with 51 facets each, whose moment-angle manifolds are diffeomorphic yet have different Betti numbers, thereby disproving the conjecture that diffeomorphic moment-angle manifolds must arise from polytopes with identical Betti numbers. The construction uses multiwedges over neighborly dual polytopes and leverages the fact that moment-angle manifolds from such polytopes are diffeomorphic to connected sums of sphere products, whose structure is determined by Betti numbers.

ABSTRACT

We describe here two simple polytopes that have different Betti numbers and whose moment-angle manifolds are diffeomorphic.

Motivation & Objective

• To investigate whether diffeomorphic moment-angle manifolds must arise from polytopes with identical Betti numbers.
• To determine if Betti numbers of a moment-angle manifold uniquely determine the Betti numbers of the underlying polytope.
• To construct a counterexample where two diffeomorphic moment-angle manifolds have polytopes with different Betti numbers.
• To explore the limits of combinatorial invariants under diffeomorphism in moment-angle constructions.
• To demonstrate that the number of vertices and f-vectors are not preserved under diffeomorphism of moment-angle manifolds.

Proposed method

• Constructing multiwedges over neighborly dual polytopes with four more facets than dimension, ensuring manageable Betti number complexity.
• Using cousin extensions and biflips to generate distinct polytopes from the same Gale diagram configuration, preserving diffeomorphism type of the moment-angle manifold.
• Applying a fixed multiindex of wedging operations to two distinct polytopes derived from the same base, ensuring identical moment-angle manifold dimension and diffeomorphism type.
• Computing Betti numbers of the moment-angle manifolds via the known decomposition into connected sums of sphere products.
• Verifying that the resulting moment-angle manifolds are diffeomorphic by matching the dimensions and multiplicities of the sphere products.
• Confirming the polytopes have different Betti numbers by comparing the values of bp,q across subsets of facets.

Experimental results

Research questions

• RQ1 Can two diffeomorphic moment-angle manifolds arise from polytopes with different Betti numbers?
• RQ2 Is the Betti number of the moment-angle manifold sufficient to determine the Betti numbers of the underlying polytope?
• RQ3 Are combinatorial invariants like the number of vertices preserved under diffeomorphism of moment-angle manifolds?
• RQ4 Can multiwedges over different neighborly dual polytopes yield diffeomorphic moment-angle manifolds with distinct Betti numbers?
• RQ5 Is the f-vector of a polytope determined by the diffeomorphism type of its moment-angle manifold?

Key findings

• The moment-angle manifolds of the constructed polytopes are diffeomorphic to the connected sum #2(S37×S61)#(S38×S60)#17(S39×S59)#19(S40×S58)#20(S41×S57)#22(S42×S56)#38(S43×S55)#27(S44×S54)#29(S45×S53)#51(S46×S52)#54(S47×S51)#75(S48×S50)#36(S49×S49).
• The two polytopes have different Betti numbers: for example, b17,19 and b28,32 are 2 and 15 for P1, but 1 and 17 for P2.
• The polytope P1 has 33,686 vertices, while P2 has 33,684, showing that the number of vertices is not preserved under diffeomorphism of moment-angle manifolds.
• The moment-angle manifolds are 98-dimensional, arising from 47-dimensional polytopes with 51 facets each, using a total of 23 wedge operations.
• The Betti number b−1,0 = b46,51 = 1 is the same for both polytopes, but all other non-zero bp,q values differ, confirming distinct polytope Betti numbers.
• The diffeomorphism type of the moment-angle manifold is preserved despite distinct Betti numbers, proving that Betti numbers of the manifold do not determine those of the polytope.