Determine the full integral saturation lattice of odd matching complexes

Determine whether the canonical divided-power lattice N^(k) equals the saturation of Bouc’s hook-cycle lattice in the top integral homology of the matching complex of K_{2k+1}, and determine all invariant factors of the quotient between the saturated lattice and the original hook-cycle lattice.

Background

For the matching complex of the complete graph K_{2k+1}, Bouc’s hook cycles define an integral lattice inside the free top homology lattice. The paper proves that the quotient of the saturation by this cycle lattice is a finite 2-group with exact exponent 2^{floor(k/2)}, and constructs divided-power cycles realizing every order up to that exponent.

The paper introduces a canonical divided-power sublattice N^k contained in the saturation but does not prove equality. Determining the invariant factors of the full quotient would refine the exponent theorem and describe the complete integral lattice rather than only its largest denominator.