Integral invariants of reduced plumbing graphs

Abstract

For each fixed rank $r\geq5$, the numbers of reduced algebraic plumbing presentations of an integral invariant system are unbounded as the system varies. More precisely, the unordered coprime factorizations of an odd integer $N>1$ give $2^{\omega(N)-1}$ distinct connected reduced spin presentations with nonzero labels and primitive normalized Pontryagin form. In dimension six, these presentations define one oriented diffeomorphism class admitting positive Ricci curvature. For fixed $r$, the resulting manifolds have a common rational homotopy type and pairwise nonisomorphic integral cohomology rings. They are indecomposable as connected sums and are not total spaces of smooth sphere bundles with positive-dimensional base and fibre. We also prove that the original invariant system determines the reduced class in rank two, whereas nonuniqueness occurs in every rank at least three. In arbitrary rank, simultaneous preservation of the restricted ambient Euclidean form, the trilinear form, and the parity class is equivalent to equality of reduced classes; the Pontryagin form is then redundant. This reconstruction uses coordinate covectors obtained from the Voronoi cell and the closed circuits of their configuration. For reduced spin graphs with nonzero labels, the fifth-order coordinate moment alone determines the reduced class.