Rational homotopy of the spectral Sah algebra (including the n=4 summand of π0)

Determine the homotopy type and rational homotopy groups of the spectral Sah algebra S = \bigvee_{n\ge 0} \widetilde K(\mathcal{P}^{S^{n−1}_{O(n)}}), including the rational structure of π0 and, in particular, the n = 4 summand of π0 related to three-dimensional spherical geometry; analyze the four-term exact sequence involving H_3(SU(2);Q) (with SU(2) given the discrete topology) to characterize this piece.

Background

The authors define the spectral Sah algebra S as a wedge of reduced spherical scissors congruence K-theory spectra across dimensions, lifting Sah’s algebraic structure to spectra. While they establish a Hopf algebra structure and various structural properties, they note that explicit computations of its homotopy groups are largely unknown.

They emphasize that odd-dimensional summands vanish rationally, but the remaining rational homotopy type is not fully understood. Specifically, the n=4 summand of π0—which is tied to three-dimensional spherical geometry—remains elusive and is connected to a four-term exact sequence involving group homology of SU(2) with the discrete topology.