We develop a space-level refinement of the 2-factor homology by constructing a stable homotopy type associated to a certain family G of planar trivalent graphs equipped with perfect matchings. Specifically, we define a cover functor from the 2-factor flow category C (Γ₌) to the cube flow category C₂ (n), where the perfect matching graph Γ₌ represents a planar trivalent graph G together with a perfect matching M, such that (G, M) G. By applying the Cohen--Jones--Segal realization to the 2-factor flow category C (Γ₌), we obtain the 2-factor spectrum. This spectrum serves as a space-level version of the 2-factor homology, analogous to the Lipshitz--Sarkar Khovanov spectrum for links. We show that the cohomology of the 2-factor spectrum with Z₂-coefficients is isomorphic to the 2-factor homology, as defined by Baldridge. We prove that the stable homotopy type of the 2-factor spectrum is an invariant of planar trivalent graphs G equipped with perfect matchings M, whenever (G, M) G. Furthermore, we show that the closed webs obtained by performing flattenings at each crossing of an oriented link diagram in the context of sl₃ link homology belong to the family G.
Nidhi S. Bhattacharyya (Sun,) studied this question.