Existence of examples satisfying the vanishing conditions on the u_* map in the S-complex

Determine whether there exist rational homology three-spheres Y equipped with metrics g for which the induced map u_* in the S-complex of monopole Floer homology satisfies the vanishing conditions required in Theorem 5.19 (for example, u_* is trivial on ker(δ_{1*}) or maps H_{q−3}(Y, m_𝔰; F) to H_{q−5}(Y, m_𝔰; F)/im(δ_{2*}) trivially), thereby yielding the inequalities λ ≥ ρ established in that theorem.

Background

Theorem 5.19 provides inequalities relating the spectral invariants λ and ρ under specific algebraic vanishing conditions on the u_* map in the S-complex associated to monopole Floer homology. These conditions involve the induced maps δ{1*}, δ{2*}, and u_* on homology.

The author points out that, at present, it is unclear whether concrete examples (Y, g) satisfying these vanishing conditions exist. Identifying such examples would validate the applicability of Theorem 5.19’s inequalities and clarify the algebraic structure of the S-complex in geometric settings.