Relative aspherical conjecture and higher codimensional obstruction to positive scalar curvature

Motivated by the solution of the aspherical conjecture up to dimension 5 [CL20][Gro20], we want to study a relative version of the aspherical conjecture. We present a natural condition generalizing the model Xᵏ to the relative aspherical setting. Such model is closely related to submanifold obstruction of positive scalar curvature (PSC), and would be in similar spirit as [HPS15][CRZ23] in codim 2 case. In codim 3 and 4, we prove results on how 3-manifold obstructs the existence of PSC under our relative aspherical condition. This could be regarded as a relative version extension of the aspherical conjecture up to dim 5.