  In this paper, we prove for $n\leq 7$ that if a differentiable $n$-manifold
contains a relatively incompressible essential hypersurface in some class
$\mathcal C_{deg}$, then it admits no complete metric with positive scalar
curvature. Based on this result, we show for $n\leq 7$ that surgeries between
orientable $n$-manifolds and $n$-torus along incompressible sub-torus with
codimension no less than $2$ still preserve the obstruction for complete
metrics with positive scalar curvature. As an application, we establish
positive mass theorem with incompressible conditions for asymptotically
flat/conical manifolds with flat fiber $F$ (including ALF and ALG manifolds),
which can be viewed as a generalization of the classical positive mass theorem
from \cite{SY79PMT} and \cite{SY2017}. Finally, we investigate Gromov's fill-in
problem and bound the total mean curvature for nonnegative scalar curvature
fill-ins of flat $2$-toruses (an optimal bound is obtained for product
$2$-toruses). This confirms the validity of Mantoulidis-Miao's definition of
generalized Brown-York mass in \cite{MM2017} for flat $2$-toruses.
