  Recently H.-L. Chang and J. Li generalized the theory of virtual fundamental
class to the setting of semi-perfect obstruction theory. A semi-perfect
obstruction theory requires only the local existence of a perfect obstruction
theory with compatibility conditions. In this paper, we generalize the torus
localization, the cosection localization and their combination, to the setting
of semi-perfect obstruction theory. As an application, we show that the
Jiang-Thomas theory of virtual signed Euler characteristic works without the
technical quasi-smoothness assumption from derived algebraic geometry.
