On the cohomology of real algebraic varieties

A class of spaces with involution introduced by the author is studied:effective spaces, whose cohomology rings of fixed-point sets are completely determined by the spectral sequence of involution. Real algebraic varieties admitting a “cellular decomposition” are effective $M$-spaces. Under certain restrictions, one calculates the spectral sequence of involution and the total $\mathbb Z_2$ Betti number of the real part for real subvarieties of real algebraic varieties that are effective $GM$-spaces.