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A class of effective spaces with involution introduced by the author is studied. The cohomology ring of the fixed-point set of an effective space is 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 ℤ2 Betti number of the real part for real subvarieties of real algebraic varieties that are effective GM-spaces. Bibliography: 14 titles.