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We compute the cotorsion product of the mod 2 cohomology of spinor groupspin(n), which is the E_2-term of the Rothenberg-Steenrod spectral sequence forthe mod 2 cohomology of the classifying space of the spinor group spin(n). As aconsequence of this computation, we show the non-collapsing of theRothenberg-Steenrod spectral sequence for n > 16.

On the Rothenberg-Steenrod spectral sequence for the mod 2 cohomology of classifying spaces of spinor groups