Abstract
A basis is presented for the R P n {\mathbf {R}}{{\mathbf {P}}^n} classes in N n ( B O ) {\mathfrak {N}_n}(BO) . This basis is used to prove that every smooth involution ( M , T ) (M,T) on a closed manifold bounds if its fixed point set is a disjoint union of odd-dimensional projective spaces of constant dimension.
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