Deficiencies of smooth manifolds

Being given a closed manifold Mⁿ, there are involutions (X²ⁿ, T) on closed manifolds of twice the dimension having fixed point set M. Kulkarni defined the deficiency of M for a class of involutions to be $\text{min}\text{(}\text{1}\text{2}\text{{}\text{dim}\text{H}{}^1\text{(X;Z}{}_{\mathrm{<mtext>2</mtext>}}\text{)?}\text{dim}\text{H}{}^1\text{(M;Z}{}_{\mathrm{<mtext>2</mtext>}}\text{)})}$ for all involutions (X, T) in the class. This paper exhibits manifolds for which the deficiency is positive for all involutions and studies the deficiencies for other classes.