Deficiencies of smooth manifolds
For a given closed manifold \(M^ n\), there are involutions \((X^{2n},T)\) on closed manifolds having the fixed point set M, which give us the integers \[ \delta (M,X)=\{\dim H^*(X;{\mathbb{Z}}_ 2)-\dim H^*(M;{\mathbb{Z}}_ 2)\}/2\geq 0; \] and the deficiency \(\eta\) (M) due to Kulkarni is the minimum of all \(\delta\) (M,X). This paper exhibits manifolds M for which \(\eta (M)>0\), and studies general conditions under which \(\delta (M,X)>0\) for any such involution (X,T) whose normal bundle to the fixed point set is the tangent bundle of M.
scientific article; zbMATH DE number 3915257
scientific article