Let \(M^ n\) be a smooth closed n-manifold, T a smooth involution on \(M^ n\). Now one considers the case in which the fixed point set of T has constant dimension n-k [R. E. Stong, Math. Z. 178, 443-447 (1981; Zbl 0469.57027)]. Let \(J^ k_ n\) be the subset of the unoriented bordism group \(MO_ n\) which consists of the bordism classes which contain a member \(M^ n\) with the above property. Let \(A^ s_ n\subset MO_ n\) be the set which consists of classes for which all characteristic numbers divisible by products \(w_{2i_ 1+1}...w_{2i_ s+1}\) are zero, where the \(w_ j\) are Stiefel-Whitney classes. Then the main result of the paper is: i) \(J^ 8_ n=A_ n^{n-15}\); ii) if \(k=9,11,13,15,17\), \(J^ k_ n=J^ 3_ n\cap A_ n^{n-2k+1}\).