Remarks on the bordism intersection map (Q864058)

The authors consider differentiable maps \(f:M\rightarrow N\) and \(g:K\rightarrow N\) where \(M, K, N\) are smooth closed manifolds of dimensions \(m, k, n\), respectively and give a characterisation of the kernel of the bordism intersection map \(I_{g}:\eta _{m}(N)\rightarrow \eta _{m+k-n_{{}}}(N) \) obtained from the bordism intersection product \(I_{m,k}\). Some related results presented in this paper are: Theorem 4.1 : ``The set of bordism classes of \(C^{\infty }\) maps \(f:M\rightarrow N\) such that \(\operatorname{rank}df(x)\leq p\) for all \(x\) is contained in \(J_{p,m-p}(N)\), where \(M\) and \(N\) are smooth closed manifolds of dimension \(m\) and \(n\), respectively, \(df\) is the differential of \(f\), \(J_{p,m-p}(N)\) is the image of the homomorphism \(l_{\ast }:\eta _{m}(N^{(p)})\rightarrow \eta _{m}(N)\) induced by the inclusion, \( 0\leq p\leq m\) and \(N^{(p)}\) is the \(p\)-skeleton of \(N\)'' and Theorem 4.2 : ``The set of bordism classes of \(C^{r}\) maps \(f:M\rightarrow N\) with \(r\geq \max \{1,(m-p)/(s+1)\}\), \(s\) and \(p\) being nonnegative integers such that \(\operatorname{rank} df(x)\leq p\) for all \(x\) is contained in \(J_{p+s,m-p+s}(N)\), where \(M\) and \(N\) are smooth closed manifolds of dimension \(m\) and \(n\), respectively.'' Interesting examples in which the conditions for these theorems are satisfied, are also given.