Extendibility and stable extendibility of the power of the normal bundle associated to an immersion of the lens space mod 4 (Q2717460)

Let \(A\) be a subspace of a space \(X\). Let \(F\) be the field of real or complex numbers. A \(k\)-dimensional \(F\)-vector bundle \(\zeta\) over \(A\) is said to be extendible (resp. stably extendible) to \(X\) if there exists a \(k\)-dimensional \(F\)-vector bundle over \(X\) whose restriction to \(A\) is equivalent (resp. stably equivalent) to \(\zeta\). Let \(\nu(q,k)\) denote the normal bundle associated to an immersion of the \((2n+1)\)-dimensional \(\bmod q\) lens space \(L^n(q)\) in \(\mathbb{R}^{2n+1+k}\) \((k>0)\). The authors are mainly concerned with the extendibility of the square of \(\nu(4,k)\) and its complexification to \(L^m(4)\) \((m>n)\). The main result of this paper consists of six theorems. Though one of them deals with a power of \(\nu(q,k)\) for any \(q>0\), it is applied to the stable non-extendibility problem of a power of \(\nu(4,k)\) consequently. However the result contains the four positive cases. In fact, for example Theorem 1 states that the square \(\nu(4,2n+2)^2\) is extendible to \(L^m(4)\) for \(2(n+1)^2>m>n\), and if \(0\leq n\leq 8\), \(\nu(4,2n+2)^2\) is so to \(L^m(4)\) for \(m>n\). The proof is based on straightforward \(K\)-theoretic arguments and also uses some results given in a series of their papers published previously.