Stable extendibility of vector bundles over real projective spaces and bounds for the Schwarzenberger numbers \(\beta(k)\)
Let \(X\) be a space, \(A\) a subspace and \(i: A\rightarrow X\) the inclusion map. Let \(F\) denote the field of real or complex numbers. A \(k\)-dimensional \(F\)-vector bundle \(\zeta\) over \(A\) is called stably extendible to \(X\) if there exists a \(k\)-dimensional \(F\)-vector bundle \(\eta\) over \(X\) such that \(i^{*}\eta\) is stably equivalent to \(\zeta .\) The authors consider the problem of stable extendibility of real (resp. complex) vector bundles over \(\mathbb R P^n\) to real (resp. complex) vector bundles over \(\mathbb R P^m\) (\(m>n\)). In each case they give four equivalent conditions characterizing when a stable real (resp. complex) \(k\)-dimensional vector bundle \(\zeta\) over \(\mathbb R P^n\) is stably extendible to \(\mathbb R P^m\) for every \(m\geq n.\) The conditions are formulated by means of the Schwarzenberger number \({\beta}(k)\) and the function \({\phi}(n)\). These arithmetic functions are defined as follows: Let \(i=(2a+1)2^{{\nu}(i)}\) then \({\beta}(k)=\min\{i-{\nu}(i)-1 \mid k<i\},\) \({\phi}(n)\) is the number of integers \(s\) with \(0 < s \leq n\) and \(s\) is congruent to \(0,1,2,4\mod 8.\)
scientific article; zbMATH DE number 2123907