Linear and projective splittings of bundles
The purpose of the paper is to record a result which relates stable splittings of linear bundles to corresponding stable splittings of projective bundles. To be more precise we need a bit of notation. Let X be a finite connected CW-complex. Let \(\xi_ 1,\xi_ 2,...,\xi_ r\) and \(\eta\) be complex vector bundles over X with dim \(\xi\) \({}_ i>0\) for each i and dim \(\eta\) \(=\dim \xi\), where \(\xi =\xi_ 1\oplus \xi_ 2\oplus...\oplus \xi_ r\). Let \(\Pi_ X(P\xi_ 1,P\xi_ 2,...,P\xi_ r)\) be the fibre product over X of the projective bundles \(P\xi_ i\) associated with \(\xi_ i\), and let \(\gamma_ i\) denote the canonical line bundle over \(P\xi_ i\). Let \(G_ r\eta\) denote the Grassmann bundle of r-planes associated with \(\eta\) and \(\gamma\) (\(\eta)\) the canonical r-plane bundle over \(G_ r\eta.\) Theorem: Suppose that there exists a map \(f: \Pi\) \({}_ X(P\xi_ 1,P\xi_ 2,...,P\xi_ r)\to G_ r\eta\) over X such that the bundle \(f^*\gamma (\eta)\) is stably equivalent to \(\gamma_ 1\oplus \gamma_ 2\oplus...\oplus \gamma_ r\). Then the bundle \(\xi\) is stably equivalent to \(\eta\). The theorem follows from a more technical result on K-theory of the bundles.
scientific article; zbMATH DE number 3986182