The vector bundle decomposition. (Q1880085)

The subject of this paper is the question whether a given vector bundle can be decomposed (or split) into a Whitney sum of its subbundles. The authors claim that in the real case the literature is sketchy at best (the reviewer will comment on this below), and they continue: ``In this paper we shall begin the study of the decomposition of real vector bundles. \dots We give a general decomposition result (Theorem 2.1.5) which relates a given vector bundle to some cohomology classes with local coefficients in the homotopy group of a Grassmann manifold; it is those classes that obstruct the decomposition. Those classes are natural with respect to the induced vector bundle by a map (see 2.1.7). For some special decompositions, we gave a relationship between those classes and the well-known characteristic classes such as Stiefel-Whitney classes and Chern classes (see 2.2.8, 2.2.9 and 2.2.10). We find applications in the study of subbundles of low codimension.'' Reviewer's remarks: (1) Contrary to the opinion expressed by the authors, there is a wide range of relevant works: \textit{M. Crabb} and \textit{B. Steer}'s [Proc. Lond. Math. Soc. 30, 1--39 (1975; Zbl 0294.57015)], \textit{U. Koschorke}'s book [Vector fields and other vector bundle morphisms -- a singularity approach (1981; Zbl 0459.57016)], his paper [Topology Appl. 75, 261--286 (1997; Zbl 0870.55011)], \textit{R. Stong}'s paper [Proc. Am. Math. Soc. 84, 576--580 (1982; Zbl 0503.55013)], or also \textit{E. Thomas}' lecture notes [Seminar on fiber spaces (1966; Zbl 0151.31604)], to name just a few. (2) For the proof of Corollary 2.1.10 (``Let \(\xi^ m\) be an \(m\)-dimensional vector bundle over a connected \(N\)-dimensional \(CW\)-complex \(X\). If \(N\leq m-k\), then \(\xi^ m\) can be decomposed as a Whitney sum \(\xi^ m=\xi^ k\oplus \xi^ {m-k}\).''), the authors refer to Theorem 2.1.5 and other results from the paper under review. But the cited result (in a stronger form) is well known; cf. for instance p. 112 in \textit{D. Husemoller}'s book [Fibre bundles. 3rd ed. (1993; Zbl 0794.55001)]. (3) In Example 3.4, the authors say: ``Let \([M_ {2k+1}]\in MO_{2k+1}\) be a \((2k+1)\)-dimensional cobordism class, then we can choose \(M_ {2k+1}\) such that \(\tau(M_ {2k+1})\approx \xi^ {2k}\oplus \mathbb{R}\).'' They give a long proof, but again, a stronger result is well known: the Hopf theorem on non-vanishing tangent vector fields (cf. 39.8 in \textit{N. Steenrond}'s book [The topology of fibre bundles (1951; Zbl 0054.07103)]) readily implies that the tangent bundle of any odd-dimensional closed connected smooth manifold has a trivial \(1\)-dimensional subbundle.