Abstract

In this paper, we studied the real vector bundle decomposition problem. We first give a general decomposition result which relates a given vector bundle to some cohomology classes with local coeficients 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. 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. We determined the local coeficients in the cohomology group which contain the decomposition obstruction classes. We find applications in the study of subbundles of low codimension. In particular, codimension 1 decomposition classes are investigated in which we find that one of the two decomposition classes for the universal bundle over BO(2n + 1) is in H²ⁿ⁺¹(BO(2n + 1),Z). This result gives rise to a new geometric interpretation for the order two elements in the integral cohomology group in odd dimension. We further make use of the cellular structure of the classifying space BO(n) to see the ‘local’ structure for the restriction of the universal bundle to each cell. In this way, we can construct the obstruction classes for the codimension 1 vector bundle decomposition. We gave an example to calculate the decomposition obstruction for the tangent bundle of RP²ⁿ, which turns out to be the generator in the cohomology of RP²ⁿ with twisted integer coeficients. On the other hand, we exhibit a trivial summand in the tangent bundle for any odd dimensional cobordism classes.

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