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
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. 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 coefficients 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 coefficients. On the other hand, we exhibit a
trivial summand in the tangent bundle for any odd dimensional cobordism
classes.
