Theory and applications of algebraic topology

The main aim of this thesis is to make some contribution to the theory of the tangent bundle of a smooth manifold. We study the possibility of reducing the group of the bundle to various specific subgroups. Such reducibility is equivalent to the existence of certain 'structures' on the manifold. In particular we deal with the trivial subgroup and (for even-dimensional manifolds) with a unitary subgroup of maximal rank; the corresponding structures are called 'parallelism' and 'almost complex structure'. The method consists of two stages: first ask the corresponding question for the stable tangent bundle; then find criteria for moving from the stable to the unstable range. The first stage uses the cohomology theories dealing with stable vector bundles over a space, while the criteria used in the second stage arise mainly from work of M.A. Kervaire. Throughout, the theory is illustrated by application to the total space of a sphere-bundle over a sphere, with its natural smooth structure. We first ask, "For which oriented sphere-bundles over spheres is the total space parallelizable?" The answer is given in terms of the structural invariants of the bundle. In particular, if the base dimension is not greater than the fibre dimension, then a necessary condition for parallelizability of the total space is that the bundle structure be trivial. However, there exist bundles in which the total space is parallelizable although the bundle structure is not even stably trivial. A complete answer to this question is not yet available. However, the method used applies to a wider class of manifolds. For example, it is proved that any Stiefel manifold which is not a sphere, is parallelizable. Next we investigate the existence of almost complex structures on certain even-dimensional manifolds, namely those whose tangent bundles can have their groups lifted to the appropriate spinor group. In particular we apply this to products of spheres and to sphere-bundles over spheres. A product of spheres containing an odd-dimensional factor was previously known to be parallelizable, hence if such a product is even-dimensional, it certainly adults an almost complex structure. A necessary condition for a product of even-dimensional spheres to admit an almost complex structure is that the dimension of each sphere involved should be 2, 4 or 6. In particular, we show that the product of a 2-sphere with a 4-sphere admits an infinite number of distinct almost complex structures. For an oriented sphere-bundle over a sphere, we obtain the following: if the base and fibre are odd-dimensional, then the total space admits an almost complex structure if and only if it is parallelizable; if the base and fibre are even-dimensional. then a necessary condition for the total space to admit an almost complex structure is that the dimensions of the base and fibre should each be 2, 4 or 6. As observed in the first paragraph, we discuss also the corresponding stable questions. One of the main tools used in the parallelizability question is the mod 2 semi-characteristic as defined by M.A. Kervaire on p.220 of 'Courbure intégrale généralisée et homotopie' (Math. Annalen, Bd. 131, (1956) 219-252). We compute this for the total space of a differentiable fibre bundle in terms of the homology of the base and fibre and the characteristic classes of the bundle, in some simple cases. We also express the mod 2 semi-characteristic of certain (4k+l)-manifolds in terms of a secondary characteristic class of the tangent bundle of the manifold. In Appendix F we reprove a theorem of Milnor and Massey on the non-existence of almost complex structure on the quaternionic protective plane; at the same time we prove the same result for certain analogous manifolds derived from multiplications on the 3-sphere other than the quaternionic multiplication.