On the automorphisms of principal fibre bundles
Let be the -th ordered configuration space of all distinct points in the Grassmannian of -dimensional subspaces of , whose sum is a subspace of dimension . We prove that is (when non empty) a complex submanifold of of dimension and its fundamental group is trivial if , and and equal to the braid group of the sphere
We prove that the space of nonempty subsets of cardinality at most k in a bouquet of m+1-dimensional spheres is (m+k-2)-connected. This, as shown by Tuffley, implies that the space is (m+k-2)-connected for any m-connected cell complex X.
Necessary and sufficient conditions for the existence of two linearly independent sections in an 8-dimensional spin vector bundle over a CW-complex of the same dimension are given in terms of characteristic classes and a certain secondary cohomology operation. In some cases this operation is computed.