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There is a conjecture that a tower of smooth subvarieties V(n) with fixed codimension l in Gk(ℂn) must be a standard example. It is shown that even under topological hypotheses, all cohomological invariants of such a tower must coincide with those of standard examples.
For every k1 0 < k < m ≧ n, there are linear spaces of real n × m matrices which have dimension (m − k)(n − k) and every nonzero element has rank greater than k. Examples of such spaces are constructed and conditions are given under which they have the largest possible dimension.
Restrictions are given on the dimensions m and k for which there is a map f: ℝ m → ℝk whose Jacobian has rank k in a neighbourhood of a singular point if f is either quadratic or even. The restrictions are shown to be best possible in the quadratic case.
Atiyah(2) defined the geometrical dimension of an element to be less than k + 1 (g dim (x) ≤ k) if there is a k-dimensional bundle over X whose stable equivalence class is x. If ξ is an n-plane complex bundle over X, we say that it has r sections if there is an (n − r)-plane bundle η such that ξ is isomorphic to η ⊕ εr where εr is the trivial r-plane bundle over X. If X has dimension 2n or less, then ξ has r sections if and only if g dim (ξ − n) ≤ n − r.
Every n-dimensional manifold admits an embedding in R2n by the result of H. Whitney . Lie groups are parallelizable and so by the theorem of M. W. Hirsch  there is an immersion of any Lie group in codimension one. However no general theorem is known which asserts that a parallelizable manifold embeds in Euclidean space of dimension less than 2n. Here we give a method for constructing smooth embeddings of compact Lie groups in Euclidean space. The construction is a fairly direct one using the geometry of the Lie group, and works very well in some cases. It does not give reasonable results for the group Spin (n) except for low values of n. We also give a method for constructing some embeddings of Spin (n), this uses the embedding of SO(n) that was constructed by the general method and an embedding theorem of A. Haefliger . Although this is a very ad hoc method, it has some interest as it seems to be the first application of Haefliger's theorem which gives embedding results appreciably below twice the dimension of the manifold. The motivation for this work was to throw some light on the problem of the existence of low codimensional embeddings of parallelizable manifolds.