Rank Constrained Homotopies

Original · EN

For any n≥ k≥ l, let S(n,k,l) be the set of all those non-negative definite matrices a∈ Mₙ(C) with l a≤ k. Motivated by applications to C*-algebra theory, we investigate the homotopy properties of continuous maps from a compact Hausdorff space X into sets of the form S(n,k,l). It is known that for any n, if k-l is approximately 4 times the covering dimension of X then there is only one homotopy class of maps from X into S(n,k,l), i.e. C(X,S(n,k,l)) is path connected. In our main Theorem we improve this bound by a factor of 8. By combining classical homotopy theory methods with C*-algebraic techniques we also show that if πᵣ(S(n,k,l)) vanishes for all r≤ d then C(X,S(n,k,l)) is path connected for any compact Hausdorff X with covering dimension not greater than d.