Rank Constrained Homotopies

Abstract

For any $n\ge k\ge l\in \mathbb{N},$ let $S(n,k,l)$ be the set of all those non-negative definite matrices $a\in M_n(\mathbb{C})$ with $l\le \text{rank }a\le k$. Motivated by applications to $C^{\ast }$-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 $C^{\ast }$-algebraic techniques we also show that if ${\pi }_r(S(n,k,l))$ vanishes for all $r\le d$ then $C(X,S(n,k,l))$ is path connected for any compact Hausdorff $X$ with covering dimension not greater than $d$.

References in corpus (1)

• Stability in the Cuntz semigroup of a commutative C*-algebra