Rank Constrained Homotopies

TL;DR

This paper studies the homotopy properties of maps into sets of non-negative definite matrices with bounded rank, improving bounds for when these maps are homotopically trivial, with applications to $C^*$-algebra theory.

Contribution

It improves the known bounds for homotopy triviality of maps into rank-constrained matrix sets and combines homotopy theory with $C^*$-algebra techniques for broader results.

Findings

Improved the bound from 4 to 8 times the dimension for homotopy triviality.

Established conditions based on vanishing homotopy groups for path connectedness.

Applied results to $C^*$-algebra theory and matrix rank constraints.

Abstract

For any $n\geq k\geq l\in\mathbb{N},$ let $S(n,k,l)$ be the set of all those non-negative definite matrices $a\in M_{n}(\mathbb{C})$ with $l\leq\text{rank }a\leq k$. 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 $\pi_{r}(S(n,k,l))$ vanishes for all $r\leq d$ then $C(X,S(n,k,l))$ is path connected for any compact Hausdorff $X$ with covering dimension not greater than $d$.