S. Zhang has suggested the study of the following question for a particular projection $p$ in a $C^{*}$-algebra $A$ : Is every projection which is unitarily equivalent to $p$ necessarily homotopic to $p$ ? It was shown by Effros, Kaminker and Zhang that the answer is yes if $A$ is a unital or non-unital purely infinite simple $C^{*}$ - algebra , and by Zhang that the answer is yes if $A$ has real rank zero and (topological) stable rank one. We show that the answer is yes whenever $A$ has stable rank one. We also give an example where $A$ is extremally rich and of real rank zero and the answer is no. A second theorem makes an additional hypothesis which rules out such examples. In addition the paper discusses the concept of extremal richness and its $K$-theoretic consequences.