Homotopy of unitaries in simple C*-algebras with tracial rank one
  Let $\epsilon>0$ be a positive number. Is there a number $\delta>0$ satisfying the following? Given any pair of unitaries $u$ and $v$ in a unital simple $C^*$-algebra $A$ with $[v]=0$ in $K_1(A)$ for which $$ \|uv-vu\|<\dt, $$ there is a continuous path of unitaries $\{v(t): t\in [0,1]\}\subset A$ such that $$ v(0)=v, v(1)=1 \and \|uv(t)-v(t)u\|<\epsilon \forall t\in [0,1]. $$ An answer is given to this question when $A$ is assumed to be a unital simple $C^*$-algebra with tracial rank no more than one.   Let $C$ be a unital separable amenable simple $C^*$-algebra with tracial rank no more than one which also satisfies the UCT. Suppose that $\phi: C\to A$ is a unital monomorphism and suppose that $v\in A$ is a unitary with $[v]=0$ in $K_1(A)$ such that $v$ almost commutes with $\phi.$ It is shown that there is a continuous path of unitaries $\{v(t): t\in [0,1]\}$ in $A$ with $v(0)=v$ and $v(1)=1$ such that the entire path $v(t)$ almost commutes with $\phi,$ provided that an induced Bott map vanishes. Other versions of the so-called Basic Homotopy Lemma are also presented. 
Huaxin Lin