  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.
