Let $Y$ be a compact, Hausdorff, contractible topological space, and $X$ be a locally compact Hausdorff space which is homeomorphic to a dense subset of $Y$. 

Question A: Is $GL_1(C(Y))\stackrel{\pi}{\longrightarrow} K_1(C(Y))$ surjective? 

Question B: If $r:C(Y)\rightarrow C(Y\setminus X)$ denotes the restriction map, then does the following diagram commute? 
$$\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}
\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex}
\newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}
\begin{array}{c}
GL_1(C(Y)) & \ra{r} & GL_1(C(Y\setminus X))    \\
\da{\pi} & & \da{\pi}     \\
K_1(C(Y)) & \ras{K_1(r)} & K_1(C(Y\setminus X))      \\
\end{array}
$$