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? $$\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} GL_1(C(Y)) & \ra{r} & GL_1(C(Y\setminus X)) \\ \da{\pi} & & \da{\pi} \\ K_1(C(Y)) & \ras{K_1(r)} & K_1(C(Y\setminus X)) \\ \end{array} $$

Let $Y$ be a compact, Hausdorff topological space, and $X$ be a locally compact, contractible, 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? $$\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}} \begin{array}{c} GL_1(C(Y)) & \ra{r} & GL_1(C(Y\setminus X)) \\ \da{\pi} & & \da{\pi} \\ K_1(C(Y)) & \ras{K_1(r)} & K_1(C(Y\setminus X)) \\ \end{array} $$