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? 
$$\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex}
\newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex}
\newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}
\begin{array}{c}
GL\_1(C(Y)) & \ra{r} & GL\_1(C(Y\setminus X)) \\
\da{\pi} & & \da{\pi} \\
K\_1(C(Y)) & \ras{K\_1(r)} & K\_1(C(Y\setminus X)) \\
\end{array}
$$



Nonstable $K$-theory question
Regarding Question A, the $3$-torus provides a counterexamle: its complex $K\_1$-group has rank $4$ but its cohomotopy has only rank $3$. (I have learnt about this example from the book of Rordam, Larsen and Laustsen.)


The diagram in Question B clearly commutes because $K\_1(r)$ just acts by picking some representative, restricting it and then taking its $K\_1$-class.


