Let $\varphi: A \to B$ be a surjective $^*$-homomorphism between unital $C^*$-algebras A and B, and let $u$ be a unitairy in $\mathcal{U}_n(B)$. I want to show that $[u]_1$ belongs to $Im(K_1(\varphi))$ if and only if there exist a natural number $m \geq n$ and $v $ in $\mathcal{U}_m (A)$ such that $\varphi(v)=u \oplus 1_{m-n}$
I am using the book "introduction to k-theory for $C^*$-algebras" by M. Rørdam so my references is for that book.
Idea of proof:
$\Leftarrow$
Assume first that there exist a natural number $m \geq n$ and $v $ in $\mathcal{U}_m (A)$ such that $\varphi(v)=u \oplus 1_{m-n}$. We want to start by showing that $\varphi$ is a unital $^*$-homomorphism i.e. by p.1 (Rørdam) we want to show that $\varphi (1_A)b=b=b \varphi (1_A) $ for all $b \in B$. Let $b \in B$. As $\varphi$ is surjective then for all $b \in B$ there exist an $a \in A$ such that $\varphi(a)=b$. Hence we obtain:
\begin{align*} \varphi (1_A)b &= \varphi (1_A) \varphi(a) \\ &= \varphi (1_A a)\\ &= \varphi (a) \\ &= b \end{align*}
And
\begin{align*} b \varphi (1_A) &= \varphi(a) \varphi (1_A) \\ &= \varphi (a 1_A )\\ &= \varphi (a) \\ &= b \end{align*} So by definition $\varphi$ is a unital $^*$-homomorphism and as A and B are assumed to be unital as well then by Rørdam (p.139) we have that $K_1(\varphi ) ([u]_1)=[\varphi (u)]_1$ for all $u$ in $\mathcal{U}_\infty(A)$. Now let $v \in \mathcal{U}_m(A)$ s.th. $\varphi(v)=u \oplus 1_{m-n}$. we then have that:
\begin{align*} K_1(\varphi ) ([v]_1) &= [\varphi(v)]_1 \\ &= [u \oplus 1_{m-n} ] \\ &\overset{8.1.4(i)}= [u]_1 +[1_{m-n}]_1 \\ &\overset{8.1.4(ii)}= [u]_1 + 0\\ &= [u]_1 \end{align*} So $[u]_1$ is in the image of $K_1(\varphi)$. Is this a correct approach?
$\Rightarrow$
In this direction I am not quite sure how to use that $[u]_1$ belongs to $Im(K_1(\varphi))$ as I am not quite sure what this implies. How would one go on with this part?
Your proof of the "if" statement is valid. For the "only if" statement, fix $u\in\mathcal U_n(A)$ and suppose $[u]_1$ lies in the image of $K_1(\varphi)$. Then there is some $v_0\in\mathcal U_m(A)$ such that $[\varphi(v_0)]_1=[u]_1.$ Thus, there is some $k\geq\max\{m,n\}$ such that $\varphi(v_0)\oplus1_{k-m}\sim_hu\oplus1_{k-n}$. Now use Lemma 2.1.7(iii) of Rørdam, Larsen, and Lausten's book to conclude that $u\oplus 1_{k-n}$ belongs to $\varphi(\mathcal U_k(A))$, that is, there is some $v\in\mathcal U_k(A)$ such that $\varphi(v)=u\oplus1_{k-n}$.