Background.
This is problem 12.4 in Rordam's C$^{*}$-algebra book. Let $A$ be a C$^{*}$-algebra.
The suspension of $A$ is defined to be $SA:=\{f\in C([0,1]): f(0)=f(1)=0\}$.
The cone of $A$ is defined to be $CA:=\{f\in C([0,1]):f(0)=0\}$.
Let $\iota\colon SA\to CA$ denote the inclusion map and let $\psi\colon CA\to A$ be given by $\psi(f):= f(1)$. We obtain the following short-exact sequence of C$^{*}$-algebras:
$$ 0\longrightarrow SA\overset{\iota}{\longrightarrow} CA\overset{\psi}{\longrightarrow} A\longrightarrow 0. $$
I am trying to compute the exponential map $\delta_{0}\colon K_{0}(A)\to K_{1}(SA)$ associated to this sequence. Based on a later problem, I believe the answer is $\delta_{0}=-\beta_{A}$, where $\beta_{A}\colon K_{0}(A)\to K_{1}(SA)$ is the Bott map.
Given a projection $p$ in $M_{n}(\widetilde{A})$, we define the projection loop $f_{p}\colon [0,1]\to M_{n}(\widetilde{A})$ by $f_{p}(t):=e^{2\pi i t}p+(1_{n}-p)=e^{2\pi i tp}$. Since $f_{p}(0)=f_{p}(1)\in M_{n}(\mathbb{C}1_{\widetilde{A}})$, we may regard $f_{p}$ as a unitary in $M_{n}(\widetilde{S\widetilde{A}})$. Let $s(p)$ denote the scalar part of $p$. A calculation done earlier in the book shows that $f_{p}f^{*}_{s(p)}$ is in fact a unitary in $M_{n}(\widetilde{SA})$ and that $\beta_{A}([p]_{0}-[s(p)]_{0})=[f_{p}f^{*}_{s(p)}]_{1}$.
Thus, I would like to show that $\delta_{0}([p]_{0}-[s(p)]_{0})=-[f_{p}f^{*}_{s(p)}]_{1}$.
My Attempt.
Define $a\in M_{n}(\widetilde{CA})$ by $a(t):=tp+(1-t)s(p)$. Note that the scalar part $s(a)=s(p)$ is constant and $a(0)=s(p)\in M_{n}(\mathbb{C}1_{\widetilde{A}})$, so $a$ is well-defined. Moreover, $a$ is self-adjoint and $\widetilde{\psi}(a)=a(1)=p$. Since $\widetilde{\iota}$ is the identity map, the standard picture of the exponential map says that $\delta_{0}([p]_{0}-[s(p)]_{0})=-[e^{2\pi i a}]_{1}$. So, to conclude that $\delta_{0}=-\beta_{A}$, I need to show that $[e^{2\pi i a}]_{1}=[f_{p}f_{s(p)}^{*}]_{1}$. I.e., that
$$[e^{2\pi i (tp+(1-t)s(p)})]_{1}=[e^{2\pi i tp}\cdot e^{-2\pi i ts(p)}]_{1}$$
This would follow if I can show that $e^{2\pi i a}$ and $f_{p}f^{*}_{s(p)}$ are connected in the set of unitary elements in $M_{n}(\widetilde{SA})$, but I am unsure how to prove this.