first I want to state which definitions I use for K-theory and reduced K-theory:
Let $X$ be a compact topological Hausdorffspace and
$V(X):=\{\text{Isomorphism classes of (complex) vector bundles over } X\}$,
$K(X)=(V(X)\times V(X))/ \sim $
where $(a,b)\sim (c,d)$ iff there exists $k\in V(X)$ with $a+d+k=b+c+k$ and $+$ denotes the direct sum of (ismoprhism classes of) vector bundles.
Given a point $pt\in X$, we have the canonical inclusion $i\colon \{pt\}\rightarrow X$, which induces a map $i^*\colon K(X)\rightarrow K(pt)$ which is given by pullback of vector bundles.
Now define $\widetilde{K}(X,pt):=kernel(i^*)$.
For a given $X$ we set $X^+:=X\sqcup \{\widetilde{pt}\}$ als the disjoint union of $X$ and a "point" $\widetilde{pt}$.
Claim: There exists an ismorphism $K(X)\cong \widetilde{K}(X^+,\widetilde{pt})$
This claim is "Observation 9.13." in "Spin Geometry" (Lawson & Michelsohn).
I want to understand how the isomorphism in the claim works or "how I should think about the isomorphism". I have thought about it and came up with a map that seems to be the desired isomorphism, but im fairly new to K-theory so it would be great if someone could tell me if my thoughts are right or wrong.
Construction attempt of the isomorphism stated in the claim:
First of all, every element in $K(X)$ can be written as $[([E],[\epsilon^k])]$ where $\epsilon^k$ denotes the trivial rank $k$ vector bundle over $X$ and $E\xrightarrow{\pi} X$ is a vector bundle over $X$. By a slight abues of notation we write
$[E]-[\epsilon^k]:=[([E],[\epsilon^k])]$.
Define a map
$F\colon K(X)\rightarrow \widetilde{K}(X^+,\widetilde{pt})$
by $F([E]-[\epsilon^k]):=[\hat{E}]-[\epsilon^k]$ where $\hat{E}:=E\sqcup\mathbb{C}^k\xrightarrow{\hat{\pi}}X^+$ and $\hat{\pi}$ is given by $\hat{\pi}(e)=\pi(e)$ for all $e\in E$, $\hat{\pi}(x)=\widetilde{pt}$ for all $x\in\mathbb{C}^k$.
This map is well defined, since $[F]-[\epsilon^k]\in K(X^+)$ is in the kernel of $i^*\colon K(X^+)\rightarrow K(\widetilde{pt})$ iff $dim_{\mathbb C} (F_{\widetilde{pt}})=k$.
Is $F$ the desired isomorphism? Is $F$ the way one should think about the isomorphism stated in the claim?