I am reading on K theory in Lawson and Michelson (Spin Geometry). One has the "exact sequence spaces" $L(X,Y)$ and there is the theorem that there is a unique equivalence of functors $\chi$ between $L$ and $K$ (the Euler characteristic) such that in the case $Y = \emptyset$
$$ \chi([V\_0, \dots, V\_n]) = \sum\_{k=0}^n (-1)^k [V\_k] \in K(X, \emptyset).$$
To give a more explicit characterization of the map in the case $n=1$, they make quite a complicated construction by defining a space $Z$ by gluying two copies of $X$ together along $Y$, defining bundles on $Z$ and using the isomorphism $K(Z, X\_1) \cong K(X, Y)$. I compared to the original paper of Atiyah, Bott and Shapiro, and they make the same construction. However, I feel that one does not need all those surgical methods.


Instead, to define $\chi([V\_0, V\_1])$, look at the exact sequence
$$0 \rightarrow K(X, Y) \stackrel{i}{\rightarrow} K(X) \stackrel{j}{\rightarrow} K(Y) \rightarrow 0$$
and notice that the element $[V\_0] - [V\_1] \in K(X)$ is in the kernel of $j$ since $V\_0$ and $V\_1$ are isomorphic when restricted to $Y$. Now define the Euler characteristic as the preimage of $[V\_0] - [V\_1]$ under $j$, which is well-defined by exactness.


Did I miss something or are the constructions that I referenced needlessly complicated? 



Euler characteristics and the difference bundle construction
You missed that the sequence is not exact at $K(X,Y)$ (neither is it exact at $K(Y)$, but that does not matter here). There is an ambiguity coming from $K^{-1} (Y)$, i.e. automorphisms of bundles. If $Y$ is a point, your construction works, and the purpose of the arguments in Atiyah-Bott-Shapiro is to extend it to $Y \neq \ast$.


