Let $X \subset Y$ be two smooth manifolds. To the inclusion $I:X \to Y$ corresponds the so called *wrong-way* map in $K-theory$ $i\_!:K(X) \to K(Y)$. It is constructed as follows: to the inclusion $X \subset Y$ corresponds inclusion of cotangent bundles $TX \subset TY$. We choose a metric on $Y$ and consider a tubular neighborhood $N$ of $X$ in $Y$. This gives us that $TN$ is tubular neighborhood of $TX$ in $TY$. One can show that it is possible to endow $TN$ with the complex structure: once we do this, we can aplly Thom isomorphism $\varphi$ in $K$-theory which goes from $K(TX)$ to $K(TN)$. The wrong-way map is defined as the composition of $\varphi$ with $h:K(TN) \to K(TY)$: the map $h$ is described as *the natural extension homomorphism*. I don't understand how this map is defined: 



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> What is the natural extension homomorphism in $K$-theory? 
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Natural extension homomorphism and wrong-way maps in K-theory
The natural extension homomorphism is defined in footnote 4 on page 7 of [the expository paper by Gregory Landweber](https://arxiv.org/abs/math/0504555) you are looking at.


In that reference, *K-theory with compact supports* is defined for a locally compact space $X$ by $K(X)=\tilde{K}(X^+)$, where $(-)^+$ denotes one-point compactification. For an inclusion $i:U\to X$ of an open subset, we get a homomorphism $i\_\*: K(U)\to K(X)$ on $K$-theory with compact supports, induced by the collapse map
$$
X\_+\to X\_+/(X\_+-U\_+)\cong U\_+.
$$
Therefore, chasing the definitions, I think the construction of $i\_!$ you give agrees with the one I described in my comment above.


