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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:

What is the natural extension homomorphism in $K$-theory?

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    $\begingroup$ Where are you reading about this construction? It seems a bit garbled. Normally you would compose the Thom isomorphism $K^*(X)\cong \tilde{K}^*(Th(N))$ for the normal bundle with the map $\tilde{K}^*(Th(N))\to K^*(Y)$ induced by collapsing the complement of the tubular neighbourhood (I suppose this could be thought of as extending a complex bundle from $Th(N)$ to $Y_+$). This requires that the normal bundle is oriented with respect to $K$-theory, i.e has a spin^c structure. $\endgroup$
    – Mark Grant
    Commented Jun 16, 2017 at 8:10
  • $\begingroup$ This is based upon expository paper by Landweber "K-theory and Elliptic operators" page 16 $\endgroup$
    – truebaran
    Commented Jun 16, 2017 at 11:14

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The natural extension homomorphism is defined in footnote 4 on page 7 of the expository paper by Gregory Landweber 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.

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