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I have some compact manifold with boundary $(M,\partial M)$, and there is a long exact sequence

$$\cdots\to KO^{-1}(\partial M)\xrightarrow{\partial} KO^{0}(M,\partial M)\to KO^0(M)\to KO^0(\partial M) $$

I want to understand the connecting homomorphism. My question is:

In general, are there any geometric descriptions of the "cycles" in $KO$ theory and the connecting homomorphism?

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  • $\begingroup$ Sort of. I suggest you look in Index theory for skew-adjoint Fredholm operators by Atiyah and Singer and Clifford modules by Atiyah, Bott and Shapiro for a description in terms of modules over a Clifford algebra. $\endgroup$
    – Denis Nardin
    Commented Mar 9, 2016 at 18:23
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    $\begingroup$ The negative K groups are the easier part: $K^{-1}(\partial M)$ is just $K^0(\Sigma \partial M)$. The connecting homomorphism is just pulling back the bundle using the standard, geometric map $M/\partial M \rightarrow \Sigma \partial M$. (Build this by viewing $M/\partial M$ as collapsing a cone on $\partial M$ and just include into the `double cone' which is the suspension.) $\endgroup$
    – Dylan Wilson
    Commented Mar 9, 2016 at 20:58

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