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What is the role of $\sum (-1)^p[\wedge^pT^*M]$ in the K-theory $K(M)$
I apologize for the vague title. Let $M$ be a compact smooth manifold, then we have $T^*M$ and hence $\wedge^pT^*M$ as vector bundles on $M$. There for we have
$$
\sum (-1)^p[\wedge^pT^*M] \in K(M).
$...
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