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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 $$ \sum (-1)^p[\wedge^pT^*M] \in K(M). $$ It seems to have close relation with the Euler class of $M$.

My question is: Is there any study on $\sum (-1)^p[\wedge^pT^*M]$? What properties do we know? Can I have some references?

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up vote 3 down vote accepted

I think the main topological significance of the element you identify is its close relationship with the Thom isomorphism $K(M) \cong K(T^*M)$. I would imagine that there is also some connection with the Euler characteristic as well, perhaps by taking Chern characters, but this is probably secondary to its role in the Thom isomorphism.

The idea is the same for any compact Hausdorff space $X$ and any vector bundle $V$ over $X$, though in the case where $X = M$ is a manifold and $V = T^*M$ I believe the result becomes a version of Poincare duality in K-theory. Recall that tensor product of bundles defines a product $K(X) \otimes K(Y) \to K(X \times Y)$. Pulling back along the diagonal map $X \to X \times X$ one has a product $K(X) \otimes K(X) \to K(X \times X) \to K(X)$ and hence a ring structure on $K(X)$. If $V$ and $W$ are vector bundles over $X$ then pullback along the diagonal map also induces a product $K(V) \otimes K(W) \to K(V \times W) \to K(V \oplus W)$ (where $V \times W$ is the product of spaces and $V \oplus W$ is the direct sum of bundles). Taking $W$ to be the zero bundle over $X$, the map $K(V) \otimes K(X) \to K(V)$ gives $K(V)$ the structure of a $K(X)$-module.

Now view $\bigwedge^p V$ as a vector bundle over $V$ and consider the element $\lambda_V = \sum_p (-1)^p [\bigwedge^p V]$ in $K(V)$. Some care must be taken as usual due to the fact that $V$ is not itself compact, but there are a number of ways to make this work out. The product with $\lambda_V$ defines a map $\varphi: K(X) \to K(V)$, and the Thom isomorphism theorem in K-theory asserts that this map is an isomorphism. To do computations one often pulls back along the zero section $i: X \to V$, and one finds that $i^* \circ \varphi$ is the map $K(X) \to K(X)$ given by multiplication by $\sum_p (-1)^p [\bigwedge^p V]$, thought of as an element of $K(X)$.

References: More detail can be found in the first section of Atiyah and Singer's "Index of Elliptic Operators I".

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Thank you Paul! – Zhaoting Wei Jul 22 '12 at 1:40

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