Contents

# Contents

## Statement

###### Proposition

(Todd class in rational cohomology is Chern character of Thom class)

Let $V \to X$ be a complex vector bundle over a compact topological space. Then the Todd class $Td(V) \,\in\, H^{ev}(X; \mathbb{Q})$ of $V$ in rational cohomology equals the Chern character $ch$ of the Thom class $th(V) \,\in\, K\big( Th(V) \big)$ in the complex topological K-theory of the Thom space $Th(V)$, when both are compared via the Thom isomorphisms $\phi_E \;\colon\; E(X) \overset{\simeq}{\to} E\big( Th(V)\big)$:

$\phi_{H\mathbb{Q}} \big( Td(V) \big) \;=\; ch\big( th(V) \big) \,.$

More generally , for $x \in K(X)$ any class, we have

$\phi_{H\mathbb{Q}} \big( ch(x) \cup Td(V) \big) \;=\; ch\big( \phi_{K}(x) \big) \,,$

which specializes to the previous statement for $x = 1$.

## References

The statement appears without proof in:

Proof is spelled out in:

Discussion in terms of representing ring spectra:

• Larry Smith, Section 1 of: The Todd character and the integrality theorem for the Chern character, Illinois J. Math. Volume 17, Issue 2 (1973), 301-310 (euclid:ijm/1256051760)

Last revised on January 12, 2021 at 10:53:01. See the history of this page for a list of all contributions to it.