rational Todd class is Chern character of Thom class

**algebraic topology** – application of higher algebra and higher category theory to the study of (stable) homotopy theory

**(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 $x = 1$.

(Karoubi 78, Chapter V, Theorem 4.4)

The statement appears without proof in:

- Pierre Conner, Edwin Floyd, p. 99 (first line) in:
*The Relation of Cobordism to K-Theories*, Lecture Notes in Mathematics**28**, Springer 1966 (doi:10.1007/BFb0071091, MR216511)

Proof is spelled out in:

- Max Karoubi, Theorem V4.4 of:
*K-Theory – An introduction*, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 (pdf, doi:10.1007%2F978-3-540-79890-3)

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.