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 $x = 1$.

###### Remark

By the discussion at universal complex orientation on MU we have:

For $V$ a complex vector bundle with Thom space $Th(V)$, its Thom class in any complex-oriented cohomology theory $E$ is classified by the composite

$Th(V) \longrightarrow M U \overset{\sigma^E}{\longrightarrow} E \,,$

where $\sigma$ represents the complex orientation as a map of homotopy-commutative ring spectra on the Thom spectrum MU.

In this perspective via classifying morphisms of ring spectra, the statement of Prop. becomes that the Todd character is the composite of the complex orientation $\sigma^E$ with the Chern character

$Td \;\colon\; M \mathrm{U} \overset{ \sigma^{KU} }{\longrightarrow} KU \overset{ ch }{\longrightarrow} H^{ev}\mathbb{Q}$

In particular, on cohomology rings $E_\bullet \coloneqq \pi_\bullet(E) \coloneqq \widetilde E(S^\bullet)$ this composite of ring spectrum maps is the Todd genus on the complex cobordism ring, factored as

$Td_{2\bullet} \;\colon\; \Omega^\mathrm{U}_{2\bullet} \;=\; (M\mathrm{U})_{2\bullet} \overset{ }{\longrightarrow} KU_{2\bullet} \overset{ch_{2\bullet}}{\longrightarrow} \mathbb{Q} \,.$

(see Smith 73, p. 303 (3 of 10), following Conner-Floyd 66, Section 6)

## References

Proof is spelled out in:

Discussion in terms of representing ring spectra:

also

Last revised on February 18, 2021 at 11:16:31. See the history of this page for a list of all contributions to it.