nLab rational Todd class is Chern character of Thom class

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 $x = 1$.

(Karoubi 78, Chapter V, Theorem 4.4)

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)

Related theorems

• e-invariant is Todd class of cobounding (U,fr)-manifold

• Grothendieck-Riemann-Roch theorem

References

• Pierre Conner, Edwin Floyd, Section 6 and 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)

• Matthias Jonas Spiegel, Section 2.3.2 of: K-Theory of Intersection Spaces, 2013 (doi:10.11588/heidok.00015738)

also

• Pierre Conner, Larry Smith, Section 9 of: On the complex bordism of finite complexes, Publications Mathématiques de l’IHÉS, Tome 37 (1969) , pp. 117-221 (numdam:PMIHES_1969__37__117_0)