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rational Todd class is Chern character of Thom class

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Contents

Statement

Proposition

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

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

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

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

ϕ H(ch(x)Td(V))=ch(ϕ K(x)), \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=1x = 1.

(Karoubi 78, Chapter V, Theorem 4.4)

Remark

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

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

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

where σ\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 σ E\sigma^E with the Chern character

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

In particular, on cohomology rings E π (E)E˜(S )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:Ω 2 U=(MU) 2KU 2ch 2. 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.