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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)

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.