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 be a complex vector bundle over a compact topological space. Then the Todd class of in rational cohomology equals the Chern character of the Thom class in the complex topological K-theory of the Thom space , when both are compared via the Thom isomorphisms :
More generally , for any class, we have
which specializes to the previous statement for .
(Karoubi 78, Chapter V, Theorem 4.4)
By the discussion at universal complex orientation on MU we have:
For a complex vector bundle with Thom space , its Thom class in any complex-oriented cohomology theory is classified by the composite
where 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 with the Chern character
In particular, on cohomology rings this composite of ring spectrum maps is the Todd genus on the complex cobordism ring, factored as
(see Smith 73, p. 303 (3 of 10), following Conner-Floyd 66, Section 6)
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)
Matthias Jonas Spiegel, Section 2.3.2 of: K-Theory of Intersection Spaces, 2013 (doi:10.11588/heidok.00015738)
also
Last revised on February 18, 2021 at 16:16:31. See the history of this page for a list of all contributions to it.