The Todd character and cohomology operations

In “On the Conflict of Bordism of Finite Complexes” J. Differential Geometry], Conner and Smith introduced a homomorphism called the Todd character, relating complex bordism theory to rational homology. Specifically the Todd character consists of a family of homomorphisms

$\text{th}{}_{\mathrm{r}}\text{: MU}{}_{\mathrm{s}}\text{(X) → H}{}_{\mathrm{s\to r}}\text{(X;}\text{Q}\text{)}$

.In L. Smith, The Todd character and the integrality theorem for the Chern character, Ill. J. Math. it was shown (note that the indexing of the Todd character is somewhat different here) that there was an integrality theorem for th analogous to the Adams integrality theorem for the Chern character J. F. Adams, On the Chern character and the structure of the unitary group, Proc. Cambridge Philos. Soc.57 (1961), 189–199; On the Chern character revisted, Ill. J. Math. Now Adams' first paper contains a wealth of information about the Chern character in addition to the integrality theorem already mentioned. Our objective in the present note is to derive analogous results for the Todd character. As in Smith these may then be used to deduce the results of Adams for the Chern character.