In the papers of Atiyah and Singer, they first show their index theorem and then derive some index formulas.

For the Fredholm index living in the integers, they use the fact that on spheres the Chern character is onto and map 1 to the top class in $H^n(S^n, \mathbb{Z})$. So for the index problem for an elliptic operator, no information is lost.

For a family of elliptic operators parametrized by say a compact space $X$, the index theorem is an equality in $K(X)$ and the index formula lives in $ ch(K(X)) \subset H^*(X, \mathbb{Q})$. So we loose the information coming from the torsion part of $K(X)$.

My question is whether or not there are some methods to recover this information.

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    $\begingroup$ The fact that the Chern character maps $K$-theory to integral cohomology for spheres is stronger than just the lack of torsion. In general even if $X$ has no torsion, we have two maximal-rank lattices in $H^*(X, \mathbb{Q})$, namely $ch(K(X))$ and $H^*(X, \mathbb{Z})$. I expect any relationship between them should reflect interesting topology in $X$ (with the spheres being an extreme case). $\endgroup$ Oct 15 '18 at 8:05
  • $\begingroup$ @OliverNash do you know a book or paper where these matters are discussed? (Meaning the relationship between the image of the Chern character and the integral cohomology) $\endgroup$
    – vap
    Oct 16 '18 at 21:17
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    $\begingroup$ @vap I'm afraid not! Years ago I learned (in an old paper of Karoubi [1]) that the fact that these two lattices coincide for spheres had strong topological consequences (obstruction to existence of almost complex structures) and I was struck by the result. I've occasionally wondered how far these ideas could go in general or how much they've been studied but I don't know. [1] archive.numdam.org/ARCHIVE/SHC/SHC_1963-1964__16_2/… $\endgroup$ Oct 17 '18 at 9:06
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    $\begingroup$ @vap You inspired me to ask about this here: mathoverflow.net/questions/313049/… Perhaps someone will illuminate us! $\endgroup$ Oct 17 '18 at 12:02

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