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

For the Fredohlm 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.

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.

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

For the Fredohlm index living in the integers, they uses the fact that on spheres the Chern character $ch : K(S^n) \to H^n(S^n, \mathbb{Z})$ is bijective. So for the index problem for one 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

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

For the Fredohlm 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.