My question relates to the proof of the Atiyah-Singer Index Theorem for families of elliptic operators, as presented in "The Index of Elliptic Operators: IV", M. F. Atiyah and I. M. Singer.


Let $A$ be a compact Hausdorff space and $q:A\times \mathbb{C}^n\rightarrow A$ be the projection, then we obtain the induced Thom isomorphism $q\_!:K\_{\text{cpt}}(A\times \mathbb{C}^n)\rightarrow K(A)$. The map $q\_!$ and the analytic index $ind:K\_{\text{cpt}}(A\times \mathbb{C}^n)\rightarrow K(A)$ coincide. According to Atiyah this follows from the case $Y$ is a point, and the fact that the analytical index is a homomorphism of $K(Y)$-modules. My question is why are these two properties enough to show the two maps coincide?


Thanks,


Tristan



Short question relating to the proof of the Atiyah-Singer Index Theorem for families
The answer should probably go something like this:


Both $q\_!$ and $ind$ are $K(A)$-module maps. Since $q\_!$ is an isomorphism, to check that these are the same maps, it suffices to check they are the same on a generator; if $u\_A\in K\_{Cpt}(A\times C^n)$ is the unique element such that $q\_!(u\_A)=1$, then we need to show that $ind(u\_A)=1$ as well.


The class $1\in K(A)$ is in the image of the tautological map $\mathbb{Z}=K(point)\to K(A)$, induced by $f:A\to point$. Both $q\_!$ and $ind$ are natural with respect to $f$, so
to prove that $ind(u\_A)=1$, it is enough to prove that $ind(u\_{point})=1$, since $ind(u\_A)=ind(f^\*(u\_{point}))= f^\*(ind(u\_{point}))$.


