Atiyah and Hirzebruch define a $c_1$-map to spell out Riemann-Roch theorem for (compact and connected) smooth manifolds. The definition is following: a map $f:Y \to X$ is called a $c_1$-map if we are given an element $c_1\in H^2(Y,\mathbb Z)$ such that
$$c_1 \equiv w_2(Y)-f^*w_2(X)\ (\mathrm{mod}\ 2). \tag{$\ast$}$$
I am confused by this definition, because the existence of $c_1$ is contained in the definition. If we distinguish $c_1$ in "$c_1$-map" and $c_1$ in the definition, then the definition says almost nothing, because the universal coefficient theorem ($H^2(Y,\mathbb Z)\otimes\mathbb Z_2 \cong H^2(Y,\mathbb Z_2)$) allows us to find an integral cohomology class $c_1$ satisfying ($*$) for each $f$. Therefore the definition seems to fail to define a class of maps (or seems to say only the choice of $c_1$).
Questions:
- What exactly do Atiyah and Hirzebruch want to define? A class of maps or a choice of $c_1$? Or do I fail to understand the definition?
- The integral cohomology class $c_1$ is contained in the Riemann-Roch theorem $$f_!(y)\cdot \hat{\mathfrak{A}}(X) = f_*(y \cdot e^{c_1/2} \cdot \hat{\mathfrak{A}}(Y)).$$ Why does not the choice of $c_1$ affect the identity?
- According to Remark following the theorem in the paper, if $X$ and $Y$ are complex manifolds and $f$ is holomorphic, then we can delete $e^{c_1/2}$ from the formula. Why can we delete it?