$\DeclareMathOperator\spin{spin}\DeclareMathOperator\ch{ch}\DeclareMathOperator\ind{ind}$In the paper [Čadek, Crabb, and Vanžura - *Obstruction theory on 8-manifolds*](https://arxiv.org/abs/0710.0734), the authors discussed the "$\spin^c$-index" for a $\spin^c$ manifold $M$ (display (3.1) of the paper):
$$y\in K^0(M)\mapsto \ind(y)=(e^{c/2}\hat{A}(\tau M)\ch(y))[M]\in\mathbb{Z}, $$
where $c$ is the $\spin^c$ class, $\tau M$ is the tangent bundle of $M$, and $\hat{A}$ is the Hirzebruch signature:
$$\hat{A}(\tau M)=1-p\_1(\tau M)+\dotsb. $$
I was wondering if there is a more coherent context in which the invariant $\ind(y)$ is discussed.



The $\operatorname{spin}^c$ index for manifolds
I suggest consulting section 26 of Hirzebruch's *Topological Methods in Algebraic Geometry* and the references therein. In particular, it contains the following statement:



> 
> Theorem 26.1.1. Let $d$ be an element of $H^2(X, \mathbb{Z})$ whose reduction mod 2 is the Whitney class $w\_2(X)$, and $\eta$ a continuous $GL(q, \mathbb{C})$-bundle over $X$. Then
> 
> 
> $$\hat{A}(X, \tfrac{1}{2}d, \eta) = \chi^m\left[e^{\frac{1}{2}d}\cdot\operatorname{ch}\eta\cdot\sum\_{j=0}^{\infty}\hat{A}\_j(p\_1, \dots, p\_j)\right]$$
> 
> 
> is an integer.
> 
> 
> 


References to proofs are given in subsections 26.3-26.5.


