The $\operatorname{spin}^c$ index for manifolds

Question

$\DeclareMathOperator\spin{spin}\DeclareMathOperator\ch{ch}\DeclareMathOperator\ind{ind}$In the paper Čadek, Crabb, and Vanžura - Obstruction theory on 8-manifolds, 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.

Answer 1

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.