The eta invariant was introduced by Atiyah, Patodi, and Singer. It roughly measures the asymmetry of the spectrum of a self-adjoint elliptic operator with respect to the origin. In the paper "Exotic structures on 4-manifolds detected by spectral invariants" Stolz showed that the eta invariant $\eta(M,g,\phi)$ of the twisted Dirac operator on a smooth closed 4 manifolds with Riemannian metric $g$ and $\text{Pin}^{+}$ structure $\phi$ is a $pin^{+}$ bordism invariant.

As we know, for an oriented 4 manifold $M$, the Hirzebruch Signature Theorem implies that $\text{index}(D)=\text{sign}(M)$, where the left hand side is the index of the signature operator of M (the analytic signature), and the right hand side is the topological signature (the signature of a quadratic form on $H^{2k}(M)$ defined by the cup product). Moreover, we have $\text{sign}(M) = \int_M L(p_1,\dots,p_n)$, where $L$ is the Hirzebruch $L$-Polynomial, and $p_i$ the Pontryagin numbers of $M$.

According to my understanding, the twisted Dirac operator is the unoriented generalizaton of the signature operator, and the eta invariant is the unoriented generalization of the signature of a manifold.

So I was wondering if one can have a similar topological description of the eta invariants $\eta(M,g,\phi)$ of a $\text{Pin}^{+}$ manifold in terms of topological data like the Stiefel Whitney numbers or Pontryagin numbers of $M$?.