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 $M$?.
Best Answer
First, in general, the eta invariant for a self-adjoint elliptic operator on closed manifold is not a topological invariant. It depends on the geometric structure of the manifold. For example in spin case, it depends on the spin structure. So we cannot expect that it can be writed only by characteristic numbers.
Next, the eta invariant is not local. In the second page of the original paper "Spectral Asymmetry and Riemannian Geometry I" by Atiyah-Patodi-Singer, they gave an nonlocal example about a suitable len space. So we cannot expect that it can be writed by an integral of some local terms.
So the result of Stolz is a surprising but not a general result.
As I know, if you want to write the eta invariant as a local expression or the spin structure, there are two point of views.
The first one followed by Atiyah-Patodi-Singer that if the manifold $M$ is the boundary of some manifold $W$, then the reduced eta invariant can be written as $$\eta(M)=\mathrm{APS-index}(D^W)-\int_W \text{some characteristic class of}\ W.$$ So after mod $\mathbb{Z}$, we can regard the eta invariant of $M$ as an integral of local terms of $W$.
The second point of view is for the spin case. In this case, the eta invariant has a geometric expression that $$\eta(M)=\frac{1}{\sqrt{\pi}}\int_0^{\infty}t^{1/2}\mathrm{Tr}[D\exp(-tD^2)]dt.$$ Here $D$ is the Dirac operator. So we can study the eta invariant using the heat kernel.