# $\eta$ invariants of Pin+ manifolds $\mathbb{RP}^{8k}$

In general, I am curious about the 'quantization' of $\eta$-invariants on Pin+ manifold, i.e., what is the 'minimal unit' of $\eta$-invariants on a manifold with certain choice of Pin+ structure.

From the paper "Exotic structures on 4-manifolds detected by spectral invariants" by Stephan Stolz, I learned a useful formula(Prop. 5.3) showing a computation of $\eta$-invariants on $\mathbb{RP}^{8k+4}$ with Pin+ structure, and it is minimal.

So I am curious about the analogous statement in $8k$ dimensional Pin+ manifolds. e.g. what is the minimal units there? Is the generator a $\mathbb{RP}^{8k}$ with certain choice of Pin+ structure?

• Recently I learned from a paper The eta invariant for even dimensional PINc manifoldshttp://www.sciencedirect.com/science/article/pii/0001870885901197 Dec 11 '15 at 1:58
• Recently I learned from a paper The eta invariant for even dimensional PINc manifolds'' Link:sciencedirect.com/science/article/pii/0001870885901197 that $\mathbb{RP}^{2l}$ saturates the `bound of minimal unit for $\eta$'. For details, see lemma 2.1 and theorem 3.3. A $Pin_+$ manifold is automatically $Pin_c$ (without gauge field), therefore, lemma 2.1 and theorem 3.3 are both applicable. Dec 11 '15 at 2:04