Atiyah–Patodi–Singer \( \eta \)-Invariant and Invariants of Finite Degree

Abstract

Let η be the Atiyah–Patodi–Singer invariant considered on smooth, compact, oriented, threedimensional submanifolds of ℝⁿ, and let A be an additive subgroup of ℝ. The problem of computing the degree of invariants of the form η mod A is examined. Here, the functional definition of invariants of finite degree is used. (A similar approach is used in S. S. Podkorytov’s paper “Quadratic property of the rational semicharacteristic.”) The main results are as follows. If 1 ∉ A, then the degree is infinite. If \( \frac{1}{3}\in A \), then the degree is equal to 1. Bibliography: 10 titles.