The η Invariant and Elliptic Operators in Subspaces

Abstract

P. Gilkey [1] observed that for differential operators satisfying the parity condition

$$ordA + \dim M \equiv 1\left( {\bmod 2} \right)$$

(1.1)

the spectral Atiyah—Patodi—Singer η invariant is rigid: for a one-parameter operator family, the corresponding family of η invariants is a piecewise constant function. In particular, the fractional part of the spectral η invariant of a self-adjoint elliptic operator is in fact a homotopy invariant depending on the principal symbol of the operator alone. Thus, we arrive at the problem of computing the fractional part of the η invariant. This fractional homotopy invariant was successfully applied in several problems of topology and differential geometry (e.g., see [2], [3], [4] and [5]).