A gluing formula for the $Z_2$-valued index of odd symmetric operators

TL;DR

This paper develops a splitting formula for the $Z_2$-valued index of Dirac-type operators with symmetry on manifolds with boundary, linking it to boundary value problems and cohomological formulas.

Contribution

It introduces a $Z_2$-valued analog of the splitting theorem for Dirac operators, connecting indices on closed manifolds to boundary value problems and cohomological expressions.

Findings

Proves a $Z_2$-valued splitting theorem for Dirac operators.

Establishes the index relation when a manifold is cut along a hypersurface.

Derives a cohomological formula for the $Z_2$-valued index.

Abstract

We investigate Dirac-type operator $D$ on involutive manifolds with boundary with symmetry, which forces the index of $D$ to vanish. We study the secondary $Z_2$-valued index of elliptic boundary value problems for such operators. We prove a $Z_2$-valued analog of the splitting theorem: the $Z_2$-valued index of an operator on a closed manifold $M$ equals the $Z_2$-valued index of a boundary value problem on a manifold obtained by cutting $M$ along a hypersurface $N$. When $N$ divides $M$ into two disjoint submanifolds $M_1$ and $M_2$, the $Z_2$-valued index on $M$ is equal to the mod 2 reduction of the usual $Z$-valued index of the Atiyah-Patodi-Singer boundary value problem on $M_1$. This leads to a cohomological formula for the $Z_2$-valued index.