FKM Invariant as Spin Structure Obstruction in Time-Reversal Insulators — E8 Intelligence Research

FINDING: The Fu-Kane-Mele (FKM) invariant is a cohomological obstruction — specifically a second Stiefel-Whitney class — that detects whether a time-reversal-invariant (class AII) insulator admits a globally consistent spin structure, with parity constraints at time-reversal invariant momenta (TRIMs). | MATH: FKM invariant \( ν ∈ Z_2 \) equals \( ∫TRIM w_2(TM) 2 \), where \( w_2 \) is the second Stiefel-Whitney class; equivalently \( ν = ∏TRIM δa_i \) with \( δa_i = ± 1 \) from parity eigenvalues. The FKMM-invariant generalizes this to quaternionic vector bundles over involutive base spaces, living in \( H^2(B,Z_2) \). | CONNECTION: The \( Z_2 \) parity structure at TRIMs mirrors the \( Z_2 \) grading of Clifford algebras \( Cl0,2 R(2) \) — the same algebra underlying quaternion units \( i,j,k \) with \( i^2=j^2=k^2=-1 \). The TRIM lattice is a hypercubic sublattice of the Bril Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com