math.GT — Geometric Topology
The dodecahedron web and its instanton homology
For the planar dodecahedron web D, the octahedral diagram of instanton homology reduces the possible values of the SO(3) instanton homology to 60 and 64. The remaining choice is determined by three coefficients of a secondary map. We give a proposed proof that all three vanish and that the dimension is 64. We express the coefficients as counts of instantons on specified foam cobordisms with one-parameter families of metrics. The index calculation requires these instantons to have negative action. An estimate uniform under neck stretching excludes them when the perturbations are sufficiently small. The construction includes local Floer complexes with zero differential, the marking calculation for excision, and a simultaneous choice of regular perturbations compatible with the families. Exact matrices and the supporting foam calculations are provided separately.