A Cohomological Obstruction Invariant for Gauge Configuration Types in Dimension Four: An Initial Contact with Yang-Mills Theory

This work presents a homotopy-theoretic framework for encoding gauge configurationspaces and constructing dimension-sensitive cohomological invariants. Using HomotopyType Theory (HoTT) as a conceptual foundation, I model gauge symmetries intrinsicallyand define mod-2 cohomology via maps into Eilenberg–MacLane spaces. Within this framework,I introduce an invariant obtained by applying Steenrod operations to characteristicclasses. Under explicit assumptions, I show that this invariant vanishes in dimensions d < 4and is nontrivial in dimension d = 4 for natural classes of examples. This provides ahomotopy-cohomological characterization of the special role of four dimensions in gaugetheory.