Visualize differential 1-forms on surfaces. Closed forms have zero exterior derivative (curl=0). Exact forms are gradients. The cohomology H¹ measures the obstruction to exactness — captured by the "holes" in the space.
Closed form on torus: dω=0 but ω≠df globally. The integral ∮ω around the generator loop is a topological invariant — it detects the hole.