8 · Keep track of winding

A closed path around a circle has an integer winding number. A continuous deformation may change the path's shape, but it cannot change that number without breaking the path.

Try this. Change the turn count and then deform the path. On the torus, choose both independent turn counts. The drawing changes shape while the selected winding data stay fixed.

The circle gives one generator in first cohomology. Proposition 3.1 describes products of circles with an exterior algebra, and the activity recomputes the displayed torus ranks from that formula.