8 · Recover homology from solidification

Derived solidification of the free condensed group on a space recovers its integral homology. The bars separate free classes from classes of finite order.

Try this. Compare all five spaces and rotate the sphere or torus. For the Klein bottle, notice the order-two class: adding that class to itself gives zero.

The activity computes homology from cellular boundary matrices using Smith normal form. Example 6.5 supplies the theorem that identifies these groups with derived solidification.