Ordinary Cohomology

Prerequisites. Direct starred prerequisites: Chapter 72. No later core chapter depends on this route.

Opening obstruction

Homotopy groups measure maps out of spheres, but many geometric distinctions are represented by maps into Eilenberg–Mac Lane spaces and organized by exact sequences.

Development contract

For abelian $A$, define ordinary cohomology by $0$-truncated unpointed maps into $K(A,n)$ and reduced cohomology by pointed maps. Treat degrees zero and one explicitly; construct the group and functorial structure and prove homotopy invariance. Prove the four axioms in Ljungström–Mörtberg, Section 3: suspension, cofiber exactness, dimension, and additivity for index types satisfying ${\mathsf{AC}}_0$. Derive the long exact sequence and Mayer–Vietoris theorem, then calculate a point, spheres, and a finite wedge.