Atlas of Fundamental Concepts

Homotopy treats continuous maps as equivalent when one deforms into another. Loop spaces produce the fundamental group and higher homotopy groups, while CW complexes and geometric realizations of simplicial sets provide workable models. Singular chains turn spaces into chain complexes; cycles modulo boundaries give homology. Dual cochains yield cohomology, whose cup product adds ring structure. Exact and long exact sequences transmit information across pairs, fibrations, and short exact sequences.