Homology and cohomology

A central theme in algebraic topology is the assignment of algebraic invariants to topological spaces. Among the most important are the homology groups \(H_n(X;\Z )\) and cohomology groups \(H^n(X;\Z )\). They are computable, satisfy convenient formal properties, and distinguish many spaces that are otherwise hard to tell apart.

More generally, a generalized (co)homology theory assigns graded abelian groups to spaces in a weakly homotopy invariant way and interacts predictably with wedges, cofiber sequences, and suspensions. We formulate these axioms directly for pointed animae, where homotopy invariance is automatic. We then explain the close relation with spectra, viewed for now as sequences of pointed animae \(E_0,E_1,E_2,\ldots \) equipped with isomorphisms \(E_n\simeq \Omega E_{n+1}\). Every spectrum represents a cohomology theory, and Brown representability gives the converse.

Ordinary homology and cohomology are singled out among generalized theories by the dimension axiom. We take their existence for granted here and construct them intrinsically in Chapter 6. The online version also develops their cellular computation and gives a constructive proof of Brown representability. This chapter thereby connects classical algebraic topology with the theory of spectra and stable \(\infty \)-categories developed in Chapter 4. Readers already comfortable with (co)homology theories may begin with Section 3.2.

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