Ordinary Homology Theory

Abstract

By an ordinary homology theory k _* we shall mean one with k _n (S⁰) = 0 unless n = 0. If k₀(S⁰) = G, then k _* will be called an ordinary homology theory with coefficients G. Reduced singular homology H̃ _* (-;G) is an ordinary homology theory with coefficients G on the category PT’. We shall show that any two ordinary homology theories with coefficients G satisfying the wedge and WHE axioms are naturally equivalent. We shall also construct the Eilenberg-MacLane spectrum H(G) with

$$ {\pi _{n}}\left( {H(G)} \right) = \left\{ {\begin{array}{*{20}{c}} {G\quad n = 0} \\ {0\quad n \ne 0.} \\ \end{array} } \right. $$