Reduced generalized homology theory

A reduced generalized homology theory on based CW complexes consists of the following data.

1. For every integer $n$ a covariant functor ${\tilde{h}}_n$ from based CW complexes and based continuous maps to abelian groups; a based map $f:X\to Y$ induces $f_{\ast }:{\tilde{h}}_n(X)\to {\tilde{h}}_n(Y)$.
2. For every based CW complex $X$ and integer $n$ a suspension isomorphism $\sigma ={\sigma }_n(X):{\tilde{h}}_n(X)\stackrel{\cong }{\to }{\tilde{h}}_{n+1}(\mathrm{\Sigma }X),$ natural in $X$.
3. For every based cellular map $f:X\to Y$, write $C_f$ for its reduced cofiber, $i:Y\to C_f$ for the structural inclusion and $q_f:C_f\to \mathrm{\Sigma }X$ for the cofiber map. The connecting homomorphism is required to be the composite ${\mathrm{\partial }}_f:={\sigma }_n(X{)}^{-1}(q_f{)}_{\ast }:{\tilde{h}}_{n+1}(C_f)⟶{\tilde{h}}_n(X)\mathrm{.}$ Thus it is natural in $f$ and lowers the homological degree by one; it is not independent extra data that may be normalized separately from suspension.

The data satisfy the following axioms.

• (H) Homotopy invariance. Based homotopic maps induce equal homomorphisms.
• (E) Exactness. For every based cellular map $f:X\to Y$ the long sequence $\cdots \to {\tilde{h}}_n(X)\stackrel{f_{\ast }}{\to }{\tilde{h}}_n(Y)\stackrel{i_{\ast }}{\to }{\tilde{h}}_n(C_f)\stackrel{{\mathrm{\partial }}_f}{\to }{\tilde{h}}_{n-1}(X)\to \cdots$ is exact. Its connecting maps have the suspension-compatible normalization in item 3 and are natural for commutative squares of based cellular maps. (The cellular restriction ensures that $C_f$ is again a based CW complex, hence is an object in the declared domain of the functors.)
• (W) Wedge axiom. For every family $(X_{\alpha })$ of based CW complexes and every $n$ the natural map ${⨁}_{\alpha }{\tilde{h}}_n(X_{\alpha })⟶{\tilde{h}}_n\left({\bigvee }_{\alpha }{X}_{\alpha }\right)$ induced by the summand inclusions is an isomorphism. The empty wedge is a point and the empty sum is the zero group, so ${\tilde{h}}_n(\ast )=0$.

No dimension axiom is imposed: ${\tilde{h}}_n(S^0)$ is an arbitrary abelian group. A generalized homology theory is a genuinely covariant notion: it is not obtained by reversing the arrows of Reduced generalized cohomology theory, and the two notions are logically independent apart from the common coefficient bookkeeping recorded in Coefficient groups of a generalized homology theory. A morphism of reduced generalized homology theories is a family of natural transformations commuting with suspension and the connecting maps.

Compare Davis–Kirk, Definition 8.27, printed pp. 227–229, for the reduced homology axioms on based CW complexes. Their exactness axiom is the three-term cofiber exactness axiom; iterating the cofiber sequence with their suspension isomorphism gives the long sequence above and fixes its boundary as $(q_f{)}_{\ast }$ followed by inverse suspension. Definition 8.28 gives the coefficient groups used in Coefficient groups of a generalized homology theory.