Reduced generalized cohomology theory

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

1. For every integer $n$ a contravariant functor ${\tilde{h}}^n$ from based CW complexes and based cellular maps to abelian groups; a based cellular map $f:X\to Y$ induces $f^{\ast }:{\tilde{h}}^n(Y)\to {\tilde{h}}^n(X)$.
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 $X$, where $\mathrm{\Sigma }$ is the reduced suspension and the sphere coordinate is written first.

For every based cellular map $f:X\to Y$, use the cofiber convention of Reduced cone suspension and cofiber sequence: $X\stackrel{f}{\to }Y\stackrel{i}{\to }{C}_f\stackrel{q_f}{\to }\mathrm{\Sigma }X\mathrm{.}$ The connecting homomorphism is normalized by the suspension and is not independent structure:

Thus its sign is fixed by the displayed cofiber sequence, the convention that the suspension coordinate is written first, and the chosen suspension isomorphism.

The data satisfy the following axioms.

• (H) Homotopy invariance. If $f{\simeq }_{\ast }g$ are based homotopic based maps, then $f^{\ast }=g^{\ast }$ on every reduced group.
• (E) Exactness. For every based cellular map $f:X\to Y$ the sequence $\cdots \to {\tilde{h}}^n(C_f)\stackrel{i^{\ast }}{\to }{\tilde{h}}^n(Y)\stackrel{f^{\ast }}{\to }{\tilde{h}}^n(X)\stackrel{{\delta }_f}{\to }{\tilde{h}}^{n+1}(C_f)\to \cdots$ is exact. The normalized connecting maps are natural for maps of based maps because the collapse maps $q_f$ and the suspension isomorphisms are natural.
• (W) Wedge axiom. For every family $(X_{\alpha })$ of based CW complexes and every $n$ the natural map ${\tilde{h}}^n\left({\bigvee }_{\alpha }{X}_{\alpha }\right)⟶{\prod }_{\alpha }{\tilde{h}}^n(X_{\alpha })$ induced by the summand inclusions is an isomorphism. For a finite index set the product is the direct sum; the empty wedge is a point and the empty product is the zero group, so in particular ${\tilde{h}}^n(\ast )=0$.

No dimension axiom is imposed: the value ${\tilde{h}}^n(S^0)$ is an arbitrary abelian group, called the coefficient group of degree $n$; the coefficient bookkeeping is recorded by the coefficient-groups definition on this page. Ordinary reduced singular cohomology with coefficients in an abelian group is the special case in which the dimension axiom holds, and the whole point of the definition is to admit theories for which ${\tilde{h}}^n(S^0)$ is nonzero in infinitely many degrees. A morphism of reduced generalized cohomology theories is a family of natural transformations commuting with the suspension isomorphisms. It then commutes with every connecting map because ${\delta }_f=q_f^{\ast }\sigma$.

Compare Loizides, §2, printed pp. 3–4, for the homotopy, wedge and exactness axioms, the suspension isomorphism $h^n\circ \mathrm{\Sigma }\simeq h^{n-1}$ and the definition $h^n:=h^n(S^0)=h^n(\mathrm{p}\mathrm{t})$ of the coefficient groups. In the paragraph ending on printed p. 3, the pair boundary is explicitly the suspension isomorphism followed by pullback along the cofiber-to-suspension map; this is the normalization imposed above. Davis--Kirk, §8.8, gives the corresponding reduced/unreduced axioms and correspondence.