Reduced homology theory and augmentation

For an unreduced theory $h$ and a nonempty based CW space $(X,x_0)$ with $x_0$ a vertex, set ${\tilde{h}}_n(X)=\ker (h_n(X)\stackrel{p_{\ast }}{\to }{h}_n(\ast ))\mathrm{.}$ The basepoint inclusion $s$ satisfies $p\circ s=\mathrm{i}\mathrm{d}$ and splits this augmentation. The underlying ordinary theory is as in Unreduced homology theory on cw pairs.

Independently, a reduced ordinary theory on based CW spaces consists of homotopy-invariant covariant functors ${\tilde{h}}_n$, natural suspension isomorphisms $\sigma :{\tilde{h}}_n(X)\to {\tilde{h}}_{n+1}(\mathrm{\Sigma }X)$, exact cofiber sequences, and arbitrary wedge additivity. More explicitly, for every based CW inclusion $A\hookrightarrow X$, ${\tilde{h}}_n(A)\to {\tilde{h}}_n(X)\to {\tilde{h}}_n(X\mathrm{/}A)$ is exact; the boundary in the extended sequence is the cofiber map to $\mathrm{\Sigma }A$ followed by ${\sigma }^{-1}$. The suspension here is reduced suspension. The dimension axiom is ${\tilde{h}}_n(S^0)=0$ for $n\mathrm{\ne }0$, with ${\tilde{h}}_0(S^0)=G$. Wedge additivity includes the empty wedge $\ast$ and gives ${\tilde{h}}_n(\ast )=0$.

The empty space is not a based object. If its reduced groups are mentioned, this library uses ${\tilde{H}}_n(\mathrm{\varnothing };G)=0$ in all degrees, as in Augmentation at 0-simplices and reduced singular homology. The augmented-chain convention ${\tilde{H}}_{-1}(\mathrm{\varnothing };G)=G$ is a different extension and is not used here.