Unreduced pair and reduced quotient axioms are equivalent on cw pairs

Start with F1. The splitting $p_{\ast }{s}_{\ast }=1$ identifies $h_n(X,\ast )$ with $\ker p_{\ast }$ by the exact sequence of $(X,\ast )$: $s_{\ast }$ is injective in every degree, so its cokernel is $h_n(X,\ast )$, and the splitting identifies that cokernel with the kernel. This is natural for based maps and is the reduced group of F2.

For a CW inclusion $i:A\hookrightarrow X$ with $A\mathrm{\ne }\mathrm{\varnothing }$, form the unreduced cone attachment $C_i=X{\cup }_{A}CA$. CW excision gives $h_n(X,A)\cong h_n(C_i,CA)$. Since $CA$ contracts to its apex, pair exactness identifies the latter with ${\tilde{h}}_n(C_i)$. Collapse $CA$ to the apex. This gives a homotopy equivalence $C_i\to X\mathrm{/}A$: extend the contraction of $CA$ over $C_i$ using F3; the terminal extension factors through the quotient, and its quotient homotopy and original homotopy exhibit the two inverse composites. Thus $h_n(X,A)\cong {\tilde{h}}_n(X\mathrm{/}A)$. For empty $A$, add a disjoint basepoint first, obtaining $h_n(X)\cong h_n(X_{+},\ast )$. This also sends the empty pair to zero.

The reduced cone sequence and contractibility of the reduced cone give ${\tilde{h}}_{n+1}(\mathrm{\Sigma }Y)\cong {\tilde{h}}_n(Y)$; take its inverse as the suspension map. To recover the unreduced boundary for nonempty $A$, view $C_i$ as the based cofiber of $A_{+}\to X_{+}$, with the cone apex as basepoint. Collapse $X$ together with that apex to obtain $C_i\to \mathrm{\Sigma }{A}_{+}$; both cone ends are now identified, as required for reduced suspension of $A_{+}$. Naturality of the pair sequences for $(C_i,X\cup \{\text{apex}\})$ and the cone pair shows that the original connecting homomorphism is this induced cofiber map followed by inverse suspension into ${\tilde{h}}_{n-1}(A_{+})\cong h_{n-1}(A)$. Thus the cofiber exact sequence is exactly the pair sequence, with its signs fixed by this convention. Dimension on $S^0$ follows from the split two-point augmentation.

Conversely, from F2 define $h_n(X,A)={\tilde{h}}_n(X_{+}\mathrm{/}{A}_{+})$ and $h_n(X)={\tilde{h}}_n(X_{+})$. When $A$ is nonempty the quotient is $X\mathrm{/}A$; when $A$ is empty it is $X_{+}$. Replacing the inclusion $A_{+}\to X_{+}$ by its mapping cylinder, its cofiber is homotopy equivalent to this quotient by the contraction argument above. Repeating the cone construction gives the sequence $A_{+}\to X_{+}\to C_i\to \mathrm{\Sigma }{A}_{+}\to \mathrm{\Sigma }{X}_{+}\to \cdots$. Each successive pair of maps is, up to homotopy, a CW inclusion and its quotient: after attaching the next cone, the previously attached contractible cone collapses by F3. The reduced exactness axiom and suspension therefore give the pair LES in every integer degree, with boundary as just specified. All these constructions respect maps, so the boundary is natural.

For $X=U\cup V$, the map $U_{+}\mathrm{/}(U\cap V{)}_{+}\to X_{+}\mathrm{/}{V}_{+}$ is a homeomorphism of based CW spaces, so reduced homotopy invariance yields CW excision. A quotient of a disjoint union of pairs is the wedge of their based quotients, with the CW weak topology; reduced wedge additivity hence gives unreduced disjoint-sum additivity. In the other direction, apply unreduced additivity to $(⨆X_{\alpha },⨆\{x_{\alpha }\})$ and the quotient formula to obtain wedge additivity. The empty wedge is a point and both empty sums are zero. The formulas also give $h_0(\ast )={\tilde{h}}_0(S^0)=G$ and the dimension vanishing.

The two recipes are inverse through the natural quotient and splitting isomorphisms above. A morphism commuting with pair boundaries commutes with the cone suspension isomorphisms, and conversely a reduced morphism commuting with suspension commutes with the reconstructed boundaries. Finally, apply the reduced cofiber sequence to $A_{+}\mathrm{/}{B}_{+}\subset X_{+}\mathrm{/}{B}_{+}$, whose quotient is $X_{+}\mathrm{/}{A}_{+}$. This gives the triple sequence. The cone map factors the usual pair boundary followed by the quotient of $A$, by naturality of the cone construction. This proves its stated formula as well as exactness, including $A=B$, $B=\mathrm{\varnothing }$, and $A=X$.