Homology Axioms Degree and Classical Applications

The singular pair sequence is exact and its connecting maps commute with every map of pairs, by F3 and F4. These assertions hold for arbitrary subspaces and hence for CW pairs.

For a homotopy of pairs, the prism chain homotopy on $C_{\ast }(X;G)$ preserves $C_{\ast }(A;G)$: every prism simplex over a simplex in $A$ stays in the target subspace. It therefore descends to relative chain quotients. The identity $f_{\mathrm{\#}}-g_{\mathrm{\#}}=\mathrm{\partial }P+P\mathrm{\partial }$ gives relative homotopy invariance, with the same absolute conclusion as F2. Negative degrees are zero.

Put $W=U\cap V$ and form $D=U{\cup }_W(W\times [0,1]){\cup }_{W}V$, attaching $W\times \{0\}$ to $U$ and $W\times \{1\}$ to $V$. Its collapse $c:D\to U\cup V$ is a homotopy equivalence relative to $V$. Here is the cofibration argument: write $D=(U{\cup }_{W}W\times [0,1]){\cup }_{W}V$. The mapping cylinder of the inclusion $W\hookrightarrow U$ retracts onto $U$, and the HEP for $(U,W)$ extends the path of $W$ from the zero end to the one end to a map $U\times [0,1]\to U{\cup }_{W}W\times [0,1]$. The resulting end map fixes the attaching copy of $W$ at the one end. The cylinder collapse and this end map are inverse up to homotopies fixed on that attaching copy, by contracting the traversed interval followed by its reverse. The HEP for the cylinder pair extends this contraction. Gluing $V$ by its identity gives the claimed relative equivalence. This uses F6 for the two CW inclusions.

In $D$ set $O=U\cup (W\times [0,2\mathrm{/}3))$ and $N=V\cup (W\times (1\mathrm{/}3,1])$. They are open and cover $D$. The closure of $D\setminus O$ is contained in $N$. Excision therefore identifies $H_n(O,O\cap N;G)$ with $H_n(D,N;G)$. Retraction of $N$ to $V$ and the pair exact sequence identify the latter with $H_n(D,V;G)$. The collapse $O\to U$ restricts on $O\cap N=W\times (1\mathrm{/}3,2\mathrm{/}3)$ to projection onto $W$. Both absolute maps are homotopy equivalences, so their commuting pair exact sequences give an isomorphism $H_n(O,O\cap N;G)\to H_n(U,W;G)$ by the exact-sequence injectivity and surjectivity chase. No inverse map of these pairs is needed. All maps commute with collapse to $(U\cup V,V)$, so the resulting isomorphism is the homomorphism of the original inclusion.

These are exactly the four structural requirements in F1. If either subcomplex or their intersection is empty the cylinder construction has the corresponding empty pieces; excision and the relative chain quotients still give the same conclusion.

In the point complex there is one singular simplex in every nonnegative degree. The boundary on its copy of $G$ is multiplication by ${\sum }_{j=0}^n(-1{)}^j$: it is the identity for positive even $n$ and zero for odd $n$; ${\mathrm{\partial }}_0=0$. Thus its homology is $G$ in degree zero and zero in every other degree, also when $G=0$.

A singular simplex has connected domain and hence its image lies in a single summand of a disjoint union. The chain complex of a union is consequently the direct sum of the chain complexes; this is the chain mechanism underlying F4. Taking the quotient by the corresponding subspace chains gives the direct sum of the relative chain complexes.

A finite-support tuple is a cycle exactly when every coordinate is a cycle. It is a boundary exactly when every coordinate is a boundary: choose a bounding chain in each of its finitely many nonzero coordinates. Thus homology commutes with this direct sum. This includes the empty family, whose chain complex is zero, and a singleton family. With F2 this verifies all axioms of F1.

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 $A=X$.

The dimension axiom gives the assertion for $(\ast ,\mathrm{\varnothing })$. On $S^0=\{-,+\}$ the split augmentation is addition $h_k(\ast )\oplus h_k(\ast )\to h_k(\ast )$, with kernel $g\mapsto (-g,g)$. Thus the reduced sphere assertion starts in degree zero, with the displayed difference orientation.

For a nonempty sphere, the contractible cone has zero reduced groups. Its exact sequence identifies the relative cone group in degree $k$ with ${\tilde{h}}_{k-1}$ of its base, including degree one where this is a reduced zero-degree group. The quotient is the suspension sphere. Iterating F1's natural suspension gives ${\tilde{h}}_k(S^n)\cong h_{k-n}(\ast )$.

For $n\ge 1$, apply the same cone boundary to $(D^n,S^{n-1})$. The dimension axiom makes $h_{k-n}(\ast )$ vanish unless $k=n$, including all negative $k$. Fix orientations by these boundary identifications starting with $(-g,g)$ on $S^0$. They remain valid for $G=0$.

By the skeletal definition F3, collapsing $F_{r-1}$ leaves a wedge of one $r$-sphere per relative cell, using the disjoint-basepoint convention when $F_{r-1}$ is empty. Quotient identification and arbitrary wedge additivity in F2, followed by F1, show $h_k(F_r,F_{r-1})$ is zero for $k\mathrm{\ne }r$ and the stated direct sum for $k=r$. This includes no cells and zero-dimensional cells.

Write ${\delta }_r:h_r(F_r,F_{r-1})\to h_{r-1}(F_{r-1},A)$ for the triple boundary and ${\rho }_r:h_r(F_r,A)\to h_r(F_r,F_{r-1})$ for the quotient map. The differential is $d_r={\rho }_{r-1}{\delta }_r$. Exactness of the triple $(F_{r-1},F_{r-2},A)$ gives ${\delta }_{r-1}{\rho }_{r-1}=0$.

Consequently $d_{r-1}{d}_r={\rho }_{r-2}({\delta }_{r-1}{\rho }_{r-1}){\delta }_r=0$ for $r\ge 2$; for $r=1$ it holds because $d_0=0$. A cellular map preserves each $F_r$ and all natural triple maps, so it commutes with these differentials. No sphere-map incidence identification has been used.

Use the filtration and notation of F1. Write $E_q^r=h_q(F_r,A)$. The triple sequence and concentration of $h_q(F_r,F_{r-1})$ in degree $r$ give $E_q^r=0$ for $q<0$ or $q>r$, inductively from $E_q^{-1}=0$. They also give $E_q^{r-1}\to E_q^r$ an isomorphism for $q<r-1$, and a surjection for $q=r-1$. Since the filtration terminates, $E_n^{n+1}\cong h_n(X,A)$, taking $F_r=X$ above the top dimension.

For $n\ge 0$, ${\rho }_n:E_n^n\to C_n^h$ is injective since $E_n^{n-1}=0$. Likewise ${\rho }_{n-1}:E_{n-1}^{n-1}\to C_{n-1}^h$ is injective when $n\ge 1$. Exactness then yields ${\rho }_n(E_n^n)=\ker d_n$; for $n=0$ this says $E_0^0=C_0^h$ since $E_{\ast }^{-1}=0$.

The surjection $i:E_n^n\to E_n^{n+1}$ has kernel $\mathrm{im}{\delta }_{n+1}$. Since ${\rho }_n{\delta }_{n+1}=d_{n+1}$, define $\alpha (x)=[{\rho }_n(y)]$ for any lift $i(y)=x$. Two lifts differ by a ${\delta }_{n+1}$ image, so the class is independent. Every cycle is ${\rho }_n(y)$ by the preceding step, giving surjectivity. If ${\rho }_n(y)=d_{n+1}(z)$, injectivity of ${\rho }_n$ implies $y={\delta }_{n+1}(z)$ and hence $x=0$, proving injectivity.

Every map of the filtered exact diagrams carries a lift to a lift and commutes with $\rho ,\delta ,i$, so it commutes with $\alpha$. This proves cellular naturality and uniqueness of the isomorphism defined by the lift rule. For pair boundaries use the same construction on the cofiber sequence $A_{+}\to X_{+}\to X_{+}\mathrm{/}{A}_{+}\to \mathrm{\Sigma }{A}_{+}$. Give each suspension cell the cone orientation. Its cellular chain group in degree $n+1$ is the reduced degree-$n$ group of $A_{+}$, and its boundary is the suspended boundary with the cone sign convention. The cofiber map sends a relative cellular cycle represented by a chain $c$ of $X$ to the suspended class of $dc\in C_{n-1}^h(A)$: the faces outside $A$ cancel because $c$ was a relative cycle. In the exact diagram this is exactly the triple connecting map. Desuspending therefore identifies the axiomatic pair boundary with the chain connecting map $[c]\mapsto [dc]$. The lift construction commutes with suspension because its $\rho$ and $\delta$ maps do.

The direct-sum decomposition by relative cells splits $0\to C_{\ast }^h(A)\to C_{\ast }^h(X)\to C_{\ast }^h(X,A)\to 0$ degreewise, so the preceding chain-boundary formula is defined for arbitrary $G$, without a flatness assumption. Negative degrees are zero by the first step. Empty pairs and pairs with no relative cells give zero on both sides.

A vertex has its prescribed coefficient map $G\to h_0(\ast )$. Inductively orient an ordered simplex $\sigma =[v_0,\dots ,v_r]$ by the relative class whose boundary is the alternating oriented boundary of $\sigma$. To justify the induction without assuming integer action, let $T$ be the union of all its faces except the face opposite $v_0$. It is a cone on the boundary of that opposite face and contracts. The exact sequence of $(\mathrm{\partial }\sigma ,T)$ identifies ${\tilde{h}}_{r-1}(\mathrm{\partial }\sigma )$ with $h_{r-1}(\mathrm{\partial }\sigma ,T)$, which by excision is the relative group of that opposite face. This specifies uniquely the boundary class from the already oriented face and its coefficient $g$. The cone pair boundary then specifies uniquely the relative $r$-class. For $r=1$ this is $(-g,g)$ at the two endpoints.

The coefficient of the opposite face is $g$. At every common codimension-two face the boundary of this boundary is zero by the triple sequence; thus the two incident face coefficients must cancel with their inductively fixed signs. The adjacency graph of the faces of a simplex is connected, so this forces all coefficients to be $(-1{)}^{j}g$ on the face omitting $v_j$. For $r=1$ the augmentation-kernel calculation supplies the same cancellation. Thus the induced differential is exactly the alternating formula of F2.

Permuting the vertices sends the alternating boundary class to its permutation sign times the old class, by induction on faces; injectivity of the relative-simplex boundary fixes the same sign upstairs. Adjacent transpositions generate all permutations, matching the oriented relation F3. A simplicial map injective on the vertices of a simplex hence acts by this signed ordered-image map. If its image has lower dimension, it factors through that lower-dimensional simplex and its relative degree-$r$ homology vanishes by F1. This proves simplicial naturality, including degenerate simplicial images.

Summing over the simplices outside $L$ yields the asserted chain isomorphism, with no tensoring of an exact sequence required: both sides are direct sums of copies of $G$ and the differential is already the same signed matrix. Apply the skeletal homology computation F1. The simplicial-to-singular comparison F4 extends to a finite pair by the natural short exact sequences of simplicial and singular chains and the resulting pair exact sequences: the absolute isomorphisms for $K,L$ give the relative isomorphism by the usual injectivity/surjectivity exact-sequence chase. This yields the desired comparison, commuting with boundaries and normalized on a vertex. Empty complexes, $K=L$, and dimension zero all give the same direct-sum formulas.

Realize the finite complexes in their barycentric Euclidean spaces. The open star of a vertex $w$ consists of points whose $w$ coordinate is positive. If several open stars intersect, their vertices belong to the support simplex of any point in the intersection; hence they span a simplex. This is the star criterion for a vertex map to extend as in F1.

For every $a\in \mathrm{\mid }L\mathrm{\mid }$, some vertex $w$ of $Q$ has positive coordinate at $f(a)$. The open set $f^{-1}({\mathrm{st}}_{P}w)$ therefore contains a metric ball $B(a,2{ϵ}_a)$. Finitely many $B(a,{ϵ}_a)$ cover the compact set $\mathrm{\mid }L\mathrm{\mid }$. Choose $\eta >0$ smaller than all their radii. Every set of diameter less than $\eta$ containing a vertex $v\in \mathrm{\mid }L\mathrm{\mid }$ lies in one of the corresponding $B(a,2{ϵ}_a)$, since $v$ lies in its inner ball. Thus any sufficiently small closed star at a vertex of $L$ maps into the star of a vertex of $Q$. If $L$ is empty this constraint is absent.

The preimages of all vertex stars in $P$ cover the compact $\mathrm{\mid }K\mathrm{\mid }$. F3 gives a Lebesgue number $\lambda >0$. By F2 choose a common iterated subdivision whose simplex diameters are less than $\min (\eta ,\lambda )\mathrm{/}3$, omitting $\eta$ if $L$ is empty. A closed vertex star has diameter at most twice the mesh. At vertices of the subdivided $L$ make the constrained choice of the preceding step; elsewhere use $\lambda$. If $K$ is zero-dimensional the stars are singletons and no subdivision is needed; if $K$ is empty the assertion is vacuous.