Relative Homology Excision and Mayer Vietoris

Inclusion sends every singular simplex of $A$ to the same simplex in $X$; its faces still have image in $A$. Hence $\mathrm{\partial }{C}_n(A;G)\subseteq C_{n-1}(A;G)$, so $[c]\mapsto [\mathrm{\partial }c]$ is well defined on the quotient.

Applying this map twice gives $[{\mathrm{\partial }}^{2}c]=0$, since the singular boundary squares to zero.

The induced singular-chain map sends $C_n(A;G)$ into $C_n(B;G)$, hence $[c]\mapsto [f_{\mathrm{\#}}c]$ defines a quotient-chain map.

It commutes with the quotient boundaries, so it maps cycles to cycles and boundaries to boundaries and hence descends to homology. The chain-level identity and composition laws persist after quotienting.

Since every singular chain group of $\mathrm{\varnothing }$ is zero, $C_{\bullet }(X,\mathrm{\varnothing };G)=C_{\bullet }(X;G)$ as complexes.

Since $C_{\bullet }(X;G)\mathrm{/}{C}_{\bullet }(X;G)$ is the zero complex, its cycles and boundaries are both zero in every degree. Taking homology proves both assertions.

Inclusion and quotient form a degreewise short exact sequence $0\to C_{\bullet }(A;G)\to C_{\bullet }(X;G)\to C_{\bullet }(X,A;G)\to 0$.

The long-exact-sequence theorem for a short exact sequence of complexes applied to step 1.1 gives precisely the displayed sequence, with the third homology group equal to relative homology by definition.

Equality of their relative classes says $c^{\mathrm{\prime }}-c=a+\mathrm{\partial }b$ for some $a\in C_n(A;G)$ and $b\in C_{n+1}(X;G)$.

Therefore $\mathrm{\partial }{c}^{\mathrm{\prime }}-\mathrm{\partial }c=\mathrm{\partial }a$, an ordinary boundary in $C_{n-1}(A;G)$; the two proposed values agree. Linearity follows from linearity of $\mathrm{\partial }$.

The maps on $A$, $X$, and quotient chains give a commuting morphism of the short exact sequences used for the two pair sequences.

Naturality of the homological long exact sequence makes every resulting square commute. In particular, chainwise $f_{\mathrm{\#}}(\mathrm{\partial }c)=\mathrm{\partial }{f}_{\mathrm{\#}}(c)$ gives commutation at the connector.

In degree $0$, both sides vanish. Assume $\mathrm{\partial }S=S\mathrm{\partial }$ on dimensions below $n$, where $n\ge 1$.

Let ${\iota }_n$ be the identity simplex of ${\mathrm{\Delta }}^n$ and put $z=S(\mathrm{\partial }{\iota }_n)$, the affine lift used in the definition of $S\sigma$. If $n=1$, then $z=[v_1]-[v_0]$, so its augmentation is zero. The degree-zero cone formula gives $\mathrm{\partial }S\sigma ={\sigma }_{\mathrm{\#}}z=S(\mathrm{\partial }\sigma )$.

If $n\ge 2$, the induction hypothesis applied in ${\mathrm{\Delta }}^n$ gives $\mathrm{\partial }z=S({\mathrm{\partial }}^2{\iota }_n)=0$ by The singular boundary squares to zero. The positive-degree cone formula therefore gives $\mathrm{\partial }S\sigma ={\sigma }_{\mathrm{\#}}z-b_{\sigma }(\mathrm{\partial }z)={\sigma }_{\mathrm{\#}}z=S(\mathrm{\partial }\sigma )$. The last equality follows directly from the recursive definition: subdivision commutes with postcomposition by $\sigma$ on each affine face.

Linearity extends these equalities to finite chains, completing the induction.

In dimension $0$, $S=1$ and $T=0$. Assume the identity on all faces of an $n$-simplex $\sigma$.

On the universal simplex, the induction hypothesis applied to $\mathrm{\partial }{\iota }_n$ gives $\mathrm{\partial }({\iota }_n-P(\mathrm{\partial }{\iota }_n))=S(\mathrm{\partial }{\iota }_n)\mathrm{.}$ The affine-cone formula and the recursive definition of subdivision therefore give $\mathrm{\partial }{P}_n={\iota }_n-P(\mathrm{\partial }{\iota }_n)-b_{n}S(\mathrm{\partial }{\iota }_n)={\iota }_n-P(\mathrm{\partial }{\iota }_n)-S({\iota }_n)\mathrm{.}$

Pushing the equality in step 2.1 forward along $\sigma :{\mathrm{\Delta }}^n\to X$ gives $\mathrm{\partial }T\sigma +T\mathrm{\partial }\sigma =\sigma -S\sigma$. Linearity proves the identity on chains.

Each barycentric subsimplex has vertices which are barycenters of nested faces; their convex-coordinate differences have diameter at most $n\mathrm{/}(n+1)$ times that of the parent.

Iterating the estimate yields $(n\mathrm{/}(n+1){)}^{r}d$. Since $0<n\mathrm{/}(n+1)<1$, this tends to $0$ as $r$ tends to infinity.

The image of each simplex of $c$ is compact, so its pulled-back cover has a Lebesgue number; mesh decay supplies a subdivision depth making every subsimplex image lie in one cover member.

There are finitely many original simplices, so the maximum of their finitely many depths works for all of them. Thus $S^{r}c$ is cover-small; no depth is claimed for all singular simplices at once.

For an ordinary cycle $z$, choose a depth with $S^{r}z$ small. Repeatedly applying $1-S=\mathrm{\partial }T+T\mathrm{\partial }$ shows $z$ and $S^{r}z$ are homologous, proving surjectivity.

If a small cycle bounds an ordinary chain $b$, choose a depth making $S^{r}b$ small. The same prism identity says the original cycle and its small subdivision differ by a boundary already in the small complex; together with $\mathrm{\partial }{S}^{r}b=S^r\mathrm{\partial }b$ this proves injectivity.

Use $S$ and $T$ with $1-S=\mathrm{\partial }T+T\mathrm{\partial }$ from Subdivision is chain homotopic to the identity. Their constructions in Barycentric subdivision operator and Subdivision prism homotopy take each singular simplex to a finite sum of its compositions with affine simplices in its own domain. Thus neither operator enlarges the image of a simplex, and both preserve $A_{\bullet }$. Also $S$ is a chain map: applying $\mathrm{\partial }$ on either side of the displayed homotopy identity gives $\mathrm{\partial }S=S\mathrm{\partial }$. Set $D_q={\sum }_{i=0}^{q-1}T{S}^i$ for $q\ge 0$, so $D_0=0$ and $1-S^q=\mathrm{\partial }{D}_q+D_q\mathrm{\partial }$ by telescoping.

For each singular simplex $\sigma$, let $a(\sigma )$ be the least $q\ge 0$ with $S^q\sigma \in A_{\bullet }$; existence follows from Finite chains eventually become cover-small with integer coefficients. Define $m$ recursively on dimension: on vertices set $m=0$, and for positive-dimensional $\sigma$ set $m(\sigma )=\max (\{a(\sigma )\}\cup \{m(\sigma \circ {\delta }_j):0\le j\le \dim \sigma \})\mathrm{.}$ Every maximum is finite and uses already defined lower-dimensional values. Since $S$ preserves small chains, $S^{m(\sigma )}\sigma$ is small. Every face $\tau$ has $m(\tau )\le m(\sigma )$ by construction, regardless of cancellations in subdivided chains. If $\sigma$ is already small, all its faces are small, so this recursion gives $m(\sigma )=0$.

Define $D\sigma =D_{m(\sigma )}\sigma$ on integral generators and extend linearly. Put $R=1-\mathrm{\partial }D-D\mathrm{\partial }$ on $C_{\bullet }$. The identity ${\mathrm{\partial }}^2=0$ gives $\mathrm{\partial }R=R\mathrm{\partial }$. On a simplex, telescoping gives $R\sigma =S^{m(\sigma )}\sigma +D_{m(\sigma )}(\mathrm{\partial }\sigma )-D(\mathrm{\partial }\sigma )\mathrm{.}$ The first term is small. For each face $\tau$, its signed correction is ${\sum }_{i=m(\tau )}^{m(\sigma )-1}T{S}^i\tau$. Each $S^i\tau$ is small for these indices, and $T$ preserves small chains. Therefore $R\sigma$ is small.

Regard $R$ as a chain map $r:C_{\bullet }\to A_{\bullet }$ and let $\iota :A_{\bullet }\hookrightarrow C_{\bullet }$. Then $1-\iota r=\mathrm{\partial }D+D\mathrm{\partial }$. On every small simplex $m=0$, hence $D=0$ on $A_{\bullet }$; its boundary is also small, so $r\iota =1$. Thus one inverse composite is the identity and the other is chain homotopic to it. In degree zero all vertices are small and $D=0$; if $X$ is empty both complexes are zero.

The group $A_n$ is the direct summand of the free abelian group $C_n$ spanned by small singular simplices. Tensoring its inclusion with $G$ identifies $A_n\otimes G$ with the cover-small subgroup of $C_n(X;G)$. Tensor $r,D$ and their identities with ${\mathrm{id}}_G$; these identities remain valid for every abelian $G$, including $G=0$, without a flatness assumption. This proves the stated chain homotopy equivalence with the page's coefficients.

Use the cover $\{X\setminus \bar{Z},\mathrm{int}A\}$: its cover-small quotient complex for $(X,A)$ consists, modulo small chains in $A$, of precisely the small chains avoiding $Z$.

The two inclusions from their small complexes to the corresponding full relative complexes are chain homotopy equivalences. The quotient identification in step 1.1 therefore induces the stated isomorphism.

Since $V$ deformation retracts onto $A$, $H_{\ast }(V,A;G)=0$. The quotient homotopy gives the same conclusion for $(V\mathrm{/}A,A\mathrm{/}A)$. The long exact sequences of the short exact chain-complex sequences $0\to C_{\ast }(V,A;G)\to C_{\ast }(X,A;G)\to C_{\ast }(X,V;G)\to 0$ and its quotient analogue therefore give $H_{\ast }(X,A;G)\cong H_{\ast }(X,V;G),H_{\ast }(X\mathrm{/}A,A\mathrm{/}A;G)\cong H_{\ast }(X\mathrm{/}A,V\mathrm{/}A;G)\mathrm{.}$

Because $A$ is closed and lies in the open set $V$, excision applies to $(X,V)$ and to $(X\mathrm{/}A,V\mathrm{/}A)$ after removing respectively $A$ and the quotient point. The resulting removed pairs are homeomorphic, so the quotient map induces $H_{\ast }(X,V;G)\cong H_{\ast }(X\mathrm{/}A,V\mathrm{/}A;G)$. Combining with step 1.1 gives $H_{\ast }(X,A;G)\cong H_{\ast }(X\mathrm{/}A,A\mathrm{/}A;G)$. Since $A\mathrm{\ne }\mathrm{\varnothing }$, $A\mathrm{/}A$ is one point, and the augmented singular complex identifies the latter groups with ${\tilde{H}}_{\ast }(X\mathrm{/}A;G)$, including degree zero.

A cover-small generator has image in $U$ or in $V$, hence belongs to the displayed sum; conversely each generator from either summand is cover-small.

The singular simplices simultaneously belonging to both free-chain subgroups are exactly those with image in $U\cap V$, so their intersection is the stated chain group.

Both $i$ and $j$ commute with boundaries; $i$ is injective and $j$ is surjective by the sum description of cover-small chains.

If $u+v=0$, then $u=-v$ is a chain in both $U$ and $V$, so $(u,v)=i(u)$. Thus the kernel of $j$ is the image of $i$.

Apply the homological long exact sequence to the short exact sequence of the preceding theorem.

The cover-small complex has the same homology as $X$, so replacing its homology term identifies that long exact sequence with the displayed one.

Exactness of the chain sequence gives $u-u^{\mathrm{\prime }}=w$ and $v-v^{\mathrm{\prime }}=-w$ for some $w\in C_n(U\cap V;G)$.

Hence $\mathrm{\partial }u-\mathrm{\partial }{u}^{\mathrm{\prime }}=\mathrm{\partial }w$, so the two overlap cycles give the same homology class. Changing $z$ by a boundary yields the same calculation one degree higher.

Composition with $f$ commutes with the singular boundary, since $(f\circ \sigma )\circ {\delta }_j=f\circ (\sigma \circ {\delta }_j)$. The cover conditions give chain maps on the overlap, on each summand, and $f_s:A_X\to A_Y$. They commute with $i(c)=(c,-c)$ and $j(u,v)=u+v$ by additivity. Thus they give a morphism of the short exact sequences in Short exact chain Mayer–Vietoris sequence, and The long exact homology sequence is natural gives a commuting ladder of their homology sequences.

Let ${\iota }_X:A_X\hookrightarrow C_{\bullet }(X;G)$ and ${\iota }_Y:A_Y\hookrightarrow C_{\bullet }(Y;G)$ be the inclusions. On every small simplex both composites are the same singular simplex $f\circ \sigma$, so $f_{\mathrm{\#}}{\iota }_X={\iota }_Y{f}_s$. The maps $H_n({\iota }_X),H_n({\iota }_Y)$ are isomorphisms by Cover-small chains compute singular homology. Therefore $H_n(f_s)H_n({\iota }_X{)}^{-1}=H_n({\iota }_Y{)}^{-1}{H}_n(f)$. Transporting the ladder of step 1.1 along these isomorphisms yields exactly the ordinary homology maps and sequences of Mayer–Vietoris sequence in singular homology.

In particular, for a small cycle $z=u+v$, its image is $f_{\mathrm{\#}}u+f_{\mathrm{\#}}v$ and $f_{\mathrm{\#}}(\mathrm{\partial }u)=\mathrm{\partial }{f}_{\mathrm{\#}}u$. Thus the connector sends the image class to the image of $[\mathrm{\partial }u]$, with the same positive U-boundary sign; independence of the chosen small representative and decomposition is supplied by Well-definedness of the Mayer–Vietoris connector and the inclusion isomorphisms. This proves every connecting square as well as the ordinary squares. The argument includes degree zero, the terminal maps to zero, empty overlap or empty cover members, and $G=0$.

Every finite complex has a total order on its finite vertex set. For a chosen order, the characteristic-simplex homomorphisms give a chain map $\mathrm{\Phi }:C_{\ast }^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K;G)\to C_{\ast }^{\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}}(\mathrm{\mid }K\mathrm{\mid };G)$ by the cited boundary lemma, and give a map of pair exact sequences for every subcomplex. No tensor-exactness is used: the chain groups in both pair sequences are degreewise split by their simplex bases.

We will use finite simplicial-subcomplex excision without invoking the later CW-homology-axioms item. Let $L$ be a nonempty subcomplex of a finite $K$. Under the canonical homeomorphism $\mathrm{\mid }\mathrm{sd}K\mathrm{\mid }\cong \mathrm{\mid }K\mathrm{\mid }$ of Barycentric subdivision realizes homeomorphically, the vertices of $\mathrm{sd}L$ are the nonempty faces of $L$. They span a full subcomplex of $\mathrm{sd}K$: in any face chain, the faces belonging to $L$ form an initial segment because $L$ is closed under taking faces. For $x\in \mathrm{\mid }\mathrm{sd}K\mathrm{\mid }$, let $s(x)$ be the sum of its barycentric coordinates at these vertices. This is a continuous function, and $U=\{s>0\}$ is an open neighborhood of $\mathrm{\mid }\mathrm{sd}L\mathrm{\mid }$. On $U$ discard the other coordinates and divide the retained ones by $s(x)$, obtaining a continuous map $r:U\to \mathrm{\mid }\mathrm{sd}L\mathrm{\mid }$. In each subdivided simplex the straight-line homotopy $(1-t)x+tr(x)$ stays in that simplex, fixes $\mathrm{\mid }\mathrm{sd}L\mathrm{\mid }$, and retracts $U$ onto it. Thus every such finite pair is a good pair in Good pairs and quotient reduced homology. If finite complexes $K=L\cup \sigma$ with $L\cap \sigma =\mathrm{\partial }\sigma$, their quotient map-out descriptions give $\mathrm{\mid }K\mathrm{\mid }\mathrm{/}\mathrm{\mid }L\mathrm{\mid }\cong \mathrm{\mid }\sigma \mathrm{\mid }\mathrm{/}\mathrm{\mid }\mathrm{\partial }\sigma \mathrm{\mid }$, so inclusion $(\mathrm{\mid }\sigma \mathrm{\mid },\mathrm{\mid }\mathrm{\partial }\sigma \mathrm{\mid })\to (\mathrm{\mid }K\mathrm{\mid },\mathrm{\mid }L\mathrm{\mid })$ induces an isomorphism on relative singular homology when $\dim \sigma >0$. The same argument applies to $\mathrm{\partial }{\mathrm{\Delta }}^r=F\cup \mathrm{\Lambda }$ with $F\cap \mathrm{\Lambda }=\mathrm{\partial }F$ for $r\ge 2$, giving $H_{\ast }(F,\mathrm{\partial }F;G)\cong H_{\ast }(\mathrm{\partial }{\mathrm{\Delta }}^r,\mathrm{\Lambda };G)$. These are exactly the two finite excisions used below.

We compute the singular relative group of an oriented simplex without using any simplicial-to-singular comparison. For $r=0$, $H_0({\mathrm{\Delta }}^0,\mathrm{\varnothing };G)=G$, generated by its vertex. For $r=1$, the pair sequence of the contractible interval and its two endpoints gives $H_1({\mathrm{\Delta }}^1,\mathrm{\partial }{\mathrm{\Delta }}^1;G)=\ker (G^2\stackrel{(a,b)\mapsto a+b}{\to }G)$; the oriented edge maps to its endpoint difference and generates this kernel. All other relative groups vanish.

We prove finite support for a compact subset $C\subseteq \mathrm{\mid }K\mathrm{\mid }$ without making choices from the set of simplices. The weak realization is Hausdorff: every vertex-coordinate function is continuous by testing its restriction on each finite simplex, and distinct points differ in some coordinate. Thus $C$ is closed. For each open simplex ${\sigma }^{\circ }$ meeting $C$ and integer $m\ge 1$, define $T_{\sigma ,m}=C\cap \{x\in \mathrm{\mid }\sigma \mathrm{\mid }:{\alpha }_v(x)\ge 1\mathrm{/}m\text{ for every }v\in \sigma \}\mathrm{.}$ Let $m_{\sigma }$ be the least $m$ for which this set is nonempty, and put $T_{\sigma }=T_{\sigma ,m_{\sigma }}$. Such an integer exists because every coordinate of a point in $C\cap {\sigma }^{\circ }$ is positive. The set $T_{\sigma }$ is closed in the compact finite-dimensional simplex $\mathrm{\mid }\sigma \mathrm{\mid }$, hence is nonempty compact, and every point of it remains in ${\sigma }^{\circ }$. For each of the finitely many total orders $o$ on the finite vertex set of $\sigma$, successively minimize the coordinates in that order on $T_{\sigma }$; finite-dimensional compactness and the extreme-value theorem give a unique lexicographic minimum $x_{\sigma ,o}\in T_{\sigma }$. Define $F_{\sigma }=\{x_{\sigma ,o}:o\text{ is a total order on }\sigma \}$. This is a canonically defined nonempty finite subset of $C\cap {\sigma }^{\circ }$, even though no order on the vertices of $K$ has been chosen.

For $r\ge 2$ let $F$ be the facet of ${\mathrm{\Delta }}^r$ opposite its first vertex and let $\mathrm{\Lambda }$ be the union of the other facets. Then $\mathrm{\partial }{\mathrm{\Delta }}^r=F\cup \mathrm{\Lambda }$ and $F\cap \mathrm{\Lambda }=\mathrm{\partial }F$. Both ${\mathrm{\Delta }}^r$ and $\mathrm{\Lambda }$ are contractible and nonempty: $\mathrm{\Lambda }$ is the cone from the first vertex on $\mathrm{\partial }F$. The pair long exact sequence gives $H_k({\mathrm{\Delta }}^r,\mathrm{\Lambda };G)=0$ for every $k$. The short exact sequence of relative singular chain complexes $0\to C_{\ast }(\mathrm{\partial }{\mathrm{\Delta }}^r,\mathrm{\Lambda };G)\to C_{\ast }({\mathrm{\Delta }}^r,\mathrm{\Lambda };G)\to C_{\ast }({\mathrm{\Delta }}^r,\mathrm{\partial }{\mathrm{\Delta }}^r;G)\to 0$ therefore gives, by its homology long exact sequence, an isomorphism $H_k({\mathrm{\Delta }}^r,\mathrm{\partial }{\mathrm{\Delta }}^r;G)\stackrel{\mathrm{\partial }}{\to }{H}_{k-1}(\mathrm{\partial }{\mathrm{\Delta }}^r,\mathrm{\Lambda };G)$ for $k\ge 1$. The explicit finite-subcomplex excision in step 1.2 identifies the right group with $H_{k-1}(F,\mathrm{\partial }F;G)$. By induction from the degree-zero and degree-one computations of step 1.3, the only nonzero relative group is $G$ in degree $k=r$. Under the two isomorphisms, the affine characteristic simplex $[v_0,\dots ,v_r]$ maps first to its alternating boundary and then, modulo $\mathrm{\Lambda }$, to the opposite facet with coefficient $+g$ (up to the fixed orientation sign); the other facets vanish in that quotient. Thus its relative class is exactly a generator, with no appeal to a prior sphere or simplicial comparison. Degree zero vanishes because $\mathrm{\partial }{\mathrm{\Delta }}^r$ is nonempty.

Put $S={\bigcup }_{{\sigma }^{\circ }\cap C\mathrm{\ne }\mathrm{\varnothing }}{F}_{\sigma }$. Each closed simplex $\mathrm{\mid }\tau \mathrm{\mid }$ meets only those $F_{\sigma }$ for which $\sigma$ is a face of $\tau$, and there are finitely many such faces; hence $S\cap \mathrm{\mid }\tau \mathrm{\mid }$ is finite. The same holds for every subset of $S$, so every subset of $S$ is closed in the weak topology of $\mathrm{\mid }K\mathrm{\mid }$. In particular $S$ is closed in the compact space $C$, hence compact, and it is discrete because its singleton complements are closed. The singleton open cover of a compact discrete space has a finite subcover, so $S$ is finite. The pairwise disjoint nonempty sets $F_{\sigma }\subseteq {\sigma }^{\circ }$ therefore occur for only finitely many occupied simplices $\sigma$. Their faces form a finite subcomplex containing $C$. The construction used finite total orders within each simplex only as an indexed finite set, never a simultaneous selection of one order per simplex.

Let $K$ be finite and build it one simplex at a time in a face-respecting order. Suppose $L$ has already been built and the next simplex is $\sigma$. Then $L\cap \sigma =\mathrm{\partial }\sigma$. The simplicial relative chain complex of $(K,L)$ is one copy of $G$ in degree $\dim \sigma$ and zero elsewhere. If $\dim \sigma >0$, the finite-subcomplex excision proved in step 1.2 identifies $H_{\ast }(\mathrm{\mid }K\mathrm{\mid },\mathrm{\mid }L\mathrm{\mid };G)$ with $H_{\ast }(\mathrm{\mid }\sigma \mathrm{\mid },\mathrm{\mid }\mathrm{\partial }\sigma \mathrm{\mid };G)$. If $\dim \sigma =0$, then $\mathrm{\mid }K\mathrm{\mid }=\mathrm{\mid }L\mathrm{\mid }\bigsqcup \mathrm{\mid }\sigma \mathrm{\mid }$ and its singular relative chain complex is directly the singular complex of the isolated vertex, giving the same relative comparison even when $L$ is empty. Step 1.3 or 2.1 shows that $\mathrm{\Phi }$ maps the relative simplicial generator to its singular generator. The commuting pair long exact sequences and the five lemma extend the isomorphism from $L$ to $K$. The empty complex starts the induction.

Every singular cycle and every singular chain witnessing a boundary is a finite sum of compact simplex images, so step 2.2 supports each in some finite subcomplex. Simplicial chains already have finite support. The finite comparison from step 3.1 is therefore surjective and injective on the directed union of finite subcomplexes, proving the isomorphism for all $K$. Finite unions make this directed system choice-free. [step 3.1, step 2.2].

It remains to make the comparison canonical and natural. On a finite complex two vertex orders give chain maps $\mathrm{\Phi },\mathrm{\Psi }$ agreeing on vertices. Construct a chain homotopy recursively on each oriented simplex $\sigma$. If it is already defined on proper faces, the chain $z=(\mathrm{\Phi }-\mathrm{\Psi })(\sigma )-H(\mathrm{\partial }\sigma )$ is a cycle in the convex $\mathrm{\mid }\sigma \mathrm{\mid }$; in degree zero the difference is zero. Coning $z$ to the barycenter of $\sigma$ gives $H(\sigma )$ with boundary $z$ (the augmented cone identity also handles a zero-chain of augmentation zero). This is a finite explicit prescription and works $G$-linearly. Hence different orders induce the same map on homology. For a simplicial map $f:K\to P$, the two chain maps $f_{\mathrm{\#}}{\mathrm{\Phi }}_K$ and ${\mathrm{\Phi }}_P{f}_{\ast }$ agree on vertices, and their values on $\sigma$ lie in the convex simplex $\mathrm{\mid }f(\sigma )\mathrm{\mid }$. The identical coning induction proves they are chain homotopic, including when $f$ collapses a simplex. On finite subcomplexes this proves order independence and naturality; step 4.1 passes them to arbitrary complexes without selecting orders simultaneously for all finite subcomplexes.

Put $D={\mathrm{\Delta }}^{n+1}$ and $K=\mathrm{\partial }D$. The integral augmented complex of $D$ has a vertex-cone contraction, as used in A simplex has zero reduced simplicial homology. Tensoring the contraction identity with $G$ preserves it, so the augmented complex with coefficients in $G$ is exact. For $k<n$, its groups and differentials computing reduced homology in degree $k$ agree with those of $K$, hence ${\tilde{H}}_k^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K;G)=0$.

In degree $n$, exactness for $D$ gives $\ker {\mathrm{\partial }}_n=\mathrm{im}{\mathrm{\partial }}_{n+1}$. The map $G=C_{n+1}(D;G)\to C_n(K;G)$ sends $g$ to the alternating sum of its facets with coefficient $g$; it is injective since any one facet has coefficient $g$ or $-g$. There are no $(n+1)$-chains in $K$, so ${\tilde{H}}_n^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K;G)\cong G$. This uses the augmentation as ${\mathrm{\partial }}_0$ when $n=0$. Above degree $n$ all simplicial groups vanish.

The boundary $K$ of the single simplex is finite, so choose an order on its finite vertex set. The characteristic-simplex comparison of Simplicial and singular homology agree sends each vertex with coefficient $g$ to the corresponding singular point with coefficient $g$; its homology map is independent of this order and commutes with augmentation to $G$. For either theory, ${\tilde{H}}_0=\ker (H_0\to G)$, directly from the kernel-in-degree-zero definition in Augmentation at 0-simplices and reduced singular homology. Thus the ordinary comparison isomorphism restricts to an isomorphism on these kernels; in positive degrees reduced and ordinary homology agree. The calculation therefore transfers to $\mathrm{\mid }K\mathrm{\mid }\cong S^n$, and negative reduced groups are zero by convention.