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 $n$.

For an $n$-simplex, the cone formula and the induction hypothesis give $\mathrm{\partial }S\sigma =\mathrm{\partial }{b}_{\sigma }S\mathrm{\partial }\sigma =S\mathrm{\partial }\sigma -b_{\sigma }\mathrm{\partial }S\mathrm{\partial }\sigma =S\mathrm{\partial }\sigma$.

Linearity extends this equality 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 $G=0$.

Extend the simplicial chains, the characteristic-simplex map, and its boundary identity $G$-linearly. For a finite-dimensional complex, filter by skeleta. The relative comparison for $(K^r,K^{r-1})$ is an isomorphism: both relative theories are a direct sum of one copy of $G$ for each $r$-simplex in degree $r$ and vanish in the other degrees.

The simplicial and singular pair long exact sequences commute with this comparison. Skeletal induction and the five lemma therefore give an isomorphism for every finite-dimensional $K$.

A finite singular cycle, and likewise a finite chain witnessing a boundary, has compact image. In a simplicial complex this image meets only finitely many open simplices and is contained in a finite-dimensional skeleton. The finite-dimensional result gives respectively surjectivity and injectivity for arbitrary $K$. The characteristic-simplex construction commutes with simplicial maps, so the isomorphism is natural.

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 characteristic-simplex comparison of Simplicial and singular homology agree sends each vertex with coefficient $g$ to the corresponding singular point with coefficient $g$, so it 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.

For $n\ge 1$, the augmentation $H_0(S^n;G)\to G$ is surjective (use any vertex) with zero kernel, hence is an isomorphism. For $n=0$, the simplicial model consists of two vertices and no edges, giving $H_0(S^0;G)\cong G\oplus G$; its augmentation kernel is $\{(g,-g):g\in G\}$. This includes $G=0$.

Write $q:X\times [0,1]\to \mathrm{\Sigma }X$ for the quotient. The sets $U=q(X\times [0,2\mathrm{/}3))$ and $V=q(X\times (1\mathrm{/}3,1])$ are open, cover $\mathrm{\Sigma }X$, and contract to their respective apices by moving the height coordinate linearly to $0$ or $1$. Their intersection is $W\cong X\times (1\mathrm{/}3,2\mathrm{/}3)$; projection $p:W\to X$ is a homotopy equivalence, with section at height $1\mathrm{/}2$. Since $X$ has its specified basepoint, these spaces are nonempty.

A point has $H_0(\ast ;G)=G$ and $H_j(\ast ;G)=0$ for $j>0$: its singular complex has one copy of $G$ in each degree, with boundary alternately zero and identity. Thus Contractible nonempty spaces have the homology of a point gives the same groups for $U,V$, with the degree-zero isomorphism induced by augmentation. For $n\ge 1$, ordinary exactness in Mayer–Vietoris sequence in singular homology gives $H_{n+1}(\mathrm{\Sigma }X;G)\cong H_n(W;G)\cong H_n(X;G)$, the last isomorphism by Homotopy equivalences induce isomorphisms on singular homology. These degrees are positive, so the groups are also reduced groups.

For $n=0$, the same exact sequence gives $0\to H_1(\mathrm{\Sigma }X;G)\stackrel{\delta }{\to }{H}_0(W;G)\stackrel{(\epsilon ,-\epsilon )}{\to }G\oplus G$. Its last kernel is exactly ${\tilde{H}}_0(W;G)$. Projection to $X$ preserves augmentation and is an ordinary homology isomorphism, so it identifies this kernel with ${\tilde{H}}_0(X;G)$ under Augmentation at 0-simplices and reduced singular homology.

Every point of $\mathrm{\Sigma }X$ has a height path to an apex, and the height path through $x_0$ joins the two apices. Hence $\mathrm{\Sigma }X$ is path connected: differences of singular points bound paths, so augmentation identifies its $H_0$ with $G$ and its reduced $H_0$ is zero. This proves the shift for $n=-1$; for $n<-1$ both sides are zero by the stated reduced-complex convention. The arguments include the one-point space and $G=0$.

A based map $f:X\to Y$ induces $[x,t]\mapsto [f(x),t]$ and preserves these covers and their projections. At chain level the connecting map takes a small cycle $z=u+v$ to $[\mathrm{\partial }u]$; applying $f_{\mathrm{\#}}$ gives $[\mathrm{\partial }{f}_{\mathrm{\#}}u]=[f_{\mathrm{\#}}\mathrm{\partial }u]$. Thus the connecting maps commute with $f$, and so do their degree-zero kernel restrictions and the projection isomorphisms. This proves naturality in every degree.