Simplicial Complexes and Simplicial Homology - Examples

For the one-vertex complex $\{\mathrm{\varnothing },\{v\}\}$, one has $C_0\cong \mathbb{Z}[v]$ and $C_n=0$ for $n\ge 1$. Hence $H_0^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}\cong \mathbb{Z}$ and $H_n^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}=0$ for $n>0$. The augmentation $\epsilon ([v])=1$ is an isomorphism, so ${\tilde{H}}_n^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}=0$ for every $n$.

For the one-edge complex with vertices $v_0,v_1$ and edge $[v_0,v_1]$, one has $C_1\cong \mathbb{Z}[v_0,v_1]$, $C_0\cong \mathbb{Z}[v_0]\oplus \mathbb{Z}[v_1]$, and $\mathrm{\partial }[v_0,v_1]=[v_1]-[v_0]$. The kernel of ${\mathrm{\partial }}_1$ is $0$, so $H_1^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}=0$. The quotient of $C_0$ by the subgroup generated by $[v_1]-[v_0]$ is $\mathbb{Z}$, so $H_0^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}\cong \mathbb{Z}$. The augmented complex is exact, hence the reduced homology also vanishes.

Let $K$ have vertices $v_0,v_1,v_2$ and edges $e_{01},e_{12},e_{20}$, with no 2-simplex. Then $C_2(K)=0$ and $C_1(K)\cong {\mathbb{Z}}^3$. If the edges are oriented cyclically, the boundary map is ${\mathrm{\partial }}_1(e_{01},e_{12},e_{20})=(v_1-v_0,v_2-v_1,v_0-v_2)\mathrm{.}$ The sum $e_{01}+e_{12}+e_{20}$ is a $1$-cycle, and every $1$-cycle is an integer multiple of this one because the three vertex equations force the three edge coefficients to agree. Since $C_2(K)=0$, one has $B_1(K)=0$, so $H_1^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)\cong \mathbb{Z}\mathrm{.}$ The complex is connected, so $H_0^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)\cong \mathbb{Z}$, and all higher homology groups vanish.

Let $K$ be the boundary of a tetrahedron. It has $4$ vertices, $6$ edges, and $4$ triangular faces. The complex is connected, hence $H_0^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)\cong \mathbb{Z}$.

Orient the faces as $[v_0,v_1,v_2]$, $[v_0,v_1,v_3]$, $[v_0,v_2,v_3]$, and $[v_1,v_2,v_3]$. The alternating sum $[v_1,v_2,v_3]-[v_0,v_2,v_3]+[v_0,v_1,v_3]-[v_0,v_1,v_2]$ is a $2$-cycle because every edge appears twice with opposite signs. Since $C_3(K)=0$, this gives a nonzero class in $H_2^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)$.

The three face boundaries $\mathrm{\partial }[v_0,v_1,v_2],\mathrm{\partial }[v_0,v_1,v_3],\mathrm{\partial }[v_0,v_2,v_3]$ are linearly independent in $C_1(K)$ because the edges $[v_1,v_2]$, $[v_1,v_3]$, and $[v_2,v_3]$ occur in only one of them. Thus $\mathrm{rank}{B}_1(K)\ge 3$. On the other hand, $\mathrm{rank}{C}_1(K)=6$ and $\mathrm{rank}{\mathrm{\partial }}_1=3$ because $H_0^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)\cong \mathbb{Z}$ and $C_0(K)\cong {\mathbb{Z}}^4$, so $\mathrm{rank}{Z}_1(K)=6-3=3$. Hence $B_1(K)=Z_1(K)$ and $H_1^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)=0$.

Now $\mathrm{rank}{\mathrm{\partial }}_2=\mathrm{rank}{B}_1(K)=3$, so $\mathrm{rank}{Z}_2(K)=4-3=1$. Since $Z_2(K)\subseteq C_2(K)\cong {\mathbb{Z}}^4$ and already contains a nonzero cycle, it follows that $H_2^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)=Z_2(K)\cong \mathbb{Z}$. Thus $H_0^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)\cong \mathbb{Z},H_1^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)=0,H_2^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)\cong \mathbb{Z},$ and $H_n^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)=0$ for $n\ge 3$.

Given a simplicial complex $K$ and a new vertex $a$, its cone $a\ast K$ is obtained by adjoining to every simplex $\sigma$ the simplex $\{a\}\cup \sigma$. The same formula used for a single simplex defines a degree-$1$ operator by setting $h[\sigma ]=[a,\sigma ]$ when $a$ is not already a vertex of $\sigma$ and $h[\sigma ]=0$ otherwise. The same cancellation check gives $\mathrm{\partial }h+h\mathrm{\partial }=\mathrm{id}$ on the augmented simplicial chain complex.

Thus the augmented chain complex of a simplicial cone is contractible, so its reduced simplicial homology vanishes. In particular, a simplicial cone is a concrete combinatorial model of a contractible space.

Let $K$ be the subdivided interval with vertices $x_0,x_1,x_2$ and edges $\{x_0,x_1\}$ and $\{x_1,x_2\}$. Let $L$ be the full $2$-simplex on vertices $a,b,c$. Define simplicial maps $f,g:K\to L$ by $f(x_0)=a,f(x_1)=b,f(x_2)=c,$ $g(x_0)=a,g(x_1)=a,g(x_2)=c\mathrm{.}$

For the edge $\{x_0,x_1\}$, the union of the image vertex sets is $\{a,b\}$; for $\{x_1,x_2\}$ it is $\{a,b,c\}$. Both are simplices of $L$, and the same is clear for the vertices, so $f$ and $g$ are contiguous. Their realizations are therefore homotopic, and they induce the same map on every simplicial homology group.

The boundary of a triangle is a finite simplicial complex with $3$ vertices and $3$ edges, so $\chi (K)=3-3=0.$ Its simplicial homology groups are $H_0^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)\cong \mathbb{Z}$ and $H_1^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)\cong \mathbb{Z}$, with no higher homology. Therefore $\mathrm{rank}{H}_0^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)-\mathrm{rank}{H}_1^{\mathrm{s}\mathrm{i}\mathrm{m}\mathrm{p}}(K)=1-1=0,$ which matches the simplex count. This is the smallest noncontractible example on the page with Euler characteristic zero.

Every function on vertex sets extends to a simplicial map.

Take the domain complex with vertices $u,v$ and edge $\{u,v\}$, and take the target complex with vertices $a,b$ and no edge $\{a,b\}$. The vertex function $u\mapsto a$, $v\mapsto b$ exists, but it sends the simplex $\{u,v\}$ to the set $\{a,b\}$, which is not a simplex of the target. Hence it is not a simplicial map.

Every finite-dimensional simplicial complex is finite and locally finite.

Let the vertex set be $\{v\}\cup \{w_n:n\ge 1\}$ and let the simplices be the empty face, all singletons, and all edges $\{v,w_n\}$. Every simplex has dimension at most $1$, so the complex is finite dimensional. But it has infinitely many edges, so it is not finite, and the vertex $v$ lies in infinitely many simplices, so it is not locally finite.

Start with a square whose opposite sides are identified to form a torus. Cut the square along one diagonal. The result is a delta-complex with two 2-simplices, three 1-simplices, and one 0-simplex after the side identifications are imposed.

This example is kept on the B page precisely because it is not an abstract simplicial complex: the same edge and vertex occur several times in the face maps of the two triangles. It is useful as a compact model of the torus, but it does not replace the A-page convention that simplices are determined by their vertex sets.