Simplicial homology

Simplicial homology computes homology from the boundary maps between free abelian groups generated by oriented simplices.

A simplicial cycle is a simplicial chain with zero boundary. Its homology class is unchanged on addition of a boundary; in the Mayer–Vietoris sequence, splitting a cycle between two subcomplexes constructs the connecting map.

The simplicial chain group $C_n(K)$ is freely generated by oriented $n$-simplices, subject to reversal of orientation changing sign. Its boundary is

In an augmented chain complex of a simplex with vertices $v_0,\dots ,v_n$, put $s(1)=[v_0]$ and $s[v_{i_0},\dots ,v_{i_q}]=[v_0,v_{i_0},\dots ,v_{i_q}]$ when $v_0$ is absent, and zero when it is present. Direct expansion of the boundary operator gives $ds+sd=1$. This proves that every positive-degree chain cycle is a chain boundary, without invoking cellular homology.

For a finite complex,

For a finite cell decomposition of the sphere into polygonal faces, the numbers of vertices, edges and faces satisfy the displayed formula. Triangulating faces preserves $V-E+F$ because each added diagonal adds one edge and one face. This is the spherical form of Euler formula for a connected planar graph, where the plane's exterior face is included.

For finite CW complexes, the product cell structure has one $(p+q)$-cell for every pair consisting of a $p$-cell and a $q$-cell. Taking the alternating cell count gives

If $p:X\to Y$ is a finite covering of degree $d$ between spaces having finite CW type, thenA lifted CW structure has exactly $d$ cells above each cell of the base.

Every closed odd-dimensional manifold has Euler characteristic zero. For an orientable manifold this follows by pairing complementary Betti numbers using Poincare duality; the general case follows from the orientable double cover and Euler characteristic under a finite covering.

If a finite simplicial complex has $f_i$ simplices of dimension $i$, thenWriting $C_i=Z_i\oplus$ a lift of $B_{i-1}$ and $Z_i=H_i\oplus B_i$ makes the boundary dimensions cancel in the alternating sum.

The vertices of the barycentric subdivision of a simplicial complex are its nonempty faces, and its simplices are strict chains of faces.

The iterated barycentric subdivisions are defined recursively by $K^{(0)}=K$ and $K^{(r+1)}=(K^{(r)}{)}^{\prime }$.

For a geometrically realized simplicial complex, its mesh isIf $K$ has dimension $n$, barycentric subdivision satisfiesso $\mu (K^{(r)})\to 0$.

The barycentric subdivision of a tetrahedron has f-vectorIts two-skeleton therefore has Euler characteristic $15-50+60=25$ and homology $H_0\cong \mathbb{Z}$, $H_1=0$, and $H_2\cong {\mathbb{Z}}^{24}$.