Category Archives: Homology

Homology 3: Cycle Groups and Boundary Groups

Let us use $\langle\cdots\rangle$ for an unoriented simplex and $(\cdots)$ for an oriented simplex. Examples. 1. $(p_0p_1)=-(p_1p_0)$. 2. \begin{eqnarray*}\sigma_2&=&(p_0p_1p_2)=(p_2p_0p_1)=(p_1p_2p_0)\\-(p_0p_2p_1)&=&-(p_2p_1p_0)=-(p_1p_0p_2).\end{eqnarray*} Let $K=\{\sigma_\alpha\}$ be an $n$-dimensional simplicial complex of oriented simplexes. Definition. The $r$-chain group $C_r(K)$ of a simplicial complex $K$ is … Continue reading

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Homology 2: Simplexes and Simplicial Complexes

Definition. A 0-simplex $\langle p_0\rangle$ is a point or a vertex. A 1-simplex $\langle p_0p_1\rangle$ is a line or an edge. A 2-simplex $\langle p_0p_1p_2\rangle$ is a triangle with its interior included. A 3-simplex $\langle p_0p_1p_2p_3\rangle$ is a solid tetrahedron. … Continue reading

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Homology 1: Free Abelian Groups

Before we discuss homology groups, we review some basics of abelian group theory. The group operation for an abelian group is denoted by $+$. The unit element is denoted by $0$. Let $G_1$ and $G_2$ be abalian groups. A map … Continue reading

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