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 Determinants II: Determinants of Order $n$  MathPhys Archive on Determinants I: Determinants of Order 2
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Category Archives: Algebraic Topology
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 0simplex $\langle p_0\rangle$ is a point or a vertex. A 1simplex $\langle p_0p_1\rangle$ is a line or an edge. A 2simplex $\langle p_0p_1p_2\rangle$ is a triangle with its interior included. A 3simplex $\langle p_0p_1p_2p_3\rangle$ 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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