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# 物理代写|拓扑物理代写Physical topology代考|Dheeraj Kulkarni

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## 物理代写|拓扑物理代写Physical topology代考|Motivating Examples

Before we begin defining the simplicial homology, let us look at a few simple and instructive examples. We will deal with symmetric objects in the Euclidean spaces namely the convex polyhedrons. We will focus on boundaries of these objects in Fig. 4.1.
Finding “Holes”‘ in the space.
First observe from Fig. $4.1$ that the boundary of a boundary of a convex polyhedron is empty. This is a very important geometric observation on which homology theory rests.

Next there may be objects without boundaries. We call objects without boundary as “cycles” (the term is justified if we look at the examples in Fig. 4.1). The figure suggests that there could be cycles in topological spaces that are not boundaries. In other words, existence of cycles which are not boundaries suggests that there are “holes” in the space. This is the second important observation for defining homology theory.

## 物理代写|拓扑物理代写Physical topology代考|Simplicial Complex

Let us define what do we mean by a symmetric polyhedron in $m$-dimensional Euclidean space.
Definition 4.3.1. Let $\left{v_{0}, v_{1}, v_{2}, \ldots, v_{n}\right}$ be $n+1$ points in $\mathbb{R}^{m}$ such that they are not contained in a hyperplane of dimension less than $n$. Then an $n$ simplex is the smallest convex set containing $\left{v_{0}, v_{1}, v_{2}, \ldots, v_{n}\right}$. We denote it by $\left[v_{0}, v_{1}, \ldots, v_{n}\right]$. Thus, we have
$$\left[v_{0}, v_{1}, \ldots, v_{n}\right]=\left{\sum_{i=0}^{n} t_{i} v_{i} \mid 0 \leq t_{i} \leq 1 \text { and } \sum_{i=0}^{n} t_{i}=1\right}$$
Thus, every point in $n$-simplex receives coordinates given by $\left(t_{0}, t_{1}, \ldots, t_{n}\right)$. They are called barycentric coordinates.

Remark 4.3.1. The points $\left{v_{0}, v_{1}, \ldots, v_{n}\right}$ do not lie in a hyperplane of dimension less than $n$ is equivalent to saying that the set of vectors $\left{v_{1}-v_{0}, v_{2}-\right.$ $\left.v_{0}, \ldots, v_{n}-v_{0}\right}$ is a linearly independent set in $\mathbb{R}^{m}$.
Remark 4.3.2. By an $n$-simplex, we really mean an ordered set of points $\left[v_{0}, v_{1}, v_{2}, \ldots, v_{n}\right]$. This naturally induces an order on the subsets of points by writing them in increasing order of subscripts. For example $\left[v_{0}, v_{1}\right],\left[v_{1}, v_{2}\right]$ and $\left[v_{0}, v_{2}\right]$ are sub-simplices of a 3 -simplex $\left[v_{0}, v_{1}, v_{2}\right]$. One more consequence of ordering on the vertices is that there is a canonical linear homeomorphism between any two $n$-simplices preserving the order of points.

## 物理代写|拓扑物理代写PHYSICAL TOPOLOGY代考|Review of Abelian Groups

Recall that a group $(G,+)$ is abelian if any two elements commute, i.e., $a+b=$ $b+a$ for all $a, b \in G$. We say that $G$ is finitely generated if there is a finite generating set for $G$. More explicitly, there is a set $\left{a_{1}, \ldots, a_{k}\right}$ such that for any element $g$ in $G$ we have $g=n_{1} a_{1}+n_{2} a_{2}+\cdots+n_{k} a_{k}$ for some integers $n_{i}$.
We say that a set $\left{g_{1}, g_{2}, \ldots, g_{r}\right}$ is linearly independent if $n_{1} g_{1}+n_{2} g_{2}+$ $\cdots+n_{r} g_{r}=0$ implies that each $n_{i}=0$.

We say that an abelian group $G$ is a free abelian group of rank $r$ if there is a linearly independent generating set with $r$ elements.

$\mathbb{Z}$ is a free abelian group of rank 1 but $\mathbb{Z}{n}$ for $n \neq 0$ is not a free abelian group. In general if $G$ is a free abelian group of rank $r$ then it is isomorphic to $$\underbrace{\mathbb{Z} \oplus \mathbb{Z} \oplus \cdots \oplus \mathbb{Z}}{r \text { copies }} .$$
Now let us recall examples of finite abelian groups. First we have the cyclic group $C_{n}$ of order $n$ which is isomorphic to $\mathbb{Z} / n \mathbb{Z}$. We can further take direct sums of cyclic abelian groups, for example, $C_{n} \oplus C_{m}$. The following theorem asserts that essentially these are the only finite abelian groups.

## 物理代写|拓扑物理代写PHYSICAL TOPOLOGY代考|SIMPLICIAL COMPLEX

\left[v_{0}, v_{1}, \ldots, v_{n}\right]=\left{\sum_{i=0}^{n} t_{i} v_{i} \mid 0 \ leq t_{i} \leq 1 \text { 和 } \sum_{i=0}^{n} t_{i}=1\right}\left[v_{0}, v_{1}, \ldots, v_{n}\right]=\left{\sum_{i=0}^{n} t_{i} v_{i} \mid 0 \ leq t_{i} \leq 1 \text { 和 } \sum_{i=0}^{n} t_{i}=1\right}

## Matlab代写

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