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# 数学代写|示性类代考Characteristic Classes代考|SF3709 Proof of the uniqueness of minimal models

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## 数学代写|示性类代考Characteristic Classes代考|Proof of the uniqueness of minimal models

Proof of the uniqueness of minimal models. In this subsection we prove the latter part of Theorem 1.41, namely the uniqueness of minimal models. Let us recall the precise statement. Let $\mathcal{A}$ be a cohomologically connected d.g.a and assume that we are given two minimal models $\rho: \mathcal{M} \rightarrow \mathcal{A}$ and $\rho^{\prime}: \mathcal{M}^{\prime} \rightarrow \mathcal{A}$. Then our task is to prove that there exists an isomorphism $\varphi: \mathcal{M} \cong \mathcal{M}^{\prime}$ such

that the diagram
\begin{aligned} &\mathcal{M} \stackrel{\rho}{\longrightarrow} \mathcal{A} \ &\begin{array}{ll} \varphi \downarrow & \ \mathcal{M}^{\prime}-\underset{\rho^{\prime}}{ } \longrightarrow & \mathcal{A} \end{array} \ & \end{aligned}
is commutative up to homotopy. Moreover we have to prove also that such map $\varphi$ is unique up to homotopy. By definition, $\mathcal{M}$ is generalized nilpotent so that it can be expressed as the union of certain increasing series
$$\text { (1.4) } \mathcal{M}0=K \subset \mathcal{M}_1 \subset \mathcal{M}_2 \subset \cdots \subset \mathcal{M}{\ell} \subset \cdots$$
of Hirsch extensions. It is natural to try to construct the desired map $\varphi$ by induction on $\ell$. Then there arise certain extension problems of d.g.a. maps. More precisely, we will apply the following proposition where we replace $\mathcal{N}, \mathcal{A}, f, \mathcal{B}$ in the statement by $\mathcal{M}_{\ell}, \mathcal{M}^{\prime}, \rho^{\prime}, \mathcal{A}$ respectively.

## 数学代写|示性类代考Characteristic Classes代考|Differential forms on simplicial complexes

Differential forms on simplicial complexes. The de Rham complex $A^*(M)$ of a $C^{\infty}$ manifold $M$ is a d.g.a. over $\mathbb{R}$ so that in principle we cannot deduce information on the rational homotopy type of $M$ from it. If there are given enough cycles of $M$ over $\mathbb{Z}$, then by investigating values of integrals over them, we can decide whether a given closed form represents a rational cohomology class or not. However, this procedure cannot be considered as an intrinsic structure of the de Rham complex. Then there appeared the de Rham theory for simplicial complexes which turn out to be utilized to obtain information about their structure over $\mathbb{Q}$.

To define the de Rham complex of simplicial complexes, first consider the $k$-dimensional standard simplex
$$\Delta^k=\left{\left(t_0, t_1, \cdots, t_k\right) \in \mathbb{R}^{k+1} ; t_i \geq 0, \sum_i t_i=1\right}$$

## 数学代写|示性类代考Characteristic Classes代考|Proof of the uniqueness of minimal models

$$\mathcal{M} \stackrel{\rho}{\longrightarrow} \mathcal{A} \quad \varphi \downarrow \quad \mathcal{M}^{\prime}-{ }{\rho^{\prime}} \longrightarrow \mathcal{A}$$ $$\text { (1.4) } \mathcal{M} 0=K \subset \mathcal{M}_1 \subset \mathcal{M}_2 \subset \cdots \subset \mathcal{M} \ell \subset \cdots$$ 应用以下命题 $\mathcal{N}, \mathcal{A}, f, \mathcal{B}$ 在声明中 $\mathcal{M}{\ell}, \mathcal{M}^{\prime}, \rho^{\prime}, \mathcal{A}$ 分别。

## 数学代写|示性类代考Characteristic Classes代考|Differential forms on simplicial complexes

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