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# 数学代写|黎曼几何代写Riemannian Geometry代考|MATH5061 Relative Cohomology and Cofibrations

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## 数学代写|黎曼几何代写Riemannian Geometry代考|Relative Cohomology and Cofibrations

There are two short topics left to discuss in this chapter, one relatively straightforward and one for discussion. We begin with relative cohomology. Given cochain complexes $(F, \mathrm{~d})$ and $(G, \mathrm{~d})$ with a cochain $\operatorname{map} \phi: F^{n} \rightarrow G^{n}$ for all $n$ (i.e., d $\phi=$ $\phi \mathrm{d})$, we form a new complex $E^{n}=F^{n} \oplus G^{n-1}$ with $\mathrm{d}(f, g)=(\mathrm{d} f, \phi(f)-\mathrm{d} g)$. Then
$$\mathrm{d}^{2}(f, g)=\mathrm{d}(\mathrm{d} f, \phi(f)-\mathrm{d} g)=\left(\mathrm{d}^{2} f, \phi(\mathrm{d} f)-\mathrm{d} \phi(f)+\mathrm{d}^{2} g\right)=(0,0) .$$
We call the cohomology of $(E$, d) the relative cohomology $\mathrm{H}(F, G, \phi)$.
Proposition 4.90 Given a cochain map $\phi: F^{n} \rightarrow G^{n}$ for all $n$ we have a long exact relative cohomology sequence
$$\ldots \mathrm{H}^{n-1}(G) \stackrel{i_{2}^{}}{\longrightarrow} \mathrm{H}^{n}(F, G, \phi) \stackrel{\pi_{1}^{}}{\longrightarrow} \mathrm{H}^{n}(F) \stackrel{\phi^{}}{\longrightarrow} \mathrm{H}^{n}(G) \stackrel{i_{2}^{}}{\longrightarrow} \mathrm{H}^{n+1}(F, G, \phi) \ldots,$$
where $\pi_{1}: E^{n} \rightarrow F^{n}$ is $\pi(f, g)=f$ and $i_{2}: G^{n} \rightarrow E^{n+1}$ is $i_{2}(g)=(-1)^{n}(0, g)$.
Proof Standard algebraic manipulation. Looking at the kernel of $\phi^{*}: \mathrm{H}^{n}(F) \rightarrow$ $\mathrm{H}^{n}(G)$ shows that $\mathrm{H}^{n}(F, G, \phi)$ is defined precisely to make this work.

## 数学代写|黎曼几何代写Riemannian Geometry代考|Quantum Principal Bundles and Framings

Vector bundles in classical geometry typically arise as associated to something deeper, a principal bundle. A connection on this then induces covariant derivatives on all associated bundles in a coherent way. This is the situation in Riemannian geometry where a ‘spin connection’ on the frame bundle induces the Levi-Civita connection on tensor fields but also a covariant derivative on the spinor bundle in the case of a spin manifold, leading to the Dirac operator. Similarly in gauge theory, a principal connection induces covariant derivatives on all associated matter fields.
Briefly, a principal $G$-bundle $P$ over a manifold $X$ is defined exactly like a vector bundle with a surjection $\pi: P \rightarrow X$ but each fibre $P_{x}=\pi^{-1}(x)$ now has the structure of a fixed group $G$. This is achieved by starting with a free right action of $G$ on the manifold $P$ such that $X=P / G$. Free here means any non-identity element of $G$ acts without fixed points, which is equivalent to saying that the map
$$P \times G \rightarrow P \times P, \quad(p, g) \mapsto\left(p, p^{g}\right)$$
is an inclusion, where $p^{g}$ denotes the right action of $g \in G$ on $p \in P$. A connection on $P$ is defined concretely as an equivariant complement in $\Omega^{1}(P)$ to the ‘horizontal forms’ (those pulled back from $\Omega^{1}(X)$ ). This is, however, equivalent to $\omega \in \Omega^{1}(P) \otimes \mathfrak{g}$ with certain properties, where $\mathfrak{g}$ is the Lie algebra of $G$. We will see details in the noncommutative case. Given this data, there is an associated vector bundle $E=P \times_{G} V$ and a connection $\nabla$ on it, for any representation $V$ of $G$. We will give the algebraic and potentially ‘quantum’ version of this notion where the structure group is now a Hopf algebra or ‘quantum group’ as in Chap. 2. We will then use this theory to understand the geometry of quantum homogeneous spaces and framed quantum manifolds more generally.

# 黎曼几何代写

## 数学代写|黎曼几何代写Riemannian Geometry代考|Relative Cohomology and Cofibrations

$$\mathrm{d}^{2}(f, g)=\mathrm{d}(\mathrm{d} f, \phi(f)-\mathrm{d} g)=\left(\mathrm{d}^{2} f, \phi(\mathrm{d} f)-\mathrm{d} \phi(f)+\mathrm{d}^{2} g\right)=(0,0) .$$

$$\ldots \mathrm{H}^{n-1}(G) \stackrel{i_{2}}{\longrightarrow} \mathrm{H}^{n}(F, G, \phi) \stackrel{\pi_{1}}{\longrightarrow} \mathrm{H}^{n}(F) \stackrel{\phi}{\longrightarrow} \mathrm{H}^{n}(G) \stackrel{i_{2}}{\longrightarrow} \mathrm{H}^{n+1}(F, G, \phi) \ldots$$

## 数学代写|黎曼几何代写Riemannian Geometry代考|Quantum Principal Bundles and Framing

$$P \times G \rightarrow P \times P, \quad(p, g) \mapsto\left(p, p^{g}\right)$$

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