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# 数学代写|拓扑学代写TOPOLOGY代考|Definition and properties of K-theory

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## 数学代写|拓扑学代写TOPOLOGY代考|Definition and properties of K-theory

The first and the most inportant generalizations of cohomology theory – $K$-theory and cobordism theory – were introduced (or more accurately first considered from the appropriate point of view) in the late $1950 \mathrm{~s}$ and 1960s by Atiyah. Subsequent development of the associated methodology very significantly augmented the algebraic apparatus of topology. Moreover for the investigation of many topological problems either $K$-theory or cobordism (bordism) theory has turned out to provide the most appropriatc context. The general axiomatics of extraordinary homology (and cohomology) theory were worked out by G. W. Whitehead in the early 1960s.

We begin with the basic concepts of $K$-theory. For a pair $(K, L)$ of finite $C W$-complexes the groups $K_{\mathbb{R}}^0(K, L)$ and $K_{\mathbb{C}}^0(K, L)$ are defined as the Grothendieck groups of classes of stably equivalent vector bundles (real and complex respectively) with base $B=K / L$, where “stable equivalcnce” of two vector bundles $\nu_1$ and $\nu_2$ means that
$$\nu_1 \oplus \varepsilon^{N_1}=\nu_2 \oplus \varepsilon^{N_2}$$
where $\varepsilon^{N_i} \quad(i=1,2)$ is the trivial vector bundle with fiber $\mathbb{R}^{N_i}$ or $\mathbb{C}^{N_i}$. Every vector bundle $\nu$ over $K / L$ has a stable inverse. This is explained in Chapter 4, §1: one first realizes $K / L$ as a deformation retract of a manifold $U \supset K / L$ over which $\nu$ can be extended to a bundle stably equivalent to the tangent bundle of $U$; the inverse $-\nu$ is then realized as the normal bundle over $U$ with respect to an embedding $U \subset \mathbb{R}^q, \quad q$ sufficiently large:
$$\nu \oplus(-\nu)=\varepsilon^N \sim 0$$

## 数学代写|拓扑学代写TOPOLOGY代考|Bordism and cobordism theory as generalized homology and cohomology

From a geometrical point of view, bordism theory is the most natural homology theory. We start with the classical situation where the cycles are taken to be any smooth manifolds. In unoriented bordism theory $\Omega_^O(\cdot)=\mathfrak{N}(\cdot)$, a singular $n$-cycle of a space $X$ is a pair $\left(M^n, f\right)$, where $M^n$ is a closed manifold and $f$ is a map of the manifold to $X$ :
$$\left(M^n, f\right) ; \quad f: M^n \rightarrow X$$
Two singular $n$-cycles $\left(M_1^n, f_1\right), \quad\left(M_2^n, f_2\right)$ are equivalent (or cobordant): $\left(M_1^n, f_1\right) \sim\left(M_2^n, f_2\right)$, if there exists a manifold $L^{n+1}$ with boundary $\partial L^{n+1}=$ $M_1^n \cup M_2^n$ (the disjoint union) and a map $g: L^{n+1} \rightarrow X$ such that
$$\left.g\right|{M_1^n}=f_1,\left.\quad g\right|_{M_2^n}=f_2$$
Here all manifolds involved are assumed to be embedded into $\mathbb{R}^N$ for $N>>n$. We now define the $n$-dimensional bordism group by
$$\Omega_n^O(X)={n \text {-cycles }} / \sim$$
where the (abelian) group structure is defined in terms of the disjoint union of $n$-cycles:
$$\left[\left(M_1^n, f_1\right)\right]+\left[\left(M_2^n, f_2\right)\right]=\left[\left(M_1^n \cup M_2^n, f_1 \cup f_2\right)\right]$$

## 数学代写|拓扑学代写TOPOLOGY代考|Definition and properties of K-theory

$$\nu_1 \oplus \varepsilon^{N_1}=\nu_2 \oplus \varepsilon^{N_2}$$

$$\nu \oplus(-\nu)=\varepsilon^N \sim 0$$

## 数学代写|拓扑学代写TOPOLOGY代考|Bordism and cobordism theory as generalized homology and cohomology

$$\left(M^n, f\right) ; \quad f: M^n \rightarrow X$$

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