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# 数学代写|拓扑学代写TOPOLOGY代考|Smooth manifolds and homotopy theory

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## 数学代写|拓扑学代写TOPOLOGY代考|Smooth manifolds and homotopy theory

In this section we shall first consider some ideas, based on the geometry of manifolds, giving access by elementary means to certain information concerning the homotopy classes of maps of manifolds (in cases other than those examined in Chapter 3 where the homotopy groups were known to be trivial for elementary reasons). In their further development these geometrical ideas become combined with the algebraic techniques described at the conclusion of Chapter 3.

We remind the reader that for smooth manifolds any continuous maps and homotopies can be approximated arbitrary closely by smooth maps coinciding with the original map wherever it happened to be smooth, so that we always assume all maps and homotopies of manifolds to be of smoothness class $C^{\infty}$ if need be.

The simplest of homotopy invariants is the degree of a map between closed orientable manifolds of the same dimension:
$$f: M^n \longrightarrow N^n$$
defined as follows: Consider a generic point $y$ (i.e. “rcgular”) in $N^n$; the Jacobian $J_f$ of $f$ is then non-zero at each point of the complete inverse image $f^{-1}(y)=\left{x_1, \ldots, x_k\right}$. We define
$$\operatorname{deg} f=\sum_{j=1}^k \operatorname{sgn} J_f\left(x_j\right)$$

## 数学代写|拓扑学代写TOPOLOGY代考|There is a complicated theory

There is a complicated theory (devised by Rohlin in the early 1950s) leading to the determination of the groups $\pi_{n+3}\left(S^n\right)$. As the first (non-trivial) step, it is shown that every non-trivial framed manifold $\left(W^3, v_n\right)$ is bordant to a framed sphere $S^3 \subset \mathbb{R}^{n+3}$, situated in standard fashion in $\mathbb{R}^{n+3}$ (since in the stable range every disposition of $S^3$ in $\mathbb{R}^{n+3}$ is equivalent to the standard one). It is shown next that the latitude in defining a frame on $S^3 \in \mathbb{R}^{n+3}$ determines a homomorphism (in fact an epimorphism)
$$J: \pi_3\left(S O_n\right) \longrightarrow \pi_{n+3}\left(S^n\right)$$
Since $\pi_3\left(S O_n\right) \cong \mathbb{Z}$ for $n>4$, it now follows that the stable group $\pi_{n+3}\left(S^n\right)$ is cyclic. Further analysis reveals that

$$\pi_{n+3}\left(S_n\right) \cong \mathbb{Z} / 24 \text { for } n>4, \quad \pi_7\left(S^4\right) \cong \mathbb{Z} \oplus \mathbb{Z} / 12, \quad \pi_6\left(S^3\right) \cong \mathbb{Z} / 12 .$$
These results are closely linked to Rohlin’s theorem to the effect that for a closed manifold $M^4$ with the property that $M^4 \backslash\left{x_0\right}$ is parallelizable, the first Pontryagin class $p_1\left(M^4\right)$ is divisible by 48 . (Recall that charactcristic classes are elements of the integral cohomology groups.)
As a generalization of (3.7) one has that the equivalence classes of framed $k$-spheres $\left(S^k, \nu_n\right)$ determine in $\pi_{n+k}\left(S^n\right)$ the image of the “Whitehead homomorphism” (see Chapter $3, \S 8$ )
$$J: \pi_k\left(S O_n\right) \longrightarrow \pi_{n+k}\left(S^n\right)$$

## 数学代写|拓扑学代写TOPOLOGY代考|Smooth manifolds and homotopy theory

$$f: M^n \longrightarrow N^n$$

$$\operatorname{deg} f=\sum_{j=1}^k \operatorname{sgn} J_f\left(x_j\right)$$

## 数学代写|拓扑学代写TOPOLOGY代考|There is a complicated theory

$$J: \pi_3\left(S O_n\right) \longrightarrow \pi_{n+3}\left(S^n\right)$$

$$\pi_{n+3}\left(S_n\right) \cong \mathbb{Z} / 24 \text { for } n>4, \quad \pi_7\left(S^4\right) \cong \mathbb{Z} \oplus \mathbb{Z} / 12, \quad \pi_6\left(S^3\right) \cong \mathbb{Z} / 12 .$$

$$J: \pi_k\left(S O_n\right) \longrightarrow \pi_{n+k}\left(S^n\right)$$

## MATLAB代写

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