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Lefschetz不动点理论
为了计数不动点$x=f(x),$ 只需数$\Delta\cap G(f)$的交点个数.
由此定义$f$的整体Lefschetz数为$L(f):=I(\Delta,G(f)).$ 它是同伦不变量.
定理 1. 设$f$为紧致定向流形$X$的光滑自映射. 若$L(f)\neq 0,$ 则$f$有不动点.
推论 2. 若映射同伦于常值映射, 则$L(f)=\pm $
由此可推出Brouwer不动点定理.
称光滑映射$f:X\rightarrow X$为Lefschetz映射, 若$\Delta\pitchfork G(f).$
由于$G(\mathrm{id})=\Delta,$ $L(\mathrm{id})=\chi(X).$
若$\mathrm{id}$同伦于无不动点映射, 则$\chi(X)=0.$
特别地对于非零欧拉示性数的流形, 无不动点映射必不同伦于恒同映射.
命题 3. 任意光滑映射$f$同伦于一个Lefschetz映射.
重复横截同伦定理的证明即可. 不能直接使用是因为要得到图像的横截.
从横截性条件来看, $f$是Lefschetz映射当且仅当$df_x$不以$1$为特征值.
可定义局部Lefschetz数$L_x(f)=1,$
若$T_{(x,x)}\Delta+T_{(x,x)}G(f)=T_{x,x}(X\times X).$ 定向不匹配为$-1.$
此时$L(f)=\sum_{f(x)=x}L_x(f).$
命题 4. $L_x(f)=\operatorname{sgn}\det(df_x-I).$
由此可通过构造同伦于恒同映射的Lefschetz映射来计算欧拉示性数.
对于非Lefschetz映射, 我们有
命题 5 (分裂命题). 设$U$为映射$f$的不动点$x$处邻域, 且$U$中不含其它不动点. 那么存在同伦映射$f_t$使得$f_1$在$U$中仅含Lefschetz不动点, 且在$U$的某个紧集外同伦保持不变.
取局部小扰动 $f_t(x)=f(x)-t\rho(x) a$ 即可. 特别地, 令扰动 $a$ 为
$f(x)-x$ 的正则值, 即有 $\det df_x-I\neq 0.$
设$x$为$f$的孤立不动点. 若$B$为以$x$为球心的小闭球, 且不含其它不动点, 则
$$
F:\partial B\rightarrow S^{k-1},\quad z\mapsto \frac{f(z)-z}{|f(z)-z|}
$$
为光滑映射. 称$\deg F$为$f$在$x$处的局部Lefschetz数, 记为$L_x(f).$
定义与$B$选取无关.
命题 6. 在Lefschetz不动点, 关于Lefschetz不动点和孤立不动点的局部Lefschetz数定义一致.
只需利用线性同痕引理, 令$F\sim F_0$为标准保定向/反定向函数即可.
命题 7. 设$f:\mathbb{R}^k\rightarrow \mathbb{R}^k$有孤立不动点$x$, 设$B$为以$x$为球心的不含$f$其它不动点的闭球. 选取满足分裂命题的函数$f_1\sim f,$ 那么$L_x(f)=\sum_{f_1(z)=z}L_z(f_1),$ $z\in B.$
前述说明都是在欧氏空间上进行的, 容易搬到流形上说明.
定义与坐标卡选取无关.
定理 8 (Lefschetz数的局部计算). 设$f:X\rightarrow X$为紧定向流形$X$上的光滑自映射, 且$f$只有有限个不动点. 则$f$的整体Lefschetz数等于局部Lefschetz数之和, 即
$$
L(f)=\sum_{f(x)=x}L_x(f)
$$
Poincaré-Hopf 定理
对$\mathbb{R}^k$中向量场$v,$ 在$x=0$处有一个孤立零点.
取充分小开球$B_\varepsilon$不包含其它零点, 定义指标
$$
\operatorname{Ind}_0(v):=\deg \left(\partial B_\varepsilon\rightarrow S^{k-1},x\mapsto \frac{v(x)}{|v(x)|}\right)=W(v,0)
$$
定理 9 (Poincaré-Hopf 定理). $\sum_{v(x)=0} \operatorname{Ind}_x(v)=\chi(X).$
只需利用管状邻域构造向量场的积分曲线, 定义映射$f_t\sim \mathrm{id},$
说明
$$
\chi(X)=L(\mathrm{id})=L(f_t)=\sum_{f_t(z)=z} L_{z}f_t=\sum_{v(x)=0} \operatorname{Ind}_x(v).
$$
由于
$$
L_z(f_t)=\deg \frac{f_t(z)-z}{|f_t(z)-z|},
$$
对$f_t(z)$做Taylor展开, 线性近似得到结论.
Hopf 映射度定理
定理 10. $f_0,f_1:X^k\rightarrow S^k$同伦当且仅当$\deg f_0=\deg f_1.$
设$f:X^k\rightarrow \mathbb{R}^{k+1},$ 定义环绕数
$$
W(f,z):=\deg (u:X^k\rightarrow S^k,x\mapsto \frac{f(x)-z}{|f(x)-z|}),\quad z\notin \operatorname{Im}f.
$$
命题 11. 设$x$为正则点, $f(x)=z.$ 令$B$为以$x$为心的充分小闭球. 记$\partial f=f|_{\partial B}:\partial B\rightarrow \mathbb{R}^k,$ 则$W(\partial f,z)=1,$ 若$df_x$保定向. 反之为$-1.$
证明和Lefschetz不动点类似.
命题 12. 设$f:B\rightarrow \mathbb{R}^k$为闭球上的光滑映射, $z$为正则值, $f^{-1}(z)\cap \partial B=\varnothing.$ 那么按定向计数有$\# f^{-1}(z)=W(\partial f,z).$
证明利用唱片引理即可.
命题 13. 设$B$为$\mathbb{R}^k$中闭球, $f:\mathbb{R}^k\setminus \operatorname{Int}B\rightarrow Y$为光滑映射. 若$\partial f:\partial B\rightarrow Y$同伦于常值映射, 则$f$可延拓为光滑映射$\mathbb{R}^k\rightarrow Y.$
证明利用Whitney逼近定理.
对于Hopf映射度定理, 首先证明映射度为零的情形. 对维数进行归纳证明.
定理 14 (延拓定理). 设$W$是$k+1$维紧致连通定向带边流形, $f:\partial W\rightarrow S^k$为光滑映射, 则$f$可延拓为$F:W\rightarrow S^k$当且仅当$\deg f=0.$
命题 15. 设$W$是任意紧致带边流形, $f:\partial W\rightarrow \mathbb{R}^{k+1}$为任意光滑映射, 则$f$可延拓到整个$W$上.
现对于任意映射度相同的$f_0,f_1,$ 将其定义在$X^k\times I$上,
延拓给出同伦.
定理 16. 紧致连通定向流形$X$上有处处非零的向量场当且仅当$\chi(X)=0.$
文章最后更新于 2023-06-13 12:33:59
Lefschetz fixed point theory
In order to count fixed points $x=f(x),$, just count the number of intersections of $\Delta\cap G(f)$.
This defines the overall Lefschetz number of $f$ as $L(f):=I(\Delta,G(f)).$ which is a homotopy invariant.
Theorem 1. Let $f$ be the smooth self-mapping of the compact oriented manifold $X$. If $L(f)\neq 0,$, then $f$ has a fixed point.
Corollary 2. If the mapping is homotopic to the constant value mapping, then $L(f)=\pm $
From this we can derive Brouwer's fixed point theorem.
Smooth map $f:X\rightarrow X$ is called Lefschetz map, if $\Delta\pitchfork G(f).$
Due to $G(\mathrm{id})=\Delta,$ $L(\mathrm{id})=\chi(X).$
If $\mathrm{id}$ is homotopic to a map without fixed points, then $\chi(X)=0.$
In particular, for manifolds with non-zero Euler characteristics, fixed-point-free maps must be inconsistent with identity maps.
Proposition 3. Any smooth map $f$ is homotopic to a Lefschetz map.
Just repeat the proof of the transverse homotopy theorem. It cannot be used directly because the transverse section of the image needs to be obtained.
From the perspective of transversality conditions, $f$ is a Lefschetz map if and only if $df_x$ does not have $1$ as an eigenvalue.
Definable local Lefschetz number $L_x(f)=1,$
If $T_{(x,x)}\Delta+T_{(x,x)}G(f)=T_{x,x}(X\times X).$ orientation does not match $-1.$
At this time $L(f)=\sum_{f(x)=x}L_x(f).$
Proposition 4. $L_x(f)=\operatorname{sgn}\det(df_x-I).$
From this, the Euler characteristic number can be calculated by constructing a Lefschetz map that is homotopic to an identity map.
For non-Lefschetz mappings, we have
Proposition 5 (Split proposition). Let $U$ be the neighborhood of the fixed point $x$ of the map $f$, and $U$ does not contain other fixed points. Then there is a homotopy map $f_t$ such that $f_1$ only contains Lefschetz fixed points in $U$, and the homotopy remains unchanged outside a certain compact set of $U$.
Just take the local small perturbation $f_t(x)=f(x)-t\rho(x) a$. In particular, let the perturbation $a$ be
The regular value of $f(x)-x$ is $\det df_x-I\neq 0.$
Let $x$ be the isolated fixed point of $f$. If $B$ is a small closed sphere with $x$ as its center and does not contain other fixed points, then
$$
F:\partial B\rightarrow S^{k-1},\quad z\mapsto \frac{f(z)-z}{|f(z)-z|}
$$
is a smooth mapping. Call $\deg F$ the local Lefschetz number of $f$ at $x$, denoted as $L_x(f).$
The definition has nothing to do with the $B$ selection.
Proposition 6. At the Lefschetz fixed point, the definitions of the local Lefschetz numbers for the Lefschetz fixed point and the isolated fixed point are consistent.
Just use the linear homotrace lemma and let $F\sim F_0$ be the standard orientation-preserving/anti-orientation function.
Proposition 7. Let $f:\mathbb{R}^k\rightarrow \mathbb{R}^k$ have an isolated fixed point $x$, let $B$ be a closed sphere with $x$ as the center and without other fixed points of $f$. Select the function $f_1\sim f,$ that satisfies the split proposition, then $L_x(f)=\sum_{f_1(z)=z}L_z(f_1),$ $z\in B.$
The foregoing explanations are all performed on Euclidean space, and can easily be transferred to manifolds for explanation.
The definition has nothing to do with coordinate card selection.
Theorem 8 (Local calculation of Lefschetz numbers). Assume $f:X\rightarrow X$ is a smooth self-mapping on the compactly oriented manifold $X$, and $f$ has only a finite number of fixed points. Then the overall Lefschetz number of $f$ is equal to the sum of the local Lefschetz numbers, that is
$$
L(f)=\sum_{f(x)=x}L_x(f)
$$
Poincaré-Hopf theorem
For the vector field $v,$ in $\mathbb{R}^k$, there is an isolated zero at $x=0$.
Take a sufficiently small kick $B_\varepsilon$ that does not include other zero points and define the indicator
$$
\operatorname{Ind}_0(v):=\deg \left(\partial B_\varepsilon\rightarrow S^{k-1},x\mapsto \frac{v(x)}{|v(x)|}\right)=W(v,0)
$$
Theorem 9 (Poincaré-Hopf theorem). $\sum_{v(x)=0} \operatorname{Ind}_x(v)=\chi(X).$
Just use the tubular neighborhood to construct the integral curve of the vector field and define the mapping $f_t\sim \mathrm{id},$
Description
$$
\chi(X)=L(\mathrm{id})=L(f_t)=\sum_{f_t(z)=z} L_{z}f_t=\sum_{v(x)=0} \operatorname{Ind}_x(v).
$$
due to
$$
L_z(f_t)=\deg \frac{f_t(z)-z}{|f_t(z)-z|},
$$
Perform Taylor expansion on $f_t(z)$ and obtain the conclusion through linear approximation.
Hopf mapping degree theorem
Theorem 10. $f_0,f_1:X^k\rightarrow S^k$ Homotopy if and only if $\deg f_0=\deg f_1.$
Let $f:X^k\rightarrow \mathbb{R}^{k+1},$ define the number of wraps
$$
W(f,z):=\deg (u:X^k\rightarrow S^k,x\mapsto \frac{f(x)-z}{|f(x)-z|}),\quad z\notin \operatorname{Im}f.
$$
Proposition 11. Let $x$ be a regular point, $f(x)=z.$ and $B$ be a sufficiently small closed sphere with $x$ as the center. Note $\partial f=f|_{\partial B}:\partial B\rightarrow \mathbb{R}^k,$, then $W(\partial f,z)=1,$ if $df_x$ maintains the orientation. Otherwise, $-1.$
The proof is similar to the Lefschetz fixed point.
Proposition 12. Let $f:B\rightarrow \mathbb{R}^k$ be the smooth mapping on the closed sphere, $z$ be the regular value, $f^{-1}(z)\cap \partial B=\varnothing.$ then according to the orientation count there is $\# f^{-1}(z)=W(\partial f,z).$
The proof can be proved by using the record lemma.
Proposition 13. Let $B$ be a closed sphere in $\mathbb{R}^k$, and $f:\mathbb{R}^k\setminus \operatorname{Int}B\rightarrow Y$ be a smooth map. If $\partial f:\partial B\rightarrow Y$ is homotopic to the constant map, then $f$ can be extended to a smooth map $\mathbb{R}^k\rightarrow Y.$
Prove using Whitney's approximation theorem.
For Hopf's mapping degree theorem, first prove the case where the mapping degree is zero. Prove it inductively on the dimensions.
Theorem 14 (The continuation theorem). Assume $W$ is a $k+1$-dimensional compact connected directed edge manifold, $f:\partial W\rightarrow S^k$ is a smooth map, then $f$ can be extended to $F:W\rightarrow S^k$ if and only if $\deg f=0.$
Proposition 15. Assume $W$ is any compact manifold with edges, $f:\partial W\rightarrow \mathbb{R}^{k+1}$ is any smooth map, then $f$ can be extended to the entire $W$.
Now for any $f_0,f_1,$ with the same mapping degree, define it on $X^k\times I$,
Continuation gives homotopy.
Theorem 16. There is a vector field everywhere non-zero on the compactly connected directed manifold $X$ if and only if $\chi(X)=0.$
The article was last updated on 2023-06-13 12:33:59
