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同伦群
$n$维同伦群$\pi_n(X,x_0)$定义为映射$f:(I^n,\partial I^n)\rightarrow (X,x_0)$的同伦类全体.
同伦$f_t$满足$f_t(\partial I^n)\equiv x_0.$
也可视为$f:(S^n,s_0)\rightarrow (X,x_0)$的同伦类全体.
考虑$I^{n-1}\subset I^n,$ $J^{n-1}:=\partial I^n\setminus I^{n-1}.$
那么相对同伦群$\pi_n(X,A,x_0)$定义为映射$(I^n,\partial I^n, J^{n-1})\rightarrow (X,A,x_0)$的同伦类全体.
也可视为$(D^n,S^{n-1},s_0)\rightarrow (X,A,x_0)$的同伦类全体.
相对同伦群有长正合列:
$$
\cdots\rightarrow \pi_n(A,x_0)\xrightarrow{i_\ast }\pi_n(X,x_0)\xrightarrow{j_\ast }\pi_n(X,A,x_0)\xrightarrow{\partial} \pi_{n-1}(A,x_0)\rightarrow\cdots\rightarrow \pi_0(X,x_0)
$$
类似于同调的情况, 我们也能证明更一般的三元组$(X,A,B)$的情况,
当$B=x_0$时即为上面的式子.
$X$中的道路$\gamma$也能诱导不同基点同伦群间的同构$\beta_\gamma,$
从而道路连通的情况可省去基点$x_0.$ 称空间$X$是$n$连通的, 若$\pi_i(X)=0,$
$\,\forall\,i\le n.$
$n$连通当且仅当所有$X$中$S^i$都是可缩的(用映射的观点来看),
$\,\forall\,i\le n.$ 相对同伦的情况类似.
CW复形同伦论
胞腔逼近
定理 1 (Whitehead). 若CW复形间的映射$f:X\rightarrow Y$诱导了同伦群上的同构$f_\ast :\pi_\ast (X)\cong \pi_\ast (Y),$ 则$f$是一个同伦等价. 若$f$还是包含$X\hookrightarrow Y,$ 则结论更强: $X$是$Y$的强形变收缩核.
定理 2 (胞腔逼近). 每个CW复形间的映射$f:X\rightarrow Y$同伦于一个胞腔映射, 即将胞腔映到(比自己维数低的)胞腔.
对给定对$(X,A),$ $A$为非空CW复形.
它的一个$n$连通CW模型指一个$n$连通CW对$(Z,A),$
$f:Z\rightarrow X$在$A$上为恒等映射,
且$f_\ast :\pi_i(Z)\rightarrow \pi_i(X)$在$i>n$时为同构, 对$i=n$为满射.
命题 3. 每个这样的对$(X,A)$都存在一个$n$连通CW模型$f:(Z,A)\rightarrow (X,A),$ $\,\forall\,n\ge 0.$ $Z$由$A$贴$n$维以上胞腔而得.
当$n=0$时, 情况最为特殊. 取$Z$包含$A$的一个连通分支,
即有CW逼近$f:Z\rightarrow X,$ 诱导了所有同伦群的同构,
是弱同伦等价. 事实上,
可以证明$(X,A)$的$n$连通CW模型是在关于$A$相对同伦等价的意义下唯一的.
特别的, 空间的CW逼近是在同伦等价意义下唯一的.
命题 4. 弱同伦等价$f:X\rightarrow Y$诱导了全体同调/上同调群的同构, 关于任意系数群$G.$
挖去定理
关于同伦我们也有挖去定理, 由如下的命题表示:
定理 5. 对于CW复形$X,$ 有子复形$A\cap B=C.$ 若$(A,C)$是$m$连通的, $(B,C)$是$n$连通的, 则由嵌入映射诱导的$\pi_i(A,C)\rightarrow \pi_i(X,B)$为同构, $\,\forall\,i<m+n.$ 对于$i=m+n$为满射.
推论 6 (Freudenthal). $\pi_i (S^n)\rightarrow \pi_{i+1}(S^{n+1})$为同构, $i<2n-1.$ $i=2n-1$为满射. 更一般的, 这对$\pi_i(X)\rightarrow\pi_{i+1}(SX)$也成立, 只要$X$为$n-1$连通CW复形.
命题 7. (Eilenberg-Maclane空间)CW复形$K(G,n)$在同伦意义下唯一, 仅由$G,n$决定.
Hurewicz定理
定理 8. 若空间$X$是$(n-1)$连通的, 则$\widetilde H_i(X)=0,$ $\,\forall\,i<n;$ $\pi_n(X)\cong H_n(X).$ 相对同伦类似.
换句话说, 只要$X$是单连通的, 那么非平凡的同伦群和同调群发生在统一维数,
且是同构的. 结合Whitehead定理我们可以证明如下的结论. (把$Y$换成$M_f$.)
推论 9. 若$f_\ast :H_\ast (X)\cong H_\ast (Y),$ 则单连通CW复形间的映射$f:X\rightarrow Y$为同伦等价.
纤维丛
纤维丛$F\rightarrow E\xrightarrow{p} B$指$\,\forall\,b\in B,$
$p^{-1}(b)\approx F,$ 且有局部平凡化$p^{-1}(U)\approx U\times F.$
纤维化还要求$p$关于$X$满足同伦提升性质:
$\,\forall\,g_t:X\rightarrow B$满足$g_0$有提升$\widetilde g_0,$
则整体存在提升$\widetilde{g}_t.$
定理 10. 若纤维丛$F\rightarrow E\rightarrow B$关于圆盘都是纤维化(称为Serre纤维化), 则有如下长正合列:
$$
\cdots \rightarrow \pi_n(F,x_0)\rightarrow \pi_n(E,x_0)\rightarrow \pi_n(B,b_0)\rightarrow \pi_{n-1}(F,x_0)\rightarrow \cdots\rightarrow \pi_0(E,x_0)\rightarrow 0.
$$
注意到这即是$\pi_n(E,F,x_0)\cong \pi_n(B,b_0).$ 事实上, 我们有如下性质,
说明纤维丛差不多都是纤维化:
命题 11. 纤维丛$p:E\rightarrow B$关于任意CW复形对$(X,A)$都有同伦提升性质.
稳定同伦群
对于$n$连通CW复形$X,$ 我们有$\pi_i(X)\cong \pi_{i+1}(SX),$ $i<2n+1.$
特别的对$i\le n$成立, 因此$SX$是$n+1$连通的. 我们有如下的suspensions序列
$$
\pi_i(X)\rightarrow \pi_{i+1}(SX)\rightarrow \pi_{i+2}(S^2X)\rightarrow \cdots
$$
当$i<2n+1$时, 它们都是同构. 记这些群为$\pi_i^s(X),$ 称为稳定同伦群.
文章最后更新于 2022-10-11 17:51:42
homotopy group
The $n$-dimensional homotopy group $\pi_n(X,x_0)$ is defined as the entire homotopy class of the mapping $f:(I^n,\partial I^n)\rightarrow (X,x_0)$.
Homotopy $f_t$ satisfies $f_t(\partial I^n)\equiv x_0.$
It can also be regarded as the entire homotopy class of $f:(S^n,s_0)\rightarrow (X,x_0)$.
Consider$I^{n-1}\subset I^n,$ $J^{n-1}:=\partial I^n\setminus I^{n-1}.$
Then the relative homotopy group $\pi_n(X,A,x_0)$ is defined as the entire homotopy class of the mapping $(I^n,\partial I^n, J^{n-1})\rightarrow (X,A,x_0)$.
It can also be regarded as the entire homotopy class of $(D^n,S^{n-1},s_0)\rightarrow (X,A,x_0)$.
The relative homotopy group has a long exact sequence:
$$
\cdots\rightarrow \pi_n(A,x_0)\xrightarrow{i_\ast }\pi_n(X,x_0)\xrightarrow{j_\ast }\pi_n(X,A,x_0)\xrightarrow{\partial} \pi_{n-1}(A,x_0)\rightarrow\cdots\rightarrow \pi_0(X,x_0)
$$
Similar to the homology case, we can also prove the more general triplet $(X,A,B)$ case,
When $B=x_0$, it is the above formula.
The road $\gamma$ in $X$ can also induce isomorphism $\beta_\gamma,$ between homotopy groups at different base points.
Therefore, if the roads are connected, the base point $x_0.$ can be omitted, and the space $X$ is said to be $n$ connected, if $\pi_i(X)=0,$
$\,\forall\,i\le n.$
$n$ is connected if and only if $S^i$ in all $X$ are shrinkable (from a mapping point of view),
$\,\forall\,i\le n.$ The situation with relative homotopy is similar.
CW complex homotopy theory
Cellular Approximation
Theorem 1 (Whitehead). If the mapping between CW complexes $f:X\rightarrow Y$ induces the isomorphism $f_\ast :\pi_\ast (X)\cong \pi_\ast (Y),$ on the homotopy group, then $f$ is a homotopy equivalent. If $f$ still contains $X\hookrightarrow Y,$, the conclusion is stronger: $X$ is the strong deformation contraction kernel of $Y$.
Theorem 2 (Cellular approximation). The mapping $f:X\rightarrow Y$ between each CW complex is homotopic to a cell mapping, that is, mapping the cell to a cell (with a lower dimension than its own).
For the given pair $(X,A),$ $A$ is a non-empty CW complex.
One of its $n$ connected CW model Refers to a $n$ connected CW pair $(Z,A),$
$f:Z\rightarrow X$ is an identity mapping on $A$,
And $f_\ast :\pi_i(Z)\rightarrow \pi_i(X)$ is isomorphic to $i>n$ and surjective to $i=n$.
Proposition 3. For each such pair $(X,A)$, there is a $n$ connected CW model $f:(Z,A)\rightarrow (X,A),$ $\,\forall\,n\ge 0.$ $Z$, which is obtained by attaching $A$ to the cell with dimensions above $n$.
When $n=0$, the situation is most special. Take a connected component of $Z$ including $A$,
There is CW approach $f:Z\rightarrow X,$ induces the isomorphism of all homotopy groups,
Yes weak homotopy equivalence.In fact,
It can be proved that the $n$ connected CW model of $(X,A)$ is unique in the sense of relative homotopy equivalence with respect to $A$.
In particular, the CW approximation of space is unique in the sense of homotopy equivalence.
Proposition 4. Weak homotopy equivalence $f:X\rightarrow Y$ induces the isomorphism of all homology/cohomology groups, with respect to any coefficient group $G.$
dig out theorem
We also have a poaching theorem about homotopy, which is expressed by the following proposition:
Theorem 5. For the CW complex $X,$, there is a subcomplex $A\cap B=C.$. If $(A,C)$ is connected to $m$ and $(B,C)$ is connected to $n$, then $\pi_i(A,C)\rightarrow \pi_i(X,B)$ induced by embedding mapping is isomorphism, and $\,\forall\,i<m+n.$ is surjective for $i=m+n$.
Corollary 6 (Freudenthal). $\pi_i (S^n)\rightarrow \pi_{i+1}(S^{n+1})$ is an isomorphism, $i<2n-1.$ $i=2n-1$ is a surjection. More generally, this is also true for $\pi_i(X)\rightarrow\pi_{i+1}(SX)$, as long as $X$ is a connected CW complex of $n-1$.
Proposition 7. (Eilenberg-Maclane space) CW complex $K(G,n)$ is unique in the homotopy sense and is determined only by $G,n$.
Hurewicz theorem
Theorem 8. If the space $X$ is connected to $(n-1)$, then $\widetilde H_i(X)=0,$ $\,\forall\,i<n;$ $\pi_n(X)\cong H_n(X).$ are relatively homotopic and similar.
In other words, as long as $X$ is simply connected, then non-trivial homotopy groups and homology groups occur in the same dimension,
And they are isomorphic. Combined with Whitehead's theorem, we can prove the following conclusion. (Replace $Y$ with $M_f$.)
Corollary 9. If $f_\ast :H_\ast (X)\cong H_\ast (Y),$, then the mapping $f:X\rightarrow Y$ between simply connected CW complexes is homotopy equivalent.
fiber plexus
Fiber plexus $F\rightarrow E\xrightarrow{p} B$ refers to $\,\forall\,b\in B,$
$p^{-1}(b)\approx F,$ and has local trivialization $p^{-1}(U)\approx U\times F.$
Fibrosis also requires that $p$ regarding $X$ be satisfied homotopy boosting properties:
$\,\forall\,g_t:X\rightarrow B$ Satisfy $g_0$ and improve $\widetilde g_0,$
Then the overall existence is improved $\widetilde{g}_t.$
Theorem 10. If the fiber bundle $F\rightarrow E\rightarrow B$ is all fibrotic about the disk (called Serre fibrosis), then there is the following long exact sequence:
$$
\cdots \rightarrow \pi_n(F,x_0)\rightarrow \pi_n(E,x_0)\rightarrow \pi_n(B,b_0)\rightarrow \pi_{n-1}(F,x_0)\rightarrow \cdots\rightarrow \pi_0(E,x_0)\rightarrow 0.
$$
Notice that this is $\pi_n(E,F,x_0)\cong \pi_n(B,b_0).$. In fact, we have the following properties,
It shows that fibrous plexuses are almost all fibrosis:
Proposition 11. The fiber bundle $p:E\rightarrow B$ has the homotopy lifting property for any CW complex pair $(X,A)$.
stable homotopy group
For $n$ connected CW complex $X,$ we have $\pi_i(X)\cong \pi_{i+1}(SX),$ $i<2n+1.$
This holds especially for $i\le n$, so $SX$ is connected to $n+1$. We have the following sequence of suspensions
$$
\pi_i(X)\rightarrow \pi_{i+1}(SX)\rightarrow \pi_{i+2}(S^2X)\rightarrow \cdots
$$
When $i<2n+1$, they are all isomorphisms. Let these groups be $\pi_i^s(X),$ and they are called stable homotopy group.
The article was last updated on 2022-10-11 17:51:42
