Machine-translated from Chinese.
有向图的连接
增广链复形
对于链复形
$$
0\xleftarrow{\partial}\Lambda_0\xleftarrow{\partial}\Lambda_1\leftarrow\cdots,
$$
我们考虑
$$
0\xleftarrow{\partial} \mathbb{K}\xleftarrow{\partial}\Lambda_0\xleftarrow \Lambda_1\leftarrow\cdots,\quad \Lambda_{-1}:=\mathbb{K},
$$
其中$\partial e_i:=e,$ 为$\mathbb{K}$的单位元.
此时的同调群记为$\widetilde H_p,$ 称为约化同调群.
除了$\widetilde H_0=H_0/\mathbb{K},$ 模掉一个$\mathbb{K}$外,
其它同调群不变, 即$\widetilde H_p=H_p,$ $\,\forall\,p\ge 1,$
$\widetilde H_{-1}=\{0\}.$
记$\widetilde\beta_p=|\widetilde H_p|$为约化Betti数,
那么$\widetilde\beta_0=0$当且仅当$G$是连通的.
道路的连接
若$u=e_{i_0\cdots i_p},v=e_{j_0\cdots j_q}$分别是$X,Y$中的两条道路,
记其连接$u\ast v=e_{i_0\cdots i_p j_0\cdots j_q},$ 为$X\sqcup Y$中的道路.
做线性扩展, 我们有
$$
\ast :\Lambda_p(X)\times \Lambda_q(Y)\rightarrow \Lambda_{p+q+1}(X\sqcup Y).
$$
引理 1. $\partial (u\ast v)=(\partial u)\ast v+(-1)^{p+1}u\ast \partial v,$ $u\in \Lambda_p(X).$
有向图的连接
有向图$Z:=X\ast Y$的点集由$X\sqcup Y$组成, 将所有$X$中的点连接到$Y$中的点,
即$x\rightarrow y,$ $\,\forall\,x\in X,$ $y\in Y.$
这样就保证了$\mathcal{A}_p(X)\ast \mathcal{A}_q(Y)\subset \mathcal{A}_{p+q+1}(X\ast Y).$
引理 2. $\Omega_p(X)\ast \Omega_q(Y)\subset \Omega_{p+q+1}(X\ast Y),$ 对于约化同调群有同样的结论.
定理 3 (Künneth公式). $\,\forall\,p,q,r\ge -1,$ 我们有$\Omega_r(Z)= \bigoplus_{p+q+1=r}\Omega_p(X)\ast \Omega_q(Y).$ 进而$\widetilde H_r(Z)=\bigoplus_{p+q+1=r} \widetilde H_p(X)\ast \widetilde H_q(Y),$ $\widetilde\beta_r(Z)=\sum_{p+q+1=r} \widetilde\beta_p(X)\widetilde\beta_q(Y).$
借助这个公式, 我们可以得到对于有向图$X,$ 锥体$X\ast \{\bullet\}$是零调的,
双锥$X\ast \{\bullet,\bullet\}$的$r$-约化Betti数恰好是$X$的$r-1$-约化Betti数.
由此可以凭借$S^0=\{\bullet,\bullet\},$ 不断做双锥得到任意维球面$S^n,$
满足$\widetilde\beta_n(S^n)=1$是唯一非平凡的约化Betti数.
文章最后更新于 2024-03-04 15:10:35
Joins of Directed Graphs
augmenting chain complex
For chain complex
$$
0\xleftarrow{\partial}\Lambda_0\xleftarrow{\partial}\Lambda_1\leftarrow\cdots,
$$
we consider
$$
0\xleftarrow{\partial} \mathbb{K}\xleftarrow{\partial}\Lambda_0\xleftarrow \Lambda_1\leftarrow\cdots,\quad \Lambda_{-1}:=\mathbb{K},
$$
Where $\partial e_i:=e,$ is the unit element of $\mathbb{K}$.
The homology group at this time is recorded as $\widetilde H_p,$ and is called the reduced homology group.
In addition to $\widetilde H_0=H_0/\mathbb{K},$ molding out one $\mathbb{K}$,
Other homology groups remain unchanged, namely $\widetilde H_p=H_p,$ $\,\forall\,p\ge 1,$
$\widetilde H_{-1}=\{0\}.$
Let $\widetilde\beta_p=|\widetilde H_p|$ be the reduced Betti number,
Then $\widetilde\beta_0=0$ is connected if and only if $G$ is connected.
road connection
If $u=e_{i_0\cdots i_p},v=e_{j_0\cdots j_q}$ are two roads in $X,Y$ respectively,
Let its connection $u\ast v=e_{i_0\cdots i_p j_0\cdots j_q},$ be the road in $X\sqcup Y$.
Doing linear expansion, we have
$$
\ast :\Lambda_p(X)\times \Lambda_q(Y)\rightarrow \Lambda_{p+q+1}(X\sqcup Y).
$$
Lemma 1. $\partial (u\ast v)=(\partial u)\ast v+(-1)^{p+1}u\ast \partial v,$ $u\in \Lambda_p(X).$
Joins of Directed Graphs
The point set of the directed graph $Z:=X\ast Y$ consists of $X\sqcup Y$, connecting all the points in $X$ to the points in $Y$,
That is $x\rightarrow y,$ $\,\forall\,x\in X,$ $y\in Y.$
This ensures $\mathcal{A}_p(X)\ast \mathcal{A}_q(Y)\subset \mathcal{A}_{p+q+1}(X\ast Y).$
Lemma 2. $\Omega_p(X)\ast \Omega_q(Y)\subset \Omega_{p+q+1}(X\ast Y),$ has the same conclusion for the reduced homology group.
Theorem 3 (Künneth formula). $\,\forall\,p,q,r\ge -1,$ we have $\Omega_r(Z)= \bigoplus_{p+q+1=r}\Omega_p(X)\ast \Omega_q(Y).$ and then $\widetilde H_r(Z)=\bigoplus_{p+q+1=r} \widetilde H_p(X)\ast \widetilde H_q(Y),$ $\widetilde\beta_r(Z)=\sum_{p+q+1=r} \widetilde\beta_p(X)\widetilde\beta_q(Y).$
With the help of this formula, we can get that for the directed graph $X,$, the cone $X\ast \{\bullet\}$ is zero-tuned,
The $r$-reduced Betti number of the bipyramid $X\ast \{\bullet,\bullet\}$ is exactly the $r-1$-reduced Betti number of $X$.
From this, we can rely on $S^0=\{\bullet,\bullet\},$ to continuously make bicones to obtain any dimensional sphere $S^n,$
Satisfying $\widetilde\beta_n(S^n)=1$ is the only non-trivial reduced Betti number.
The article was last updated on 2024-03-04 15:10:35
