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乘积空间
对于$X,Y$中的$p,q$道路$e_x,e_y,$ 希望定义$e_x\times e_y\in Z=X\times Y.$
称正则道路$z=z_0\cdots z_r$是阶梯型的, 若$\,\forall\,k=0,\cdots,r-1,$
$a_k=a_{k+1}$或$b_k=b_{k+1}.$
令$x=\{x_i\}$为$z$在$X$上的投影, 缩并重复的点. $y$同理.
记$z_k=(x_i,y_j)\mapsto (i,j)\in\mathbb{Z}^2.$
$z$对应了一条$\mathbb{Z}^2$中的折线$S(z).$
记$L(z)$为$S(z)$下方的方块个数.
记$\prod_{x,y}$为$Z$中所有的阶梯型道路$z.$
定义 1. 给定$X,Y$中的正则道路$x,y,$ 定义$e_x\times e_y:=\sum_{z\in \prod_{x,y} } (-1)^{L(z)}e_z.$ 这样便可以将$\times$延拓到全体$\mathcal{R}_p(X),\mathcal{R}_q(Y)$中, 使得$u\times v\in\mathcal{R}_{p+q}(Z).$
引理 2. 若$u\in\mathcal{R}_p(X),$ $v\in \mathcal{R}_q(Y),$ 那么$\partial(u\times v)=\partial u\times v+(-1)^p u\times \partial v.$
阶梯型道路拐角处取$\partial$会消掉部分项, 只剩水平与垂直道路,
分别构成了等号后的两部分.
回忆对于$u,u'\in \mathcal{R}_\ast (X),$
$\left<{}u,u'\right>=\sum_{x\in R(X)} u^x(u')^x.$
引理 3. 若$u\in\mathcal{R}_p(X),$ $\varphi\in \mathcal{R}_{p'}(X),$ $v\in \mathcal{R}_q(Y),$ $\psi\in \mathcal{R}_{q'}(Y),$ 那么$\left<{}u\times v,\varphi\times \psi\right>=\binom{p+q}{p}\left<{}u,\varphi\right>\left<{}v,\psi\right>.$
令$X,Y$为有向图, 定义$Z=X\square Y$为以$X\times Y$为顶点的(最小)有向图,
使得所有阶梯型道路都是准许的, 若其在$X,Y$中的投影都是准许的.
引理 4. $\Omega_p(X)\times \Omega_q(Y)\subset \Omega_r(Z),$ 且$\times$延拓到同调群上.
引理 5. $\,\forall\,w\in \Omega_\ast (Z),$ 存在唯一表示$w=\sum_{\begin{subarray}{c} x\in A(X)\\ y\in A(Y) \end{subarray} }c^{xy}e_x\times e_y.$
引理 6. $w\in \Omega_\ast (Z)$有表示$w=\sum_{x\in A(X)}e_x\times a^x,$ $a^x\in \Omega_\ast (Y),$ $w\in \sum_{y\in A(Y)}\ell^y\times e_y,$ $\ell^y\in\Omega_\ast (X).$
引理 7. $r=p+q,$ $u\in\Omega_p^\perp(X),$ $y\in \mathcal{A}_q(Y),$ $u\times v\in \Omega_r^\perp (Z).$
定理 8. $\,\forall\,w\in\Omega_r(Z),$ 有表示$w=\sum u_i\times v_i,$ $u_i\in \Omega_{p_i}(X),$ $v_i\in \Omega_{q_i}(Y),$ $p_i+q_i=r.$
定理 9 (Künneth formula). $\Omega_\ast (X\square Y)\cong \Omega_\ast (X)\otimes \Omega_\ast (Y),$ $\otimes\mapsto \times.$ 进而同调群也有同样的表示.
文章最后更新于 2024-11-04 17:38:36
product space
For $p,q$ road $e_x,e_y,$ in $X,Y$ wish to define $e_x\times e_y\in Z=X\times Y.$
The regular road $z=z_0\cdots z_r$ is said to be stepped, if $\,\forall\,k=0,\cdots,r-1,$
$a_k=a_{k+1}$ or $b_k=b_{k+1}.$
Let $x=\{x_i\}$ be the projection of $z$ on $X$, shrink and merge repeated points. The same goes for $y$.
Note$z_k=(x_i,y_j)\mapsto (i,j)\in\mathbb{Z}^2.$
$z$ corresponds to a polyline $S(z).$ in $\mathbb{Z}^2$
Record $L(z)$ as the number of blocks below $S(z)$.
Denote $\prod_{x,y}$ as all the stepped roads $z.$ in $Z$
Definition 1. Given the regular road $x,y,$ in $X,Y$, define $e_x\times e_y:=\sum_{z\in \prod_{x,y} } (-1)^{L(z)}e_z.$ so that $\times$ can be extended to the whole $\mathcal{R}_p(X),\mathcal{R}_q(Y)$, so that $u\times v\in\mathcal{R}_{p+q}(Z).$
Lemma 2. If $u\in\mathcal{R}_p(X),$ $v\in \mathcal{R}_q(Y),$ then $\partial(u\times v)=\partial u\times v+(-1)^p u\times \partial v.$
Selecting $\partial$ at the corner of a stepped road will eliminate some items, leaving only horizontal and vertical roads.
They respectively constitute the two parts after the equal sign.
Memories for $u,u'\in \mathcal{R}_\ast (X),$
$\left<{}u,u'\right>=\sum_{x\in R(X)} u^x(u')^x.$
Lemma 3. If $u\in\mathcal{R}_p(X),$ $\varphi\in \mathcal{R}_{p'}(X),$ $v\in \mathcal{R}_q(Y),$ $\psi\in \mathcal{R}_{q'}(Y),$ then $\left<{}u\times v,\varphi\times \psi\right>=\binom{p+q}{p}\left<{}u,\varphi\right>\left<{}v,\psi\right>.$
Let $X,Y$ be a directed graph, define $Z=X\square Y$ as the (minimum) directed graph with $X\times Y$ as the vertex,
Makes all stepped roads allowed if their projections in $X,Y$ are allowed.
Lemma 4. $\Omega_p(X)\times \Omega_q(Y)\subset \Omega_r(Z),$ and $\times$ are extended to the homology group.
Lemma 5. $\,\forall\,w\in \Omega_\ast (Z),$ There is a unique representation $w=\sum_{\begin{subarray}{c} x\in A(X)\\ y\in A(Y) \end{subarray} }c^{xy}e_x\times e_y.$
Lemma 6. $w\in \Omega_\ast (Z)$ indicates $w=\sum_{x\in A(X)}e_x\times a^x,$ $a^x\in \Omega_\ast (Y),$ $w\in \sum_{y\in A(Y)}\ell^y\times e_y,$ $\ell^y\in\Omega_\ast (X).$
Lemma 7. $r=p+q,$ $u\in\Omega_p^\perp(X),$ $y\in \mathcal{A}_q(Y),$ $u\times v\in \Omega_r^\perp (Z).$
Theorem 8. $\,\forall\,w\in\Omega_r(Z),$ Indicated $w=\sum u_i\times v_i,$ $u_i\in \Omega_{p_i}(X),$ $v_i\in \Omega_{q_i}(Y),$ $p_i+q_i=r.$
Theorem 9 (Künneth formula). $\Omega_\ast (X\square Y)\cong \Omega_\ast (X)\otimes \Omega_\ast (Y),$ $\otimes\mapsto \times.$ Then the homology group also has the same expression.
The article was last updated on 2024-11-04 17:38:36
