Talk:Poincaré duality II (Ex)
- Note: There is a sign error in this solution to be fixed!! Give the triangulation with -simplices and -simplices . Let us choose the fundamental class
The diagonal approximation maps to
Capping with the fundamental class sends to
Similarly, sends
Thus we obtain the chain equivalence
Where
Note: this doesn't quite work - this is not a chain map, composing one way gives minus the composition the other way. Fixing the sign error we can take the inverse permutation matrices as the chain inverse.
$-simplices $\{v_0,v_1,v_2\}$ and S^1 the triangulation with -simplices and -simplices . Let us choose the fundamental classThe diagonal approximation maps to
Capping with the fundamental class sends to
Similarly, sends
Thus we obtain the chain equivalence
Where
Note: this doesn't quite work - this is not a chain map, composing one way gives minus the composition the other way. Fixing the sign error we can take the inverse permutation matrices as the chain inverse.
$-simplices $\{[v_0,v_1], [v_1,v_2] , [v_2,v_0] \}$. Let us choose the fundamental class $$ [S^1]=[v_0,v_1]\cup[v_1,v_2]\cup[v_2,v_0].$$ The diagonal approximation $\tau$ maps $[v_0,v_1]$ to $$\begin{aligned} && \sum_{p+q=1} {_p[v_0,v_1]\otimes [v_0,v_1]_q} \&=& {_0[v_0,v_1]}\otimes [v_0,v_1]_1 + {_1[v_0,v_1]}\otimes [v_0,v_1]_0 \&=& \{v_0\}\otimes [v_0,v_1] + [v_0,v_1]\otimes\{v_1\}.\end{aligned}$$ Thus the diagonal approximation of the fundamental class $\tau([S^1])$ is $$ \{v_0\}\otimes [v_0,v_1] + [v_0,v_1]\otimes\{v_1\}+ \{v_1\}\otimes [v_1,v_2] + [v_1,v_2]\otimes\{v_2\} + \{v_2\}\otimes [v_2,v_0] + [v_2,v_0]\otimes\{v_0\}. $$ Capping with the fundamental class sends $v_0^*$ to $$\begin{aligned} && E(v_0^*\otimes\tau([S^1])) \ &=& E(v_0^*\otimes ([v_2,v_0]\otimes\{v_0\}+ \mathrm{other}\;\mathrm{terms})) \ &=& v_0^*(v_0)\otimes[v_2,v_0]\ &=& [v_2,v_0]. \end{aligned}$$ Similarly, $-\cap[S^1]$ sends $$\begin{aligned} v_1^* &\mapsto& [v_0,v_1], \ v_2^* &\mapsto& [v_1,v_2], \ [v_0,v_1]^* &\mapsto& v_0, \ [v_1,v_2]^* &\mapsto& v_1, \ [v_2,v_0]^* &\mapsto& v_2. \end{aligned}$$ Thus we obtain the chain equivalence $$\xymatrix{The diagonal approximation maps to
Capping with the fundamental class sends to
Similarly, sends
Thus we obtain the chain equivalence
Where
Note: this doesn't quite work - this is not a chain map, composing one way gives minus the composition the other way. Fixing the sign error we can take the inverse permutation matrices as the chain inverse.