Poincaré duality II (Ex)

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For a simplicial complex X, define the front p-face of an n-simplex \sigma = [v_0\ldots v_n] as _p\sigma:=[v_0\ldots v_p] and the back q-face as \sigma_q := [v_{n-q}\ldots v_n].

The Alexander-Whitney diagonal approximation is given by

\displaystyle \begin{aligned}  \tau: C_n(X) &\to (C(X)\otimes C(X))_n \\   \sigma &\mapsto \sum_{p+q=n}{_p\sigma\otimes\sigma_q} \end{aligned}

and the partial evaluation map is defined as

\displaystyle  \begin{aligned}  E: C^r(X)\otimes C_p(X)\otimes C_q(X) &\to C_q(X)\\  a\otimes z \otimes w &\mapsto \left\{ \begin{array}{cc} a(w)\otimes z, & r=q \\ 0, & \mathrm{otherwise} \end{array}\right. ~. \end{aligned}
Recall, we define the cap product on the chain level by
\displaystyle a\cap z := E(a\otimes \tau(z))
and this descends to a well defined product on (co)homology. Consider S^1 as a simplicial complex with three 0-simplices and three 1-simplices. Compute explicitly, using the Alexander-Whitney diagonal approximation, the map
\displaystyle  -\cap[S^1]: C^{1-*}(S^1) \to C_*(S^1)
thus verifying that S^1 has Poincaré duality.
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