Poincaré duality II (Ex)

For a simplicial complex $X$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}X$, define the front $p$$p$-face of an $n$$n$-simplex $\sigma = [v_0\ldots v_n]$$\sigma = [v_0\ldots v_n]$ as $_p\sigma:=[v_0\ldots v_p]$$_p\sigma:=[v_0\ldots v_p]$ and the back $q$$q$-face as $\sigma_q := [v_{n-q}\ldots v_n].$$\sigma_q := [v_{n-q}\ldots v_n].$

The Alexander-Whitney diagonal approximation is given by

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

and the partial evaluation map is defined as

$\displaystyle \begin{aligned} E: C^r(X)\otimes C_p(X)\otimes C_q(X) &\to C_q(X)\\ a\otimes z \otimes w &\mapsto \left\{ \begin{array}{cc} a(w)\otimes z, & r=q \\ 0, & \mathrm{otherwise} \end{array}\right. ~. \end{aligned}$ $$\displaystyle \begin{aligned} E: C^r(X)\otimes C_p(X)\otimes C_q(X) &\to C_q(X)\\ a\otimes z \otimes w &\mapsto \left\{ \begin{array}{cc} a(w)\otimes z, & r=q \\ 0, & \mathrm{otherwise} \end{array}\right. ~. \end{aligned}$$

Recall, we define the cap product on the chain level by

$\displaystyle a\cap z := E(a\otimes \tau(z))$ $$\displaystyle a\cap z := E(a\otimes \tau(z))$$

and this descends to a well defined product on (co)homology. Consider $S^1$$S^1$ as a simplicial complex with three $0$$0$-simplices and three $1$$1$-simplices. Compute explicitly, using the Alexander-Whitney diagonal approximation, the map

$\displaystyle -\cap[S^1]: C^{1-*}(S^1) \to C_*(S^1)$ $$\displaystyle -\cap[S^1]: C^{1-*}(S^1) \to C_*(S^1)$$

thus verifying that $S^1$$S^1$ has Poincaré duality.