Talk:Elementary invariants of Heegaard diagrams (Ex)

The Heegaard diagram $(\Sigma_g,\alpha,\beta)$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}(\Sigma_g,\alpha,\beta)$ naturally gives a handle structure on $Y$$Y$. The handlebody $H_\alpha$$H_\alpha$ consists of the $0$$0$-handle and $g$$g$ $1$$1$-handles, whose belt spheres are the $\alpha$$\alpha$ curves. The handlebody $H_\beta$$H_\beta$ consists of the $3$$3$-handle and $g$$g$ $2$$2$-handles (this is a relative handle decomposition of $H_\beta$$H_\beta$). The attaching spheres of the $2$$2$-handles are the $\beta$$\beta$ curves.

We will give group presentations for $H_1(Y;\mathbb{Z})$$H_1(Y;\mathbb{Z})$ and $\pi_1(Y)$$\pi_1(Y)$ with generators corresponding to the $\alpha$$\alpha$ curves (which are in correspondence with the $1$$1$-handles). We obtain a relation for each $2$$2$-handle (which are in correspondence with the $\beta$$\beta$ curves). The $\beta$$\beta$ curves describe the attaching maps of the $2$$2$-handles; thus we can compute $H_1(Y;\mathbb{Z})$$H_1(Y;\mathbb{Z})$ and $\pi_1(Y)$$\pi_1(Y)$ by investigating the intersections of the $\alpha$$\alpha$ and $\beta$$\beta$ curves, as each intersection marks a $2$$2$-handle attaching circle traveling geometrically once over a $1$$1$-handle.

Let $\gamma_{ij}=\gamma([\alpha_i],[\beta_j])$$\gamma_{ij}=\gamma([\alpha_i],[\beta_j])$ indicate the algebraic intersection of $\alpha_i$$\alpha_i$ with $\beta_j$$\beta_j$. Then $H_1(Y;\mathbb{Z})\cong\langle x_1,\ldots, x_g\mid [x_i,x_j]\forall(i,j), \sum_{i=1}^g\gamma_{ij}x_i\forall j\rangle$$H_1(Y;\mathbb{Z})\cong\langle x_1,\ldots, x_g\mid [x_i,x_j]\forall(i,j), \sum_{i=1}^g\gamma_{ij}x_i\forall j\rangle$.

Now pick a basepoint $b_i$$b_i$ for each $\beta_i$$\beta_i$. Traveling once around $\beta_i$$\beta_i$ starting at $b_i$$b_i$, say that $\beta_i$$\beta_i$ intersects $\alpha_{i_1},\alpha_{i_2},\ldots,\alpha_{i_{n_i}}$$\alpha_{i_1},\alpha_{i_2},\ldots,\alpha_{i_{n_i}}$ in order (with each entry indicating one intersection), with sign $\epsilon_{i_j}\in\{1,-1\}$$\epsilon_{i_j}\in\{1,-1\}$. Let $w_i=y_{i_1}^{\epsilon_{i_1}}y_{i_2}^{\epsilon_{i_2}}\cdots y_{i_n}^{\epsilon_{i_n}}$$w_i=y_{i_1}^{\epsilon_{i_1}}y_{i_2}^{\epsilon_{i_2}}\cdots y_{i_n}^{\epsilon_{i_n}}$. Then $\pi_1(Y)\cong\langle y_1,\ldots, y_g\mid w_1,\ldots, w_g\rangle$$\pi_1(Y)\cong\langle y_1,\ldots, y_g\mid w_1,\ldots, w_g\rangle$.

In particular, under these identifications of $H_1(Y;\mathbb{Z})$$H_1(Y;\mathbb{Z})$ and $\pi_1(Y)$$\pi_1(Y)$ with group presentations, we can write the Hurewicz map $\rho:\pi_1(Y)\to H_1(Y;\mathbb{Z})$$\rho:\pi_1(Y)\to H_1(Y;\mathbb{Z})$ as the map

$\displaystyle \rho:\left\langle x_1,\ldots, x_g\mid [x_i,x_j]\forall(i,j), \sum_{i=1}^g\gamma_{ij}x_i\forall j\right\rangle\to\langle y_1,\ldots, y_g\mid w_1,\ldots, w_g\rangle$ $$\displaystyle \rho:\left\langle x_1,\ldots, x_g\mid [x_i,x_j]\forall(i,j), \sum_{i=1}^g\gamma_{ij}x_i\forall j\right\rangle\to\langle y_1,\ldots, y_g\mid w_1,\ldots, w_g\rangle$$

given by $x_i\mapsto y_i$$x_i\mapsto y_i$ For each $i=1,\ldots, g$$i=1,\ldots, g$.