Talk:Elementary invariants of Heegaard diagrams (Ex)

The Heegaard diagram $(\Sigma_g,\alpha,\beta)$$<wikitex>; The Heegaard diagram $(\Sigma_g,\alpha,\beta)$ naturally gives a handle structure on $Y$. The handlebody $H_\alpha$ consists of the <script type="math/tex">(\Sigma_g,\alpha,\beta)$ naturally gives a handle structure on $Y$$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$.

$-handle and $g$ (\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$.

$-handles, whose belt spheres are the $\alpha$ curves. The handlebody $H_\beta$ consists of the $-handle and $g$ $-handles (this is a relative handle decomposition of $H_\beta$). The attaching spheres of the $-handles are the $\beta$ curves. We will give group presentations for $H_1(Y;\mathbb{Z})$ and $\pi_1(Y)$ with generators corresponding to the $\alpha$ curves (which are in correspondence with the (\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$.

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

$-handle. Let $\gamma_{ij}=\gamma([\alpha_i],[\beta_j])$ indicate the algebraic intersection of $\alpha_i$ with $\beta_j$. Then $H_1(Y;\mathbb{Z})\cong\langle x_1,\ldots, x_g\mid [x_i,x_j]\forall(i,j), \sum_{i=1}^g\gamma_{ij}x_i\forall j\rangle$. Now pick a basepoint $b_i$ for each $\beta_i$. Traveling once around $\beta_i$ starting at $b_i$, say that $\beta_i$ intersects $\alpha_{i_1},\alpha_{i_2},\ldots,\alpha_{i_{n_i}}$ in order (with each entry indicating one intersection), with sign $\epsilon_{i_j}\in\{1,-1\}$. Let $w_i=y_{i_1}^{\epsilon_{i_1}}y_{i_2}^{\epsilon_{i_2}}\cdots y_{i_n}^{\epsilon_{i_n}}$. Then $\pi_1(Y)\cong\langle y_1,\ldots, y_g\mid w_1,\ldots, w_g\rangle$. In particular, under these identifications of $H_1(Y;\mathbb{Z})$ and $\pi_1(Y)$ with group presentations, we can write the Hurewicz map $\rho:\pi_1(Y)\to H_1(Y;\mathbb{Z})$ as the map $$\rho:\left\langle x_1,\ldots, x_g\mid [x_i,x_j]\forall(i,j), \sum_{i=1}^g\gamma_{ij}x_i\forall j\right\rangle\to\langle y_1,\ldots, y_g\mid w_1,\ldots, w_g\rangle$$ given by $x_i\mapsto y_i$ For each $i=1,\ldots, g$. (\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$.