Plumbing (Ex)

Latest revision as of 14:55, 1 April 2012

The exercises below are about plumbing manifolds. For the details of the construction, see the page Plumbing.

In this page we use slightly different notation. For $<wikitex>; The exercises below are about plumbing manifolds. For the details of the construction, see the page [[Plumbing]]. In this page we use slightly different notation. For $i = 1, \dots, n$ let $N_i$ be a closed connected oriented manifold of dimension $p_i$ and let $$ \zeta_i \colon E_i \to N_i$$ be an oriented $D^{q_i}$-bundle over $N_i$ where $p_i + q_i = n$ is fixed and each $q_i \geq 2$. Let $G$ be a graph with verticies $\{v_1, \dots, v_k \}$ where the edge set between $v_i$ and $v_j$ is non-empty only if $p_i = q_j$ and $i \neq j$. Starting from the dijoint union of the total spaces $E_i$ we form a manifolds as follows: given an edge in $G$ connecting $v_i$ and $v_j$, let $x_i\in N_i$ and let $D^q_i \times D^p_i\subseteq E_i$ be a neighbourhood of $x_i$, such that $D^q_i \times \{0\}\subseteq N_i$ and $y\times D^p_i$ are the fibers of $E_i$. Let $h_{\pm}: D_i^{q_i} \rightarrow D_j^{p_j}$ and $k_{\pm}: D_j^{p_j} \rightarrow D_i^{q_i}$ be orientation preserving (resp. reversing) diffeomorphisms. We define the plumbing $E_i \diamond E_j$ of $E_i$ and $E_j$ at $x_i$ and $x_j$ by taking $E_i \sqcup E_j$ and identifying $D_i^{q_i} \times D_i^{p_i}$ and $D_j^{q_j} \times D_j^{q_j}$ via $$I_\pm(x,y)=(k_{\pm}(y), h_{\pm}(x)).$$ Proceeding in this way for each edge of $G$ we obtain the plumbing manifold $$ X := X(G; \{\zeta_i\}).$$ {{beginthm|Exercise}} Let $X = X(G; \{\zeta_k \}$ be connected. Show the following: #$\pi_1(\partial X)\cong\pi_1(X)$ is free. #$H_i(\partial X)=H_i(X)=0$ for i = 1, \dots, n$ let

$$\displaystyle \zeta_i \colon E_i \to S^q_i$$

be an oriented $D^{q}$-bundle over $S^q$. Let $G$ be a graph with verticies $\{v_1, \dots, v_k \}$.

Starting from the dijoint union of the total spaces $E_i$ we form the plumbing manifold $X(G; \{\zeta_i\}$ as follows: given an edge in $G$ connecting $v_i$ and $v_j$, for $k = i$ and $j$, let $x_k \in E_k$ and let $D^{q} \times D^{q} \subseteq E_k$ be a neighbourhood of $x_k$, such that $D^{q} \times \{0\}\subseteq S^q_k$ and such that $y \times D^{q}$ if the fiber of $E_k \to S^q_k$. Let $h_{\pm}: D^{q} \rightarrow D^{q}$ and $k_{\pm}: D^{q} \rightarrow D^{q}$ be orientation preserving (resp. reversing) diffeomorphisms. We define the plumbing $E_i \diamond E_j$ of $E_i$ and $E_j$ at $x_i$ and $x_j$ by taking $E_i \sqcup E_j$ and identifying $D^q \times D^q \subset E_i$ and $D^q \times D^q \subset E_j$ via

$$\displaystyle I_\pm(x,y)=(k_{\pm}(y), h_{\pm}(x)).$$

Proceeding in this way for each edge of $G$ we obtain the plumbing manifold

$$\displaystyle X := X(G; \{\zeta_i\}).$$

let

$$\displaystyle \zeta_i \colon E_i \to S^q_i$$

be an oriented $D^{q}$-bundle over $S^q$. Let $G$ be a graph with verticies $\{v_1, \dots, v_k \}$.

Starting from the dijoint union of the total spaces $E_i$ we form the plumbing manifold $X(G; \{\zeta_i\}$ as follows: given an edge in $G$ connecting $v_i$ and $v_j$, for $k = i$ and $j$, let $x_k \in E_k$ and let $D^{q} \times D^{q} \subseteq E_k$ be a neighbourhood of $x_k$, such that $D^{q} \times \{0\}\subseteq S^q_k$ and such that $y \times D^{q}$ if the fiber of $E_k \to S^q_k$. Let $h_{\pm}: D^{q} \rightarrow D^{q}$ and $k_{\pm}: D^{q} \rightarrow D^{q}$ be orientation preserving (resp. reversing) diffeomorphisms. We define the plumbing $E_i \diamond E_j$ of $E_i$ and $E_j$ at $x_i$ and $x_j$ by taking $E_i \sqcup E_j$ and identifying $D^q \times D^q \subset E_i$ and $D^q \times D^q \subset E_j$ via

$$\displaystyle I_\pm(x,y)=(k_{\pm}(y), h_{\pm}(x)).$$

Proceeding in this way for each edge of $G$ we obtain the plumbing manifold

$$\displaystyle X := X(G; \{\zeta_i\}).$$

Exercise 0.1. Let $X = X(G; \{\zeta_i \})$ be connected. Show the following:

1. $\pi_1(\partial X)\cong\pi_1(X)$ is free.
2. $H_i(\partial X)=H_i(X)=0$ for $1<i<q-1$

Hint 0.2. The statement is trivial for $X$, since $X$ is homotopy equivalent to a wedge of 1-spheres and $q$-spheres. Now use van Kampen's theorem for $\pi_1(\partial X)$ and for $H_i(\partial X)$ use the Mayer-Vietories Sequence with $\partial E_i\backslash (D_i^q\times S^{q-1})$ and show that all components involved are $(q-2)$ connected.

Exercise 0.3. Choose $S^1\subseteq \partial X$ representing a generator of $\pi_1(\partial X)$ and let $V$ be the trace of a surgery on this $S^1$. Define $X':=X\cup_{\partial X}V$. Show the following:

1. $H_i(X')\cong H_i(X)$ for $i\neq 1$.
2. $H_i(\partial X')\cong H_i(\partial X)$ for $1<i<2q-2$.

Hint 0.4. For (1) use the long exact sequence of the pair $(X',X)$ and $X'\simeq X\cup D^2$. For (2) use the long exact sequence of the pair $(V,\partial X')$ as well as Poincaré Duality and the Universal Coefficient Theorem.

Exercise 0.5. Assume now that each bundle $\zeta_i$ is some multiple of $\tau_{S^q}$, the unit disc bundle of the tangent bundle of the $q$-sphere. Show that there is a degree 1 normal map $(f,b),f:(X',\partial X')\rightarrow (D^{2q},S^{2q-1})$.

Hint 0.6. Use that $X$ is homotopy equivalent to a wedge of 1-spheres and $q$-spheres and that the tangent bundle of $X$ is stabily trivial (since it is so on every component of the wedge), so that the normal bundle of $(X,\partial X)\subseteq (D^{2q+k},S^{2q+k-1})$ is trivial for $k$ large. Then show that the map can be extend over $V$.

The exercises and hints on this page were sent by Fabian Hebestreit, Daniel Kasprowski and Christoph Winges.

== References == {{#RefList:}} [[Category:Exercises]]i = 1, \dots, n let

$$\displaystyle \zeta_i \colon E_i \to S^q_i$$

be an oriented $D^{q}$-bundle over $S^q$. Let $G$ be a graph with verticies $\{v_1, \dots, v_k \}$.

Starting from the dijoint union of the total spaces $E_i$ we form the plumbing manifold $X(G; \{\zeta_i\}$ as follows: given an edge in $G$ connecting $v_i$ and $v_j$, for $k = i$ and $j$, let $x_k \in E_k$ and let $D^{q} \times D^{q} \subseteq E_k$ be a neighbourhood of $x_k$, such that $D^{q} \times \{0\}\subseteq S^q_k$ and such that $y \times D^{q}$ if the fiber of $E_k \to S^q_k$. Let $h_{\pm}: D^{q} \rightarrow D^{q}$ and $k_{\pm}: D^{q} \rightarrow D^{q}$ be orientation preserving (resp. reversing) diffeomorphisms. We define the plumbing $E_i \diamond E_j$ of $E_i$ and $E_j$ at $x_i$ and $x_j$ by taking $E_i \sqcup E_j$ and identifying $D^q \times D^q \subset E_i$ and $D^q \times D^q \subset E_j$ via

$$\displaystyle I_\pm(x,y)=(k_{\pm}(y), h_{\pm}(x)).$$

Proceeding in this way for each edge of $G$ we obtain the plumbing manifold

$$\displaystyle X := X(G; \{\zeta_i\}).$$

Exercise 0.1. Let $X = X(G; \{\zeta_i \})$ be connected. Show the following:

1. $\pi_1(\partial X)\cong\pi_1(X)$ is free.
2. $H_i(\partial X)=H_i(X)=0$ for $1<i<q-1$

Hint 0.2. The statement is trivial for $X$, since $X$ is homotopy equivalent to a wedge of 1-spheres and $q$-spheres. Now use van Kampen's theorem for $\pi_1(\partial X)$ and for $H_i(\partial X)$ use the Mayer-Vietories Sequence with $\partial E_i\backslash (D_i^q\times S^{q-1})$ and show that all components involved are $(q-2)$ connected.

Exercise 0.3. Choose $S^1\subseteq \partial X$ representing a generator of $\pi_1(\partial X)$ and let $V$ be the trace of a surgery on this $S^1$. Define $X':=X\cup_{\partial X}V$. Show the following:

1. $H_i(X')\cong H_i(X)$ for $i\neq 1$.
2. $H_i(\partial X')\cong H_i(\partial X)$ for $1<i<2q-2$.

Hint 0.4. For (1) use the long exact sequence of the pair $(X',X)$ and $X'\simeq X\cup D^2$. For (2) use the long exact sequence of the pair $(V,\partial X')$ as well as Poincaré Duality and the Universal Coefficient Theorem.

Exercise 0.5. Assume now that each bundle $\zeta_i$ is some multiple of $\tau_{S^q}$, the unit disc bundle of the tangent bundle of the $q$-sphere. Show that there is a degree 1 normal map $(f,b),f:(X',\partial X')\rightarrow (D^{2q},S^{2q-1})$.

Hint 0.6. Use that $X$ is homotopy equivalent to a wedge of 1-spheres and $q$-spheres and that the tangent bundle of $X$ is stabily trivial (since it is so on every component of the wedge), so that the normal bundle of $(X,\partial X)\subseteq (D^{2q+k},S^{2q+k-1})$ is trivial for $k$ large. Then show that the map can be extend over $V$.

The exercises and hints on this page were sent by Fabian Hebestreit, Daniel Kasprowski and Christoph Winges.