Plumbing (Ex)

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$$\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}{^}i = 1, \dots, n$ let

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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