# Plumbing

## 1 Introduction

Plumbing is a gluing construction which takes as input some disk bundles over manifolds (frequently just spheres) with $n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}n$-dimensional total space and produces another $n$$n$-manifold with boundary. It identifies fibers of one bundle with disks in the base manifold of the other bundle and vice versa.

## 2 Construction

As special case of the following construction goes back at least to [Milnor1959].

Let $i \in \{1, \dots, k\}$$i \in \{1, \dots, k\}$, let $(p_i, q_i)$$(p_i, q_i)$ be pairs of positive integers such that $p_i + q_i = n$$p_i + q_i = n$ and let $\alpha_i$$\alpha_i$ be an oriented $q_i$$q_i$-dimensional vector bundle over the $p_i$$p_i$-dimensional oriented manifold $M_i$$M_i$. We consider the corresponding disk bundles $\displaystyle D^{q_i} \to D(\alpha_i) \to M_i.$

Let $G$$G$ be a graph with vertices $\{v_1, \dots, v_n\}$$\{v_1, \dots, v_n\}$ such that the edge set between $v_i$$v_i$ and $v_j$$v_j$, is non-empty only if $p_i = q_j$$p_i = q_j$ and $i \neq j$$i \neq j$. We choose disjoint disks $D_{ij}$$D_{ij}$ in $M_i$$M_i$ (one for each edge incident to $v_i$$v_i$) and trivializations $D(\alpha_i)|_{D_{ij}}\cong D^{p_i} \times D^{q_i}$$D(\alpha_i)|_{D_{ij}}\cong D^{p_i} \times D^{q_i}$ (preserving orientations). Finally we form the manifold $W = W(G;\{\alpha_i\})$$W = W(G;\{\alpha_i\})$ from the disjoint union of the $D(\alpha_i)$$D(\alpha_i)$ by identifying, for each edge of $G$$G$, the corresponding $D^{p_i} \times D^{q_i} \subseteq D(\alpha_i)$$D^{p_i} \times D^{q_i} \subseteq D(\alpha_i)$ and $D^{p_j} \times D^{q_j}\subseteq D(\alpha_j)$$D^{p_j} \times D^{q_j}\subseteq D(\alpha_j)$ with the standard diffeomorphism $(x,y)\mapsto (y,x)$$(x,y)\mapsto (y,x)$, which interchanges base and fiber of the two bundles.

The manifold $W$$W$ is the result of the plumbing, often one is mainly interested in its boundary $\partial W$$\partial W$.

...

## 4 An important special case

If $G$$G$ is simply connected and all the base manifolds are spheres then $\displaystyle \Sigma(G, \{\alpha_i \}) : = \partial W$

is often a homotopy sphere. We establish some notation for graphs, bundles and define

• let $T$$T$ denote the graph with two vertices and one edge connecting them and define $\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$$\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$,
• let $E_8$$E_8$ denote the $E_8$$E_8$-graph,
• let $\tau_{n} \in \pi_{n-1}(SO(n))$$\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$$n$-sphere,
• let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$$\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$, $k > 4s$$k > 4s$, denote a generator,
• let $\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$$\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$, denote a generator:
• let $S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$$S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$ be the suspension homomorphism,
• $S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$$S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$ for $k = 1, 2$$k = 1, 2$ and $S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$$S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$ for $k > 2$$k > 2$,
• let $\eta_n : S^{n+1} \to S^n$$\eta_n : S^{n+1} \to S^n$ be essential.

The plumbing construction can be used to produce exotic spheres:

• $\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$$\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$, the Milnor sphere, generates $bP_{4k}$$bP_{4k}$, $k>1$$k>1$.
• $\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$$\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$, the Kervaire sphere, generates $bP_{4k+2}$$bP_{4k+2}$.
• $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$$\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$$k = 1, 2$.
• For general $k$$k$, $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$$\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic.
• $\Sigma^8(\gamma_3^5, \eta_3\tau_4)$$\Sigma^8(\gamma_3^5, \eta_3\tau_4)$, generates $\Theta_8 = \Zz_2$$\Theta_8 = \Zz_2$.
• $\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$$\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$, generates $\Theta_{16} = \Zz_2$$\Theta_{16} = \Zz_2$.