Plumbing

Latest revision as of 21:04, 23 March 2012

[edit] 1 Introduction

Plumbing is a gluing construction which takes as input some disk bundles over manifolds (frequently just spheres) with ${{Stub}} == Introduction == <wikitex>; Plumbing is a gluing construction which takes as input some disk bundles over manifolds (frequently just spheres) with $n$-dimensional total space and produces another $n$-manifold with boundary. It identifies fibers of one bundle with disks in the base manifold of the other bundle and vice versa. </wikitex> == Construction == <wikitex>; As special case of the following construction goes back at least to {{cite|Milnor1959}}. Let $i \in \{1, \dots, k\}$, let $(p_i, q_i)$ be pairs of positive integers such that $p_i + q_i = n$ and let $\alpha_i$ be an oriented $q_i$-dimensional vector bundle over the $p_i$-dimensional oriented manifold $M_i$. We consider the corresponding disk bundles $$ D^{q_i} \to D(\alpha_i) \to M_i.$$ Let $G$ be a graph with vertices $\{v_1, \dots, v_n\}$ such that the edge set between $v_i$ and $v_j$, is non-empty only if $p_i = q_j$ and $i \neq j$. We choose disjoint disks $D_{ij}$ in $M_i$ (one for each edge incident to $v_i$) and trivializations $D(\alpha_i)|_{D_{ij}}\cong D^{p_i} \times D^{q_i}$ (preserving orientations). Finally we form the manifold $W = W(G;\{\alpha_i\})$ from the disjoint union of the $D(\alpha_i)$ by identifying, for each edge of $G$, the corresponding $D^{p_i} \times D^{q_i} \subseteq D(\alpha_i)$ and $D^{p_j} \times D^{q_j}\subseteq D(\alpha_j)$ with the standard diffeomorphism $(x,y)\mapsto (y,x)$, which interchanges base and fiber of the two bundles. The manifold $W$ is the result of the plumbing, often one is mainly interested in its boundary $\partial W$. </wikitex> == Invariants == <wikitex>; ... </wikitex> == An important special case == <wikitex>; If $G$ is simply connected and all the base manifolds are spheres then $$\Sigma(G, \{\alpha_i \}) : = \partial W$$ is often a homotopy sphere. We establish some notation for graphs, bundles and define * let $T$ denote the graph with two vertices and one edge connecting them and define $\Sigma(\alpha, \beta) : = \Sigma(T; \{\alpha, \beta\})$, * let $E_8$ denote the $E_8$-graph, * let $\tau_{n} \in \pi_{n-1}(SO(n))$ denote the tangent bundle of the $n$-sphere, * let $\gamma_{4s-1}^k \in \pi_{4s-1}(SO(k)) \cong \Zz$, $k > 4s$, denote a generator, * let $\gamma_{4s-1}' \in \pi_{4s-1}(SO(4s-1)) \cong \Zz$, denote a generator: * let $S : \pi_k(SO(j)) \to \pi_k(SO(j+1))$ be the suspension homomorphism, **$S^2(\gamma'_{4k-1}) = \pm 2 \gamma_{4k-1}^{4k+1}$ for $k = 1, 2$ and $S^2 (\gamma'_{4k-1}) = \pm \gamma_{4k-1}^{4k+1}$ for $k > 2$, * let $\eta_n : S^{n+1} \to S^n$ be essential. The plumbing construction can be used to produce exotic spheres: * $\Sigma^{4k-1}(E_8; \{\tau_{2k}, \dots \tau_{2k}\}) =: \Sigma_M$, the Milnor sphere, generates $bP_{4k}$, $k>1$. * $\Sigma^{4k+1}(\tau_{2k+1}, \tau_{2k+1}) =: \Sigma_K$, the Kervaire sphere, generates $bP_{4k+2}$. * $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is the inverse of the Milnor sphere for $k = 1, 2$. **For general $k$, $\Sigma^{4k-1}(S\gamma_{4k-1}', S\gamma_{4k-1}')$ is exotic. * $\Sigma^8(\gamma_3^5, \eta_3\tau_4)$, generates $\Theta_8 = \Zz_2$. * $\Sigma^{16}(\gamma_{7}^9, \eta_7\tau_8)$, generates $\Theta_{16} = \Zz_2$. </wikitex> == References == {{#RefList:}} [[Category:Theory]]n$-dimensional total space and produces another $n$-manifold with boundary. It identifies fibers of one bundle with disks in the base manifold of the other bundle and vice versa.

[edit] 2 Construction

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

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

$$\displaystyle D^{q_i} \to D(\alpha_i) \to M_i.$$

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

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

[edit] 3 Invariants

...

[edit] 4 An important special case

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

The plumbing construction can be used to produce exotic spheres:

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

[edit] 5 References

• [Milnor1959] J. Milnor, Differentiable structures on spheres, Amer. J. Math. 81 (1959), 962–972. MR0110107 (22 #990) Zbl 0111.35501