1.6. Pairs-of-pants in higher dimensions [04S3]

Original official author HTML, exact retained edition. Historical TeX conversion verdicts remain unchanged. Cited-edition alignment and mathematical self-containment are not assessed.

Complete original source context · Original author HTML

1.6. Pairs-of-pants in higher dimensions

Definition 8.

Let $\mathcal{H}\subset{\mathbb{C}}{\mathbb{P}}^{n}$ be the union of the $n+2$ generic hyperplanes in ${\mathbb{C}}{\mathbb{P}}^{n}$ . Let $\mathcal{U}\subset{\mathbb{C}}{\mathbb{P}}^{n}$ be the union of their $\epsilon$ -neighborhoods for a very small $\epsilon>0$ .

The complement $\bar{\mathcal{P}}_{n}={\mathbb{C}}{\mathbb{P}}^{n}\smallsetminus\mathcal{U}$ is a manifold with boundary and corners. We call $\bar{\mathcal{P}}_{n}$ the $n$ -dimensional pair-of-pants. We call $\mathcal{P}_{n}={\mathbb{C}}{\mathbb{P}}^{n}\smallsetminus\mathcal{H}$ the $n$ -dimensional open pair-of-pants

Immediately we have the following proposition.

Proposition 1.22.

A pair-of-pants is a compact manifold with boundary. An open pair-of-pants is diffeomorphic to the pair-of-pants minus its boundary.

Remark 1.23.

Note that the choice of $n+2$ generic hyperplane in ${\mathbb{C}}{\mathbb{P}}^{n}$ is unique up to the action of $PSL_{n+1}(\mathbb{C})$ . Thus $\mathcal{P}_{n}$ can be given a canonical complex structure.

Note that $\mathcal{P}_{1}$ is diffeomorphic to the Riemann sphere punctured 3 times, while $\bar{\mathcal{P}}_{1}$ is diffeomorphic to a closed disk with 2 holes. Thus Definition 8 agrees with the classical, one-dimensional, pair-of-pants definition.

The following proposition describes a natural stratification of the boundary $\partial\bar{\mathcal{P}}$ .

Proposition 1.24.

We have the following canonical decomposition of the boundary $\partial\bar{\mathcal{P}_{n}}=\bigcup\limits_{j=0}^{n-1}\partial_{j}\bar{\mathcal{P}_{n}}$ , where $\partial_{j}\bar{\mathcal{P}_{n}}$ is a $(2n-j)$ -dimensional smooth manifold such that each its connected component is a trivial $T^{j}$ -fibration over $\mathcal{P}_{n-j}$ . Different parts do not intersect: $\partial_{j}\bar{\mathcal{P}_{n}}\cap\partial_{k}\bar{\mathcal{P}_{n}}=\emptyset$ , if $j\neq k$ , but the closure of $\partial_{j}\bar{\mathcal{P}_{n}}$ contains $\partial_{k}\bar{\mathcal{P}_{n}}$ for all $k\leq j$ . The number of connected components is $\begin{pmatrix}n+2\\ j+2\end{pmatrix}$ .

Proof.

Connected components of the manifold $\partial_{j}\bar{\mathcal{P}_{n}}$ can be obtained as the intersections of the boundaries of the $\epsilon$ -neighborhoods of $j$ different hyperplanes from $\mathcal{H}$ . ∎