1.4. Stratified fibrations [04RQ]

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1.4. Stratified fibrations

Let $V$ and $F$ be smooth manifolds, $\Delta\subset\mathbb{R}^{n+1}$ be a lattice polyhedron of full dimension and $\Pi$ be a maximal dual $\Delta$ -complex.

Definition 7.

A smooth map $\lambda:V\to\bar{\Pi}$ is called a stratified $F$ -fibration if

•

The restriction of $\lambda$ to any open $n$ -cell $e\subset\bar{\Pi}$ is a trivial fibration with the fiber $F$ ;
•

for each integer pair $(l,k)$ , $0\leq k\leq l\leq n$ there exists a smooth “model” map $\lambda_{l,k}:V_{l,k}\to\Pi_{l,k}$ , where $\Pi_{l,k}\approx\mathbb{R}^{k}\times\Sigma^{l-k}_{0}\times[0,+\infty)^{n-l}$ , such that any $(l,k)$ -point of $\bar{\Pi}$ has a neighborhood $U\supset x$ such that

$\lambda|_{U}:\lambda^{-1}(U)\to U$

is diffeomorphic to the model map. The model map depends only on $l$ and $k$ .

The map $\lambda_{l,k}$ is called the $(l,k)$ -fiber degeneration; the fiber $F_{l,k}=\lambda^{-1}_{l,k}(x)$ is called the $(l,k)$ -fiber of $\lambda$ .

The following proposition is a direct corollary of Definition 7.

Proposition 1.16.

Let $\lambda:V\to\bar{\Pi}$ be a stratified fibration over the compactification of a maximal dual $\Delta$ -polygon $\Pi$ . For any open $(l,k)$ -cell $e$ of $\bar{\Pi}$ the restriction of $\lambda$ to $e$ is a trivial fibration over $e$ with the fiber $F_{l,k}$ .

Remark 1.17.

Definition 7 can be generalized in a straightforward way to the case when the base is any space with a prescribed stratification. Here we used the stratification given by Proposition 1.14.