# Proper maps and also family members of portable facility manifolds

Kodaira specifies an intricate analytic family members of portable facility manifolds as the information $(E,B,\pi)$, where $E$ and also $B$ are intricate manifolds, and also $\pi$ is a surjective holomorphic submersion such that the preimage $\pi^{-1}(x)$ of any kind of factor $x \in B$ is a portable facility submanifold of $E$. (Here intricate manifolds are called for to be attached.)

A key building of this definition is that this defines a differentiable fiber package. This reality can virtually be gotten from the Ehresmann fibration theory, which could be mentioned as adheres to:. " Let $X$, $Y$ be differentiable manifolds, and also allow $f: X \rightarrow Y$ be a correct surjective submersion. After that $f$ is the estimate of a differentiable fiber package."

Specifically, the map $\pi$ specifying a family members does not have the properness required to use Ehresmann is theorem. It is, nonetheless, an effect of the reality that a family members offers a fiber package that $\pi$ is without a doubt correct.

Exists an extra primary means of seeing that $\pi$ must appertain?

I assume the adhering to argument solutions your inquiry. It advises me a great deal of the workouts in the standard geography training course I took throughout my basic. Enjoyable times.

Remember that $\pi$ is a submersion, so for any kind of $x$ in $E$ there exists an area of the kind $U \times V$, such that $\pi$ understands the estimate $U \times V \to V$. Intend we just recognize that $E_t := \pi^{-1}(t)$ is portable for any kind of $t$ in $B$, and also allow is show that $\pi$ appertains.

Allow $K \subset B$ be portable and also set $L = \pi^{-1}(K)$. Allow $(\mathcal U_\alpha)$ be a treatment of $L$ by open parts of $E$. We will certainly show that there exists a limited treatment of $E$ by open parts of $(\mathcal U_\alpha)$, and also hence by collections of the initial treatment.

Take $t \in K$. Find areas $\mathcal U_1, \ldots \mathcal U_{n(t)}$ that cover $E_t$. For any kind of factor $x \in E_t$, find areas $U_{t,x} \times V_{t,x} \subset \mathcal U_{j(x)}$ for some $j(x)$ (the $\mathcal U_{j(x)}$ being just one of the above), such that the map $\pi$ understands the estimate $U_{t,x} \times V_{t,x} \to V_{t,x}$.

The density of $E_t$ offers finitely several such $U_{t,x_\nu} \times V_{t,x_\nu}$ which cover $E_t$ and also such that there is a $V_t \subset B$ had in the photo of any kind of $V_{t,x_\nu}$ by $\pi$ (the junction of the finitely several such $V$ has $t$).

Currently : $K$ is portable, so it is covered by finitely most of those $V_t$. The equivalent limited collection $(U_{t,x_\nu} \times V_{t,x_\nu})$ covers $L$, and also contains parts of components of $(\mathcal U_\alpha)$. Hence finitely several components of $(\mathcal U_\alpha)$ cover $L$.

I can not address your inquiry yet if we take $E$ to be the topologist's sine curve and also its estimate to $B=[0,1]$ we have a topological (i.e. continual) family members of factors (portable subspaces, analogs of the portable facility submanifolds) which nonetheless is not correct as a map of topological rooms, given that $E=\pi^{-1}(B)$ is not portable while $B$ is.

We might get comparable nonproper instances of topological family members with portable fibers changing the fibers or the base.

Below the base is a manifold yet not the complete space/family, and also I do not see quickly such counterexamples with family members of portable topological manifolds.

In fact I do not recognize just how Ehresmann is theory is made use of in your thinkings, specifically whether you get that $\pi$ appertains from it.

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