fibre Fibre 2002-08-07 17:34:56 2005-06-10 14:37:06 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Given a function $f\colon X \longrightarrow Y$, a \emph{fibre} is an inverse image of an element of $Y$. That is given $y \in Y$, $f^{-1}(\{y\}) = \{ x \in X \mid f(x) = y \}$ is a fibre. {\bf Example:} Define $f\colon \mathbb{R}^2 \longrightarrow \mathbb{R}$ by $f(x,y) = x^2 + y^2$. Then the fibres of $f$ consist of concentric circles about the origin, the origin itself, and empty sets depending on whether we look at the inverse image of a positive number, zero, or a negative number respectively. {\bf Example:} Suppose $M$ is a manifold, and $\pi\colon TM\to M$ is the canonical projection from the tangent bundle $TM$ to $M$. Then fibres of $\pi$ are the tangent spaces $T_x(M)$ for $x\in M$.