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\title The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 2
\endtitle
\author Professor Steven Bradlow \\ Class Notes From Math 433
\endauthor
\affil University of Illinois at Urbana-Champaign
\endaffil
\address 273 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801
\endaddress
\email bradlow\@math.uiuc.edu
\endemail
\date January 23, 1998
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\endtopmatter
\document
In the previous lecture, we gave various bundles without a formal
definition.
\definition{Definition 1} A bundle is a quadruple, $(E,B,F,\pi)$, where
$E,B,F$ are spaces and $\pi:E \to B$ is a continuous map, called the
projection, such that for every $x \in B$ we have that $\pi^{-1}(x) \cong F$
and for every $b \in B$ there is an open neighborhood $U \subseteq B$ of $b$ such that $\pi^{-1}(U) \cong U \times F$
in a fiber preserving way. $E$ is called the total space, $B$ the base
spaces, and $F$ the fiber.
\enddefinition
While the definition of a bundle is a very general one, we will be
applying the definition of a bundle to several specialized categories.
\vskip.25in
\itemitem{(a)}{\bf Smooth}: $E,B,F$ are smooth manifolds, maps are
smooth.
\itemitem{(b)}{\bf TopM}: $E,B,F$ are manifolds, maps are continous
maps.
\itemitem{(c)}{\bf Holomorphic}: $E,B,F$ are smooth complex manifolds,
maps are holomorphic.
\vskip.25in
Suppose that $(E,B,F,\pi)$ is a bundle. We identify two special cases by
placing restrictions on $F$ and $\pi$.
\definition{Definition 2}If $F$ is a linear vector space (eg $\Bbb R^n,\Bbb C^n$) and the identifications $\pi^{-1}(U) \cong U \times F$ are linear
maps, then we call $(E,B,F,\pi)$ a vector bundle.
\enddefinition
The tangent, normal, and tautological bundles are all vector bundles ({\it cf. Lecture 1}).
\definition{Definition 3} If $F$ is a Lie group which has a smooth right action of $E$ such that \itemitem{(a)} The action is free (i.e, $e \cdot g=e$ if and only if $g$ is the identity element) \itemitem{(b)} The action preserves the fibers of $E \to B$.
We then call $(E,B,F,\pi)$ a principal $F-$bundle.
\enddefinition
The Hopf bundle is an example of a principal $S^1$ bundle and the
homogeneous bundle $O(n) \to O(n)/O(n-1)$ is a principal $O(n-1)$ bundle.
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