 
Summary: Agebraic Geomerty I
Lectures and
Chris Portwood
November 12, 2008
1 Some Motivation and Perspective
Definition 1.1. Let M be a topological nmanifold. If (U, ), (V, ) are
two charts such that U V = the composite map,
1
: (U V ) (U V )
is called a transition map from to . Since this is a composition of
homeomorphisms, it is itself a homeomorphism. Two cahrts, (U, ) and
(V, ), are said to be smoothly compatible if either U V = or the transition
map 1 is a diffeomorphism (bijective smooth map with smooth inverse).
An atlas for M is a collection of charts that cover M. An atlas A is called
a smooth atlas if any two charts of A are smoothly compatible with each
other. A smooth atlas A on M is called maximal if it is not strictly contained
in any other smooth atlas on M (i.e. any chart that is smoothly compatible
with every other chart in A is already in A). A smooth manifold is a pair
(M, A) where M is a topological nmanifold and A is a smooth structure on
M, that is a maximal smooth atlas on M.
