By global structure one means some general
properties of the underlying spacetime. The research of a global
structure is equivalent to studying space,
time and gravitation in a more general way.
The major ideas of that study are in a few (10) definitions (given
in 
page ), some theorems, that show the
relations between the definitions, and examples, which show what
does not hold between the definitions.
An important thing is the large place given to example and
counter-example compared to the theorem. There is no key
theorem; the proofs are long and require much technical work ! But the example
are here to have an idea, before trying to prove something you
try to find a counter-example. If you cannot find one, then you
would research a proof.
The idea of global structure is to find what is the appropriate
manifold (definition page ) for our universe. The aim was to rule out most of the manifolds
by using some physical argument. It turns out to be not so easy.
What General Relativity requires is true for most of the manifold!
(By manifold we mean a manifold without boundary , Hausdorff ,
connected and paracompact . definitions , , page )
General Relativity require a Lorentz metric (signature n-2). All the paracompact manifold have a such metric. But we don't want a Lorentz metric, we want something flat enough to model our universe. Again, most of the manifolds are still possible.
The aim of studying global structure is to understand and prove the
singularity theorems that we will study
in chapter page .