  In a quantum mechanical treatment of gauge theories (including general
relativity), one is led to consider a certain completion, $\agb$, of the space
$\ag$ of gauge equivalent connections. This space serves as the quantum
configuration space, or, as the space of all Euclidean histories over which one
must integrate in the quantum theory. $\agb$ is a very large space and serves
as a ``universal home'' for measures in theories in which the Wilson loop
observables are well-defined. In this paper, $\agb$ is considered as the
projective limit of a projective family of compact Hausdorff manifolds,
labelled by graphs (which can be regarded as ``floating lattices'' in the
physics terminology). Using this characterization, differential geometry is
developed through algebraic methods. In particular, we are able to introduce
the following notions on $\agb$: differential forms, exterior derivatives,
volume forms, vector fields and Lie brackets between them, divergence of a
vector field with respect to a volume form, Laplacians and associated heat
kernels and heat kernel measures. Thus, although $\agb$ is very large, it is
small enough to be mathematically interesting and physically useful. A key
feature of this approach is that it does not require a background metric. The
geometrical framework is therefore well-suited for diffeomorphism invariant
theories such as quantum general relativity.
