1 Introduction

A surface that appears smooth to the naked eye, looks rough when seen under a microscope. With increasing power of resolution, that seemingly $2$-dimensional surface may reveal its thickness and eventually its atomic ``graininess'', at which point the very definition of dimensionality becomes fuzzy.
It has long been conjectured that the same concept applies to the spacetime manifold: the short distance spacetime geometry and topology are subject to violent quantum fluctuations [1]. By ``short distance'', in this context, one usually means the Planck scale, and, by ``violent'', we mean that the quantum fluctuations in the geometry are of the same order of magnitude as the background metric, so that the smooth spacetime manifold of classical physics breaks down into a quantum foam of virtual gravitational bubbles [2] and other ``extendons'', or $p$-branes [3].
In the above picture, spacetime is not a primitive concept. Rather than being a pre-existing, albeit dynamical, background as in General Relativity, spacetime is an``emergent property'', that is, a low energy, effective construct of the truly fundamental objects it is made of. Let us call ``$p$-cells'', as opposed to $p$-branes, the elementary simplexes that constitute the pre-geometric quanta of spacetime. $p$-cells are model manifolds that form the basis of algebraic topology, and can be defined independently of any spacetime background. Physical spacetime emerges from a mapping, $X$, of $p$-cells into a target manifold. The image of a $p$-cell under the mapping $X$ corresponds to a physical $p$-brane⁴. Thus, a $0$-cell, that is a point with no spatial dimensions, corresponds to a world-line in spacetime; a $1$-cell, or line element, corresponds to a ``spacetime pixel'', or world-sheet element, and so on, all the way up to the maximal rank $(D-1)$-brane enclosing an elementary spacetime $D$-volume. Note that the role of ``time'', in such a dynamical mapping, is played by the areal distance between two extremal configurations of the world history of the $p$-brane [10].
By analogy with a superconducting medium, which consists of a condensate of Cooper pairs, spacetime may be regarded as a sort of string or brane condensate [4], [5]. Near some critical scale, presumably the Planck energy, the spacetime manifold ``evaporates'' into its fundamental constituents, i.e., strings and branes of various dimensionality. In this extreme phase, spacetime consists of a ``gas of branes'', each subject to quantum fluctuations of its shape. The shape quantum mechanics of $p$-branes was discussed in detail in a previous communication.