The visible matter field exists "now" at the surface of the 4-sphere while [M.sub.T], as defined from [R.sub.U], is causal and occupies the light cone.

P2: The observable matter field (particles) rests at the surface of a 4-sphere.

P3: A mechanism exists inflating the 4-sphere and expanding masses and energy; both effects are simultaneous.

P4: The metric expansion includes inflation of the 4-sphere radius and a reduction of particles wavelengths; energy condenses permanently and progressively.

* [[rho].sub.V] lies at the 4-sphere surface and non-homogeneity creates deviations to the flat metric.

DeMichelis, The fixed point set of a finite group action on a homology 4-sphere, Enseign.

[E]) imply that finite group actions with a single fixed point do not exist on (homology) spheres of dimension 1 or 2, and more recent work shows that smooth action of finite groups with one fixed point cannot exist on simply connected homology spheres of dimension 3 (see [BKS]); the results of [BKS] also show the nonexistence of smooth actions with exactly one fixed point on homology 4- and 5-spheres (in the case of homology 4-spheres see also [DeM] and [Mmto]).

It looks to the observer like an energy contained in a 3-sphere, but it is actually a conic 4-dimensional structure intersecting the present, the surface of the 4-sphere. Then consider the constancy of [LAMBDA]: with respect to the 4-sphere volume, and in order to reduce to its surface, we divide [M.sub.S] by the 4-sphere surface coefficient, namely 2 [[pi].sup.2]; we get:

It looks as though a 4-sphere at the surface of which observable energy lies is either inflated or heated by a constant feed; in other words, it replaces the big bang singularity by a constant power and the correlation is such that we must conjecture the following identification: energy is the expansion; meaning that [M.sub.S] and [M.sub.V] increase linearly in time.

P2: The matter field (all particles) is the surface of a 4-sphere.

Considering P3 and P4 the wavelength of massive particles reduce in time while the 4-sphere expands, the product of this reduction by this expansion gives a linear increase of the universe radius.

* [[OMEGA].sub.V] the matter density is the proportion of their sum at the 4-sphere surface.