  In the first part of this paper, the extension of the Lieb-Schultz-Mattis
theorem to dimensions larger than one is discussed. A counter example to the
original formulation of Lieb-Schultz-Mattis and Affleck is exhibited and a more
precise statement is formulated. The degeneracy of the ground-state in symmetry
breaking phases with long-range order is analyzed. The second and third parts
of the paper concern resonating valence-bond (RVB) spin liquids. In these
phases the relationship between various authors approaches: Laughlin-Oshikawa,
Sutherland, Rokhsar and Kivelson, Read and Chakraborty and the
Lieb-Schultz-Mattis-Affleck proposal is studied. The deep physical relation
between the degeneracy property and the absence of stiffness is explained and
illustrated numerically. A new conjecture is formed concerning the absolute
absence of sensitivity of the spin liquid ground-states to any twist of the
boundary conditions (thermodynamic limit). In the third part of the paper the
relations between the quantum numbers of the degenerate multiplets of the spin
liquid phases are obtained exactly. Their relationship with a topological
property of the wave functions of the low lying levels of this spin liquid
phase is emphasized. In spite of the degeneracy of the ground-state, we explain
why these phases cannot exhibit spontaneous symmetry breaking.
