  In previous publications, we have started investigating some possible
applications to the retraction theory in gravitational physics and showed that
it can be very useful in providing proofs and explaining the topological bases.
In the current work, we investigate the role of the retraction theory in the
mathematical study of space-time singularities and suggest a topological
restriction on the formation of singularities. Although we present a toy model
using a 5D cosmological metric as an example, the current study applies for any
collapsing system under the influence of gravity such as a massive star or the
whole universe. After showing that the cosmological space-time $M$ can be
retracted into lower dimensional circles $S_i \subset M $, we prove the
existence of a homotopy between this retraction and the identity map which
defines a deformation retract on $M$. Since a circle doesn't deformation
retract onto a point but it does retract to a point, the defined deformation
retract stops any circle $S_i$ from retracting into a point which means no
singularity is formed. The paper underlines the role of algebraic topology in
the mathematical study of space-time singularities and represents a new
application of the retraction theory in mathematical physics.
