Space-time singularities and the theory of retracts
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.