In this paper, we suggest a mathematical representation to the holographic principle through the theory topological retracts. We found that the topological retraction is the mathematical analogs of the hologram idea in modern quantum gravity and it can be used to explore the geometry of the hologram boundary. An example has been given on the five dimensional (5D) wormhole space-time W which we found it can retract to lower dimensional circles S i ⊂ W . In terms of the holographic principle, the description of this volume of space-time W is encoded on the lower-dimensional circle which is the region boundary.

The deformation retract is, by definition, a homotopy between a retraction and the identity map. We show that applying this topological concept to Ricci-flat wormholes/black holes implies that such objects can get deformed and reduced to lower dimensions. The homotopy theory can provide a rigorous proof to the existence of black holes/wormholes deformations and explain the topological origin. The current work discusses such possible deformations and dimensional reductions from a global topological point of view, it also represents a new application of the homotopy theory and deformation retract in astrophysics and quantum gravity.

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