all this information together we can construct an infinite chain of conformal diagrams as shown in Fig.2. In the present case the spacelike variable r does not become timelike when we cross the event horizon. As we did in the non extreme case with respect to the thick-line rectangle we can identify the square at the bottom with the square at the top. Proceeding like that we obtain a manifold which is cyclic in the timelike coordinate. 

## VII. CONCLUSIONS 

If we divide the quantum black hole physics into the three categories: (i) problems zeroing around the central singularity (final stage of a black hole) [3, 5], (ii) those zeroing around the horizon (Hawking evaporation) [21] and (iii) those around the question regarding the existence of bound orbits in the outer regions [22], this article addresses the first issue in the framework of noncommutative geometry. We found a maximal singularity-free extension of the noncommutative geometry inspired Schwarzschild metric. The new coordinate chart we derived in the text has the advantage that we can illustrate more clearly the overall topology of the non extreme and extreme noncommutative geometry inspired Schwarzschild manifold. The Penrose diagrams are shown in Figs.1 and 2, respectively. The most striking feature of the manifold is black hole tunnels connecting different universes and/or a cyclic structure of the manifold in the time coordinate after identification of two parts of the Penrose diagram (see Fig. 1) is performed. This together with the absence of a central singularity reveals the main difference as compared to the classical Schwarzschild black hole structure. By some minor modification the method we used can be applied to derive the maximal extension of the noncommutative geometry inspired Reissner-Nordstr¨ om black hole [9]. Finally, the stability issue of the tunnels has been examined in [23], where it has been shown that the Cauchy horizon of the noncommutative geometry inspired Schwarzschild black hole is stable under massless scalar perturbations governed by a wave equation modified accordingly to noncommutative geometry. 

[1] P. Nicolini, “Noncommutative black holes, the final appeal to quantum gravity: A Review”, Int. J. Mod. Phys. A24 : 1229, 2009 

