We show that if K is a closed bounded subset of a Riemannian manifold M of dimension m > 3 and the fractal dimension of K is less than m − 3, then the fundamental groups of M and M − K are isomorphic.
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C. Matheus and K. K. Olivera, “Geometrical versus topological properties of manifolds,” J. Inst. Math. Jusseiu, 4, No. 4, 639–651 (2005).
K. Falconer, Fractal Geometry: Mathematical Foundations, Wiley, New York (1990).
M. P. do Carmo, Riemannian Geometry, Birkhäuser, Boston–Berlin (1992).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 6, pp. 854–858, June, 2011.
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Mirzaie, R. On the fundamental group of Riemannian manifolds with omitted fractal subsets. Ukr Math J 63, 993–998 (2011). https://doi.org/10.1007/s11253-011-0559-z
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DOI: https://doi.org/10.1007/s11253-011-0559-z