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.
Similar content being viewed by others
References
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).
Author information
Authors and Affiliations
Additional information
Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 6, pp. 854–858, June, 2011.
Rights and permissions
About this article
Cite this article
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
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-011-0559-z