The inverse limit of the fundamental groups of branched cyclic coverings

Author:
Michael Dellomo

Journal:
Proc. Amer. Math. Soc. **104** (1988), 321-326

MSC:
Primary 57M25; Secondary 55P25

DOI:
https://doi.org/10.1090/S0002-9939-1988-0958092-1

MathSciNet review:
958092

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Abstract: Cowsik and Swarup [**CS**] have shown that the homology groups of the infinite cyclic cover of a knot inject into the inverse limit of the homology groups of the branched cyclic covers. They also give conditions under which the injection is an isomorphism. We prove an analogous result for the fundamental group and generalize it to the case of links.

**[BK]**A. K. Bousfield and D. M. Kan,*Homotopy limits, completions and localizations*, Lecture Notes in Math., vol. 304, Springer-Verlag, Berlin and New York, 1972.**[BZ]**G. Burde and H. Zieschang,*Eine Kennzeichnung der Torus Knoten*, Math. Ann.**167**(1966), 169-176.**[CS]**R. C. Cowsik and G. A. Swarup,*A remark on infinite cyclic covers*, J. Pure Appl. Algebra**11**(1977), 131-138.**[Dl]**M. Dellomo,*On the inverse limit of the finite branched cyclic covers of a knot*, J. Pure Appl. Algebra**40**(1986), 15-26.**[D2]**-,*Through the non-simply connected looking glass, or the inverse limit of finite branched covers*, Ph.D. Thesis, Johns Hopkins Univ., 1984.**[DS]**J. Dydak and J. Segal,*Shape theory: An introduction*, Lecture Notes in Math., vol. 688, Springer-Verlag, Berlin, 1978.**[H]**J. Hempel,*Residual finiteness for**-manifolds*, Ann. of Math. Studies (to appear).**[MKS]**W. Magnus, A. Karras and D. Solitar,*Combinatorial group theory*, Dover, New York, 1976.**[MS1]**Sibe Mardešić and Jack Segal,*Shape theory*, North-Holland Mathematical Library, vol. 26, North-Holland Publishing Co., Amsterdam-New York, 1982. The inverse system approach. MR**676973****[R]**Dale Rolfsen,*Knots and links*, Publish or Perish, Inc., Berkeley, Calif., 1976. Mathematics Lecture Series, No. 7. MR**0515288**

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0958092-1

Article copyright:
© Copyright 1988
American Mathematical Society