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There exists a polyhedron with infinitely many left neighbors


Author: Danuta Kolodziejczyk
Journal: Proc. Amer. Math. Soc. 129 (2001), 303-309
MSC (2000): Primary 55P55, 55P15
DOI: https://doi.org/10.1090/S0002-9939-00-05812-3
Published electronically: August 30, 2000
MathSciNet review: 1784026
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Abstract:

We show that there exists a finite polyhedron $P$ homotopy dominating infinitely many finite polyhedra $K_i$ of different homotopy types such that there isn't any homotopy type between $P$ and $K_i$. This answers negatively the question raised by K. Borsuk in 1975: Does every FANR have only finitely many left neighbors?


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  • [BeDu] P. H. Berridge, M. J. Dunwoody, Non free projective modules for torsion-free groups, J. London Math. Soc. (2) 19 (1979), 433-436. MR 80k:20041
  • [B1] K. Borsuk, Theory of Retracts, Polish Scientific Publishers 44, Warsaw, 1967. MR 35:7306
  • [B2] K. Borsuk, Theory of Shape, Polish Scientific Publishers 59, Warsaw, 1975. MR 54:6132
  • [Br] K. Brown, Cohomology of Groups, Springer, Berlin, 1982. MR 83k:20002
  • [DS] J. Dydak, J. Segal, Shape Theory: An Introduction, Lecture Notes in Math. 688, Springer, Berlin, 1978. MR 80h:54020
  • [Dy1] M. N. Dyer, Homotopy classification of $(\pi,m)$-complexes, J. Pure Appl. Alg. 7 (1976), 249-282. MR 58:2798
  • [Dy2] M. N. Dyer, Trees of homotopy types of $(\pi,m)$-complexes, Homological Group Theory, London Math. Soc. Lecture Notes 36, 1979, 251-254. MR 81f:55003
  • [HaHe1] H. M. Hastings, A. Heller Homotopy idempotents on finite-dimensional complexes split, Proc. Amer. Math. Soc. 85 (1982), 4, 619-622. MR 83j:55010
  • [HaHe2] H. M. Hastings, A. Heller, Splitting homotopy idempotents, Shape Theory and Geom. Top. Proc. (Dubrovnik, 1981), Lecture Notes in Math. 870, Springer, Berlin, 1981, 25-36. MR 83a:55017
  • [Ka] I. Kaplansky, Fields and Rings, University of Chicago Press, 1972. MR 50:2139
  • [KS] A. Karras, D. Solitar, On free products, Proc. Amer. Math. Soc. 9, (1958), 217-221. MR 20:2373
  • [MWh] S. Mac Lane, J. H. C. Whitehead, On the 3-type of a complex, Proc. Nat. Acad. Sci. U.S.A., 36 (1950), 41-48. MR 11:450h
  • [Ma] A. I. Malcev, On isomorphic representations of infinite groups by matrices, Mat. Sb. 8 (1940), 405-422.
  • [MS] S. Mardesic, J. Segal, Shape Theory. The Inverse System Approach, North-Holland Math. Library vol. 26, Amsterdam, 1982. MR 84b:55020
  • [Pa] D. S. Passman, Idempotents in group rings, Proc. Amer. Math. Soc., (2) 28 (1971), 371-374. MR 44:334
  • [R] D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, 1982. MR 84k:20001
  • [Wa] C. T. C. Wall, Finiteness conditions for CW-complexes, Ann. of Math. 81 (1965), 56-69. MR 30:1515
  • [Wd] F. Waldhausen, Whitehead groups of generalized free products, Ann. of Math. 108 (1978) 135-256.
  • [Wh1] J. H. C. Whitehead, Combinatorial homotopy I (and II), Bull. Amer. Math. Soc. 55 (1949), 213-245 (453-496).
  • [Wh2] J. H. C. Whitehead, Simple homotopy types, Amer. J. Math. 72 (1952), 1-57. MR 11:735c

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Additional Information

Danuta Kolodziejczyk
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-950 Warsaw, Poland; Address for correspondence: ul. Jasna 8/18, 00-013 Warsaw, Poland
Address at time of publication: Department of Mathematics and Informational Sciences, Warsaw University of Technology, pl. Politechniki 1, 00-661 Warsaw, Poland
Email: dkolodz@mimuw.edu.pl

DOI: https://doi.org/10.1090/S0002-9939-00-05812-3
Keywords: Shape, homotopy type, FANR, polyhedron, shape domination, homotopy domination, left neighbor
Received by editor(s): February 28, 1999
Published electronically: August 30, 2000
Additional Notes: The author would like to thank the Institute of Mathematics of the Polish Academy of Sciences for its support while this work was done.
Communicated by: Ralph Cohen
Article copyright: © Copyright 2000 American Mathematical Society

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