On contractible polyhedra that are not simply contractible

Authors:
Umed H. Karimov and Dusan Repovs

Translated by:

Journal:
Proc. Amer. Math. Soc. **132** (2004), 2159-2162

MSC (2000):
Primary 55M15; Secondary 54G20

DOI:
https://doi.org/10.1090/S0002-9939-04-07241-7

Published electronically:
February 12, 2004

MathSciNet review:
2053990

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Abstract | References | Similar Articles | Additional Information

Abstract: In answer to a question of Michael, Dydak, Segal and Spiez have constructed a contractible polyhedron that is not strictly contractible. In the present note we prove a related result; by using alternative methods we show that there exist contractible polyhedra that are not simply (hence not strictly) contractible.

**[1]**W. H. Beckman,*A certain class of nonaspherical**-complexes*, J. Pure Appl. Algebra**16**(1980), 243-244. MR**81d:57003****[2]**J. Dydak, J. Segal, and S. Spiez,*On questions of strict contractibility*, Topology Appl.**120**(2002), 67-75. MR**2003c:55010****[3]**G. Higman,*A finitely generated infinite simple group*, J. London Math. Soc.**26**(1951), 61-64. MR**12:390c****[4]**C. Hog-Angeloni and W. Metzler,*Geometric aspects of two-dimensional complexes*, Two-dimensional Homotopy and Combinatorial Group Theory, London Math. Soc. Lecture Notes**197**, Cambridge University Press, Cambridge, 1993, pp. 1-50.**[5]**U. H. Karimov and D. Repovs,*On suspensions of noncontractible compacta of trivial shape*, Proc. Amer. Math. Soc.**127**(1999), 627-632. MR**99c:54023****[6]**E. Michael,*Closed retracts and perfect retracts*, Topology Appl.**121**(2002), 451-468. MR**2003i:54015****[7]**H. Seifert and W. Threlfall,*A Textbook of Topology*, Pure and Appl. Math., vol.**89**, Academic Press, New York, 1980. MR**82b:55001****[8]**E. H. Spanier,*Algebraic Topology*, Springer-Verlag, New York, 1966; Corrected reprint, 1981. MR**83i:55001**

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Additional Information

**Umed H. Karimov**

Affiliation:
Institute of Mathematics, Academy of Sciences Tajikistan, Ul. Ainy $299^{A}$, Dushanbe, Tajikistan 734063

Email:
umed-karimov@mail.ru

**Dusan Repovs**

Affiliation:
Institute of Mathematics, Physics and Mechanics, University of Ljubljana, P.O. Box 2964, Ljubljana, Slovenia 1001

Email:
dusan.repovs@uni-lj.si

DOI:
https://doi.org/10.1090/S0002-9939-04-07241-7

Keywords:
Absolute retract,
polyhedron,
strong retract,
strict contractibility,
simple contractibility,
collapsibility

Received by editor(s):
April 30, 2002

Received by editor(s) in revised form:
January 28, 2003, and April 4, 2003

Published electronically:
February 12, 2004

Communicated by:
Alan Dow

Article copyright:
© Copyright 2004
American Mathematical Society