Coherent and strong expansions of spaces coincide

Sibe Mardešić

Fundamenta Mathematicae (1998)

  • Volume: 158, Issue: 1, page 69-80
  • ISSN: 0016-2736

Abstract

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In the existing literature there are several constructions of the strong shape category of topological spaces. In the one due to Yu. T. Lisitsa and S. Mardešić [LM1-3] an essential role is played by coherent polyhedral (ANR) expansions of spaces. Such expansions always exist, because every space admits a polyhedral resolution, resolutions are strong expansions and strong expansions are always coherent. The purpose of this paper is to prove that conversely, every coherent polyhedral (ANR) expansion is a strong expansion. This result is obtained by showing that a mapping of a space into a system, which is coherently dominated by a strong expansion, is itself a strong expansion.

How to cite

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Mardešić, Sibe. "Coherent and strong expansions of spaces coincide." Fundamenta Mathematicae 158.1 (1998): 69-80. <http://eudml.org/doc/212303>.

@article{Mardešić1998,
abstract = {In the existing literature there are several constructions of the strong shape category of topological spaces. In the one due to Yu. T. Lisitsa and S. Mardešić [LM1-3] an essential role is played by coherent polyhedral (ANR) expansions of spaces. Such expansions always exist, because every space admits a polyhedral resolution, resolutions are strong expansions and strong expansions are always coherent. The purpose of this paper is to prove that conversely, every coherent polyhedral (ANR) expansion is a strong expansion. This result is obtained by showing that a mapping of a space into a system, which is coherently dominated by a strong expansion, is itself a strong expansion.},
author = {Mardešić, Sibe},
journal = {Fundamenta Mathematicae},
keywords = {coherent expansion; coherent homotopy; inverse system; strong expansion; strong shape; expansion},
language = {eng},
number = {1},
pages = {69-80},
title = {Coherent and strong expansions of spaces coincide},
url = {http://eudml.org/doc/212303},
volume = {158},
year = {1998},
}

TY - JOUR
AU - Mardešić, Sibe
TI - Coherent and strong expansions of spaces coincide
JO - Fundamenta Mathematicae
PY - 1998
VL - 158
IS - 1
SP - 69
EP - 80
AB - In the existing literature there are several constructions of the strong shape category of topological spaces. In the one due to Yu. T. Lisitsa and S. Mardešić [LM1-3] an essential role is played by coherent polyhedral (ANR) expansions of spaces. Such expansions always exist, because every space admits a polyhedral resolution, resolutions are strong expansions and strong expansions are always coherent. The purpose of this paper is to prove that conversely, every coherent polyhedral (ANR) expansion is a strong expansion. This result is obtained by showing that a mapping of a space into a system, which is coherently dominated by a strong expansion, is itself a strong expansion.
LA - eng
KW - coherent expansion; coherent homotopy; inverse system; strong expansion; strong shape; expansion
UR - http://eudml.org/doc/212303
ER -

References

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  1. [G] B. Günther, Comparison of the coherent pro-homotopy theories of Edwards-Hastings, Lisica-Mardešić and Günther, Glas. Mat. 26 (1991), 141-176. Zbl0795.55010
  2. [LM1] Ju. T. Lisica and S. Mardešić, Steenrod-Sitnikov homology for arbitrary spaces, Bull. Amer. Math. Soc. 9 (1983), 207-210. Zbl0532.55003
  3. [LM2] Ju. T. Lisica and S. Mardešić, Coherent prohomotopy and strong shape category of topological spaces, in: Proc. Internat. Topology Conference (Leningrad, 1982), Lecture Notes in Math. 1060, Springer, Berlin, 1984, 164-173. 
  4. [LM3] Ju. T. Lisica and S. Mardešić, Coherent prohomotopy and strong shape theory, Glas. Mat. 19 (39) (1984), 335-399. 
  5. [M1] S. Mardešić, Approximate polyhedra, resolutions of maps and shape fibrations, Fund. Math. 114 (1981), 53-78. Zbl0411.54019
  6. [M2] S. Mardešić, Strong expansions and strong shape theory, Topology Appl. 38 (1991), 275-291. Zbl0715.55008
  7. [M3] S. Mardešić, Resolutions of spaces are strong expansions, Publ. Inst. Math. Beograd 49 (63) (1991), 179-188. Zbl0774.54015
  8. [M4] S. Mardešić, Strong expansions and strong shape for pairs of spaces, Rad Hrvat. Akad. Znan. Umjetn. Matem. Znan. 456 (10) (1991), 159-172. Zbl0790.54019
  9. [M5] S. Mardešić, Coherent homotopy and localization, Topology Appl. (1998) (to appear). 
  10. [MS] S. Mardešić and J. Segal, Shape Theory - The Inverse System Approach, North-Holland, Amsterdam, 1982. Zbl0495.55001
  11. [T] H. Thiemann, Strong shape and fibrations, Glas. Mat. 30 (1995), 135-174. Zbl0870.55007
  12. [V] R. M. Vogt, A note on homotopy equivalences, Proc. Amer. Math. Soc. 32 (1972), 627-629. Zbl0241.55009

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