Carrier and nerve theorems in the extension theory
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Abstract:
We show that a regular cover of a general topological space provides structure similar to a triangulation. In this general setting we define analogues of simplicial maps and prove their existence and uniqueness up to homotopy. As an application we give simple proofs of sharpened versions of nerve theorems of K. Borsuk and A. Weil, which state that the nerve of a regular cover is homotopy equivalent to the underlying space. Next we prove a nerve theorem for a class of spaces with uniformly bounded extension dimension. In particular we prove that the canonical map from a separable metric $n$-dimensional space into the nerve of its weakly regular open cover induces isomorphisms on homotopy groups of dimensions less than $n$.References
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Additional Information
- Andrzej Nagórko
- Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland
- Email: amn@impan.gov.pl
- Received by editor(s): May 16, 2005
- Received by editor(s) in revised form: August 19, 2005
- Published electronically: August 1, 2006
- Additional Notes: The author is grateful to Professor Henryk Toruńczyk for his advice while preparing this paper.
- Communicated by: Alexander Dranishnikov
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 551-558
- MSC (2000): Primary 54C20; Secondary 54F45, 55P10
- DOI: https://doi.org/10.1090/S0002-9939-06-08477-2
- MathSciNet review: 2255302