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Carrier and nerve theorems in the extension theory

Author: Andrzej Nagórko
Journal: Proc. Amer. Math. Soc. 135 (2007), 551-558
MSC (2000): Primary 54C20; Secondary 54F45, 55P10
Published electronically: August 1, 2006
MathSciNet review: 2255302
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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$.

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Additional Information

Andrzej Nagórko
Affiliation: Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956 Warszawa, Poland

Keywords: Carrier theorem, nerve theorem, regular cover, absolute extensor
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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.