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2009 Dold spaces in homotopy theory
Eugenia Schwamberger, Rainer M Vogt
Algebr. Geom. Topol. 9(3): 1585-1622 (2009). DOI: 10.2140/agt.2009.9.1585


We study a class of spaces whose importance in homotopy theory was first highlighted by work of Dold in the 1960s, and that we accordingly call Dold spaces. These are the spaces that possess a partition of unity supported in sets that are contractible to a point within the ambient space. Dold spaces form a broader class than spaces homotopy equivalent to CW complexes, but share the feature that a number of well known weak equivalences are genuine ones if Dold spaces are involved. In this paper we give a first systematic investigation of Dold spaces. After listing their elementary properties, we study homotopy pullbacks involving Dold spaces and simplicial objects in the category of Dold spaces. In particular, we show that the homotopy colimit of a diagram of Dold spaces is a Dold space and that the topological realization functor preserves fibration sequences if the base is a path-connected Dold space in each dimension. It follows that the loop space functor commutes with realization up to homotopy for Dold spaces. Finally, we give simple conditions which assure that free algebras over a topological operad are Dold spaces.


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Eugenia Schwamberger. Rainer M Vogt. "Dold spaces in homotopy theory." Algebr. Geom. Topol. 9 (3) 1585 - 1622, 2009.


Received: 31 October 2008; Revised: 3 April 2009; Accepted: 30 June 2009; Published: 2009
First available in Project Euclid: 20 December 2017

zbMATH: 1177.55012
MathSciNet: MR2530126
Digital Object Identifier: 10.2140/agt.2009.9.1585

Primary: 55P99
Secondary: 55P35, 55P48, 55U10

Rights: Copyright © 2009 Mathematical Sciences Publishers


Vol.9 • No. 3 • 2009
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