Volume 7, issue 2 (2007)

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On a conjecture of Gottlieb

Thomas Schick and Andreas Thom

Algebraic & Geometric Topology 7 (2007) 779–784

arXiv: math.AT/0702826

Abstract

We give a counterexample to a conjecture of D H Gottlieb and prove a strengthened version of it.

The conjecture says that a map from a finite CW–complex X to an aspherical CW–complex Y with non-zero Euler characteristic can have non-trivial degree (suitably defined) only if the centralizer of the image of the fundamental group of X is trivial.

As a corollary we show that in the above situation all components of non-zero degree maps in the space of maps from X to Y are contractible.

We use L2–Betti numbers and homological algebra over von Neumann algebras to prove the modified conjecture.

Keywords
degree of map, $L^2$–Betti numbers, Gottlieb's theorem, Gottlieb's conjecture, mapping spaces
Mathematical Subject Classification 2000
Primary: 55N99, 55N25, 55N25, 54C35
Secondary: 57P99, 55Q52
References
Publication
Received: 26 April 2007
Accepted: 2 May 2007
Published: 30 May 2007
Authors
Thomas Schick
Mathematisches Institut
Bunsenstr. 3-5
37073 Göttingen
Germany
http://www.uni-math.gwdg.de/schick/
Andreas Thom
Mathematisches Institut
Bunsenstr. 3-5
37073 Göttingen
Germany
http://www.uni-math.gwdg.de/thom/