On a conjecture of Gottlieb

Schick, Thomas; Thom, Andreas

doi:10.2140/agt.2007.7.779

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On a conjecture of Gottlieb

Thomas Schick and Andreas Thom

Algebraic & Geometric Topology 7 (2007) 779–784

DOI: 10.2140/agt.2007.7.779

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 $L^{2}$ –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/

Volume 7, issue 2 (2007)