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 ${L}^{2}$–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