On a conjecture of Gottlieb

Schick, Thomas¹ ; Thom, Andreas¹

¹Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, Germany

Algebraic and Geometric Topology, Tome 7 (2007) no. 2, pp. 779-784

Résumé

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.

Détail

DOI : 10.2140/agt.2007.7.779

Keywords: degree of map, $L^2$–Betti numbers, Gottlieb's theorem, Gottlieb's conjecture, mapping spaces

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