Algebraic & Geometric Topology

On a conjecture of Gottlieb

Thomas Schick and Andreas Thom

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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.

Article information

Source
Algebr. Geom. Topol., Volume 7, Number 2 (2007), 779-784.

Dates
Received: 26 April 2007
Accepted: 2 May 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1513796705

Digital Object Identifier
doi:10.2140/agt.2007.7.779

Mathematical Reviews number (MathSciNet)
MR2308964

Zentralblatt MATH identifier
1149.55003

Subjects
Primary: 55N99: None of the above, but in this section 55N25: Homology with local coefficients, equivariant cohomology 55N25: Homology with local coefficients, equivariant cohomology 54C35: Function spaces [See also 46Exx, 58D15]
Secondary: 57P99: None of the above, but in this section 55Q52: Homotopy groups of special spaces

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

Citation

Schick, Thomas; Thom, Andreas. On a conjecture of Gottlieb. Algebr. Geom. Topol. 7 (2007), no. 2, 779--784. doi:10.2140/agt.2007.7.779. https://projecteuclid.org/euclid.agt/1513796705


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References

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