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Singer Conjecture

The Singer conjecture asserts that the L2-Betti numbers of the universal cover of a closed aspherical n-dimensional manifold vanish outside the middle degree. In symbols,

b_j^((2))(M^~)=0

for every j!=n/2. For odd n, this means vanishing in every degree. For even n=2m, the asserted concentration would imply (-1)^mchi(M)=b_m^((2))(M^~)>=0.

See also

Aspherical Space, Euler Characteristic, L2-Betti Number, Poincaré Complex, Poincaré Duality, Universal Cover

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References

Lück, W. "Survey on L^2-Invariants and 3-Manifolds." Bull. London Math. Soc. 53, 1583-1620, 2021. https://doi.org/10.1112/blms.12536.

Cite this as:

Weisstein, Eric W. "Singer Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SingerConjecture.html

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