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L^2-Betti Number

An L^2-Betti number of a cover with deck group G is the dimension of its L^2-homology over the group von Neumann algebra L(G). For the universal cover Q^~ of a finite CW-complex Q, one algebraic definition is

b_j^((2))(Q^~)=dim_(L(G))H_j(L(G) tensor _(Z[G])C_*(Q^~)),

where G=pi_1(Q) and C_* is the cellular chain complex. The dimension is the extended von Neumann dimension, rather than ordinary vector space dimension. For projective modules represented by projections, it uses the sum of diagonal canonical traces.

These invariants need not be integers in general. For finite CW-complexes their alternating sum gives the Euler characteristic. The Singer conjecture asks for concentration in the middle degree for universal covers of closed aspherical manifolds.

See also

Betti Number, Chain Complex, Euler Characteristic, Group von Neumann Algebra, Homology Group, Singer Conjecture

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References

Lück, W. "Dimension Theory of Arbitrary Modules over Finite von Neumann Algebras and L^2-Betti Numbers I: Foundations." J. reine angew. Math. 495, 135-162, 1998. https://arxiv.org/abs/dg-ga/9707011.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. "L^2-Betti Number." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/L2-BettiNumber.html

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