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In this paper, we show how to construct examples of closed manifolds with explicitly computed irrational, even transcendental    L  2    Betti numbers, defined via the universal covering.  We show that every non‐negative real number shows up as an    L  2   ‐Betti number of some covering of a compact manifold, and that many computable real numbers appear as an    L  2   ‐Betti number of a universal covering of a compact manifold (with a precise meaning of computable given below).  In algebraic terms, for many given computable real numbers (in particular for many transcendental numbers) we show how to construct a finitely presented group and an element in the integral group ring such that the    L  2   ‐dimension of the kernel is the given number.  We follow the method pioneered by Austin [‘Rational group ring elements with kernels having irrational dimension’,  Proc. London Math. Soc . (3) 107 (2013) 1424–1448], but refine it to get explicit calculations which make the above statements possible.

Closed manifolds with transcendental L 2 ‐Betti numbers