High Quality Content by WIKIPEDIA articles! In 1976 Michael Atiyah introduced L cohomology, using von Neumann algebras to define the L2-Betti numbers of manifolds with non-trivial fundamental group. A priori the L2-Betti numbers are real numbers. Atiyah asked if it is possible for these numbers to be irrational. Nevertheless, the rationality of the L2-Betti numbers became known as the Atiyah Conjecture. In 2009 Tim Austin constructed examples of manifolds with transcendental L2-Betti numbers, answering in the affirmative the original Atiyah question.
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