An algorithmically unsolvable problem in analysis

Lenard, A.; Stillwell, J.

doi:10.1090/S0002-9939-1983-0691292-2

An algorithmically unsolvable problem in analysis
HTML articles powered by AMS MathViewer

by A. Lenard and J. Stillwell PDF

Proc. Amer. Math. Soc. 88 (1983), 129-130 Request permission

Abstract:

The decision problem of distinguishing between the cases when the Laplace-Beltrami operator on the covering space of a compact manifold has 0 in its spectrum or is bounded away from 0 is algorithmically unsolvable in any class of manifolds that includes all $4$-dimensional ones. The proof depends on a result of Brooks connecting the spectrum with the amenability of the fundamental group.

References

John Stillwell, The word problem and the isomorphism problem for groups, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 33–56. MR 634433, DOI 10.1090/S0273-0979-1982-14963-1
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
Robert Brooks, Amenability and the spectrum of the Laplacian, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 87–89. MR 634438, DOI 10.1090/S0273-0979-1982-14973-4
Frederick P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies, No. 16, Van Nostrand Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0251549
W. W. Boone, W. Haken, and V. Poenaru, On recursively unsolvable problems in topology and their classification, Contributions to Math. Logic (Colloquium, Hannover, 1966) North-Holland, Amsterdam, 1968, pp. 37–74. MR 0263090
William S. Massey, Algebraic topology: An introduction, Harcourt, Brace & World, Inc., New York, 1967. MR 0211390