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Cohomology for Anyone

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Crystallography has proven a rich source of ideas over several centuries. Among the many ways of looking at space groups, N. David Mermin has pioneered the Fourier-space approach. Recently, we have supplemented this approach with methods borrowed from algebraic topology. We now show what topology, which studies global properties of manifolds, has to do with crystallography. No mathematics is assumed beyond what the typical physics or crystallography student will have seen of group theory; in particular, the reader need not have any prior exposure to topology or to cohomology of groups.

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REFERENCES

  1. P. Alexandroff, Elementary Concepts of Topology, translated by Alan E. Farley (Dover, New York, 1961).

    Google Scholar 

  2. E. Ascher and A. Janner, “Algebraic aspects of crystallography. Space groups as extensions, ” Helv. Phys. Acta 38, 551–572 (1965).

    Google Scholar 

  3. E. Ascher and A. Janner, “Algebraic aspects of crystallography. II. Non-primitive translations in space groups, ” Comm. Math. Phys. 11, 138–167 (1968/1969).

    Google Scholar 

  4. A. Bienenstock and P. P. Ewald, “Symmetry of Fourier space, ” Acta Crystallogr. 15, 1253–1261 (1962).

    Google Scholar 

  5. K. S. Brown, Cohomology of Groups, Graduate Texts in Mathematics, Vol. 87 (Springer, New York, 1982).

    Google Scholar 

  6. J. Dräger and N. D. Mermin, “Superspace groups without the embedding: The link between superspace and Fourier-space crystallography, ” Phys. Rev. Lett. 76, 1489–1492 (1996).

    Google Scholar 

  7. B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov, Modern Geometry—Methods and Applications. Part II, Graduate Texts in Mathematics, Vol. 104 (Springer, New York, 1985).

    Google Scholar 

  8. B. N. Fisher and D. A. Rabson, in preparation.

  9. B. N. Fisher and D. A. Rabson, “Applications of group cohomology to the classification of quasicrystals in Fourier space, ” to appear in J. Phys. A.

  10. T. Hahn, ed., International Tables for Crystallography. Vol. A, 2nd edn. (published for the International Union of Crystallography, Chester, 1987).

    Google Scholar 

  11. H. Hiller, “Crystallography and cohomology of groups, ” Amer. Math. Monthly 93, 765–779 (1986).

    Google Scholar 

  12. D. Jabon, “IntegerSmithNormalForm, ” http://library.wolfram.com/database/MathSource/682 (1994).

  13. T. Janssen and A. Janner, “Superspace groups and representations of ordinary space groups: Alternative approaches to the symmetry of incommensurate crystal phases, ” Physica A 126, 163–176 (1984).

    Google Scholar 

  14. A. König and N. D. Mermin, “Electronic level degeneracy in nonsymmorphic periodic or aperiodic crystals, ” Phys. Rev. B 56, 13607–13610 (1997).

    Google Scholar 

  15. A. König and N. D. Mermin, “Screw rotations and glide mirrors: Crystallography in Fourier space, ” Proc. Nat. Acad. Sci. USA 96, 3502–3506 (1999).

    Google Scholar 

  16. A. König and N. D. Mermin, “Symmetry, extinctions, and band sticking, ” Am. J. Phys. 68, 525–530 (2000).

    Google Scholar 

  17. A. LeClair (1991), private communication.

  18. R. Lifshitz, “The symmetry of quasiperiodic crystals, ” Physica A 232, 633–647 (1996).

    Google Scholar 

  19. R. Lifshitz, “Theory of color symmetry for periodic and quasiperiodic crystals, ” Rev. Mod. Phys. 69, 1181–1218 (1997).

    Google Scholar 

  20. R. Lifshitz, “Symmetry of magnetically ordered quasicrystals, ” Phys. Rev. Lett. 80, 2717–2720 (1998).

    Google Scholar 

  21. N. D. Mermin, “Bringing home the atomic world: Quantum mysteries for anybody, ” Am. J. Phys. 49, 940–943 (1981).

    Google Scholar 

  22. N. D. Mermin, “Quantum mysteries for anyone, ” J. Philos. 78, 397–408 (1981).

    Google Scholar 

  23. N. D. Mermin, “Copernican crystallography, ” Phys. Rev. Lett. 68, 1172–1175 (1992).

    Google Scholar 

  24. N. D. Mermin, “The space groups of icosahedral quasicrystals and cubic, orthorhombic, monoclinic, and triclinic crystals, ” Rev. Mod. Phys. 64, 3–49 (1992).

    Google Scholar 

  25. N. D. Mermin, D. A. Rabson, D. S. Rokhsar, and D. C. Wright, “Stacking quasicrys-tallographic lattices, ” Phys. Rev. B 41, 10498–10502 (1990).

    Google Scholar 

  26. N. D. Mermin and D. S. Rokhsar, “Beware of 46-fold symmetry: The classification of two-dimensional quasicrystallographic lattices, ” Phys. Rev. Lett. 58, 2099–2101 (1987).

    Google Scholar 

  27. S. A. Piunikhin, “The relationship between various definitions of quasicrystallographic groups, ” Mat. Zametki 52, 74–80, 159 (1992), translation in Math. Notes 52 (1992), Nos. 5–6, 1220–1224 (1993).

    Google Scholar 

  28. D. A. Rabson and B. N. Fisher, “Fourier-space crystallography as group cohomology, ” Phys. Rev. B 65, 024201(2002).

    Google Scholar 

  29. D. A. Rabson, T.-L. Ho, and N. D. Mermin, “Space groups of quasicrystallographic tilings, ” Acta Crystallogr. Sec. A 45, 538–547 (1989).

    Google Scholar 

  30. D. A. Rabson, N. D. Mermin, D. S. Rokhsar, and D. C. Wright, “The space groups of axial crystals and quasicrystals, ” Rev. Mod. Phys. 63, 699–733 (1991).

    Google Scholar 

  31. D. S. Rokhsar, D. C. Wright, and N. D. Mermin, “Scale equivalence of quasicrystallo-graphic space groups, ” Phys. Rev. B 37, 8145–8149 (1988).

    Google Scholar 

  32. D. S. Rokhsar, D. C. Wright, and N. D. Mermin, “The two-dimensional quasicrystal-lographic space groups with rotational symmetries less than 23-fold, ” Acta Crystallogr. Sec. A 44, 197–211 (1988).

    Google Scholar 

  33. R. L. E. Schwarzenberger, n-Dimensional Crystallography (Pitman, Boston, Mass., 1980).

    Google Scholar 

  34. K. Spindler, Abstract Algebra with Applications, Vol. I (Marcel Dekker, New York, 1994).

    Google Scholar 

  35. M. Spivak, Calculus on Manifolds. A Modern Approach to Classical Theorems of Advanced Calculus (Benjamin, New York/Amsterdam, 1965).

    Google Scholar 

  36. D. Thouless, “Topological considerations, ” in The Quantum Hall Effect, S. M. G. Richard and E. Prange, eds., 2nd edn. (Springer, 1990).

  37. L. T. K. Tkhang, S. A. Piunikhin, and V. A. Sadov, “The geometry of quasicrystals, ” Uspekhi Mat. Nauk 48, 41–102 (1993), translation in Russian Math. Surveys 48(1), 37–100 (1993).

    Google Scholar 

  38. Wolfram Research, Mathematica, Version 4 (Champaign, Illinois, 1999).

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Rabson, D.A., Huesman, J.F. & Fisher, B.N. Cohomology for Anyone. Foundations of Physics 33, 1769–1796 (2003). https://doi.org/10.1023/A:1026281621848

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