Brieskorn manifold

In mathematics, a Brieskorn manifold or Brieskorn–Phạm manifold, introduced by Egbert Brieskorn (1966, 1966b), is the intersection of a small sphere around the origin with the singular, complex hypersurface

x 1 k 1 + ⋯ + x n k n = 0 {\displaystyle x_{1}^{k_{1}}+\cdots +x_{n}^{k_{n}}=0}

studied by Frédéric Pham (1965).

Brieskorn manifolds give examples of exotic spheres.[1][2]

References

  1. ^ Greuel, Gert-Martin; Purkert, Walter (2018). "Life and work of Egbert Brieskorn (1936–2013)". Journal of Singularities. 18: 1–28. arXiv:1711.09600. doi:10.5427/jsing.2018.18a. MR 3899530. (See pp. 19–23.)
  2. ^ van der Meer, J. W. (2017). On the Construction and Classification of Exotic Structures on the 7-Sphere (PDF). Utrecht University; Bachelor's thesis{{cite book}}: CS1 maint: postscript (link)
  • Brieskorn, Egbert V. (1966), "Examples of singular normal complex spaces which are topological manifolds", Proceedings of the National Academy of Sciences of the United States of America, 55 (6): 1395–1397, doi:10.1073/pnas.55.6.1395, MR 0198497, PMC 224331, PMID 16578636
  • Brieskorn, Egbert (1966b), "Beispiele zur Differentialtopologie von Singularitäten", Inventiones Mathematicae, 2 (1): 1–14, doi:10.1007/BF01403388, MR 0206972, S2CID 123268657
  • Hirzebruch, Friedrich; Mayer, Karl Heinz (1968), O(n)-Mannigfaligkeiten, Exotische Sphären und Singularitäten, Lecture Notes in Mathematics, vol. 57, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0074355, ISBN 978-3-540-04227-3, MR 0229251 This book describes Brieskorn's work relating exotic spheres to singularities of complex manifolds.
  • Milnor, John (1975). "On the 3-dimensional Brieskorn manifolds M ( p , q , r ) {\displaystyle M(p,q,r)} ". In Neuwirth, Lee P. (ed.). Knots, Groups and 3-Manifolds: Papers Dedicated to the Memory of R.H. Fox. Annals of Mathematics Studies. Vol. 84. Princeton University Press. pp. 175–225. ISBN 978-0-691-08167-0. MR 0418127.
  • Pham, Frédéric (1965), "Formules de Picard-Lefschetz généralisées et ramification des intégrales", Bulletin de la Société Mathématique de France, 93: 333–367, doi:10.24033/bsmf.1628, ISSN 0037-9484, MR 0195868