Brieskorn_sphere

Brieskorn manifold

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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}}+\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 $M(p,q,r)$ ". 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

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[1] [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] [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)

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