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

$x_1^{k_1}+\cdots +x_n^{k_n}=0$

studied by Frédéric Pham (1965).

Brieskorn manifolds give examples of exotic spheres.

References

• 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, PMID 16578636 
• Brieskorn, Egbert (1966b), "Beispiele zur Differentialtopologie von Singularitäten", Inventiones Mathematicae 2 (1): 1–14, doi:10.1007/BF01403388 
• Hirzebruch, Friedrich; Mayer, Karl Heinz (1968), O(n)-Mannigfaligkeiten, Exotische Sphären und Singularitäten, Lecture Notes in Mathematics, 57, Berlin-New York: Springer-Verlag, doi:10.1007/BFb0074355, ISBN 978-3-540-04227-3  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.. Knots, Groups and 3-Manifolds: Papers Dedicated to the Memory of R.H. Fox. Annals of Mathematics Studies. 84. Princeton University Press. pp. 175–225. ISBN 978-0-691-08167-0. https://www.jstor.org/stable/j.ctt1b7x7xd.16. 
• 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