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, 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.
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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