Brieskorn manifold

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

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

studied by 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", Invent. Math. 2 (1): 1–14, doi:10.1007/BF01403388, MR 0206972
• 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, 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, ISSN 0037-9484, MR 0195868