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, for example the Gromoll–Meyer sphere.

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, 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, hdl:21.11116/0000-0004-39D9-8, 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.