From Wikipedia, the free encyclopedia
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 .
[1]
[2]
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 ,
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. 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