Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

A new construction of $ 6$-manifolds

Author(s): Ahmet Beyaz
Journal: Trans. Amer. Math. Soc. 360 (2008), 4409-4424.
MSC (2000): Primary 57R55, 57R65
Posted: March 12, 2008
MathSciNet review: 2395178
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: This paper provides a topological method to construct all simply-connected, spin, smooth $ 6$-manifolds with torsion-free homology using simply-connected, smooth $ 4$-manifolds as building blocks. We explicitly determine the invariants that classify these $ 6$-manifolds from the intersection form and specific homology classes of the $ 4$-manifold building blocks.

References:

[Be-96]

A. Beauville, Complex algebraic surfaces, Translated from the 1978 French original by R. Barlow, with assistance from N. I. Shepherd-Barron and M. Reid. Second edition. London Mathematical Society Student Texts, 34. Cambridge University Press, Cambridge, 1996. MR 1406314 (97e:14045)

[BT-82]
R. Bott and L. W. Tu, Differential forms in algebraic topology, Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982. MR 658304 (83i:57016)

[Br-93]
G. E. Bredon, Topology and geometry, Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1993. MR 1224675 (94d:55001)

[Br-66]
W. Browder, Manifolds with $ \pi \sb{1}=Z$, Bull. Amer. Math. Soc. 72 (1966), 238-244. MR 0190940 (32:8350)

[D-83]
S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), 279-315 MR 710056 (85c:57015)

[DL-05]
H. Duan and C. Liang, Circle bundles over 4-manifolds, Arch. Math. (Basel) 85 (2005), 278-282. MR 2172386 (2006e:57038)

[Fr-82]
M. H. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357-453. MR 679066 (84b:57006)

[Ha-62]
A. Haefliger, Knotted $ (4k-1)$-spheres in $ 6k$-space, Ann. of Math. (2) 75 (1962), 452-466. MR 0145539 (26:3070)

[Ha-66]
A. Haefliger, Differential embeddings of $ S\sp{n}$ in $ S\sp{n+q}$ for $ q>2$, Ann. of Math. (2) 83 (1966), 402-436. MR 0202151 (34:2024)

[Ha-02]
A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354 (2002k:55001)

[MS-75]
J. W. Milnor and J. Stasheff, Characteristic classes, Ann. of Math. Studies 76. Princeton Univ. Press, 1975. MR 0440554 (55:13428)

[MY-66]
D. Montgomery and C. T. Yang, Differentiable actions on homotopy seven spheres, Trans. Amer. Math. Soc. 122 (1966), 480-498. MR 0200934 (34:820)

[OV-95]
C. Okonek and A. Van de Ven, Cubic forms and complex $ 3$-folds, Enseign. Math. (2) 41 (1995), 297-333. MR 1365849 (97b:32035)

[Ru-94]
Y. Ruan, Symplectic topology on algebraic 3-folds, J. Diff. Geom. 39 (1994), 215-227. MR 1258920 (95a:14037)

[Sc-97]
A. Schmitt, On the classification of certain $ 6$-manifolds and applications to algebraic geometry, Topology 36 (1997), 1291-1315. MR 1452852 (99c:57056)

[Sh-68]
J. L. Shaneson, Wall's surgery obstruction groups for $ Z\times G$, for suitable groups $ G$, Bull. Amer. Math. Soc. 74 (1968), 467-471. MR 0224102 (36:7149)

[Sm-62]
S. Smale, On the structure of $ 5$-manifolds, Ann. of Math. (2) 75 (1962), 38-46. MR 0141133 (25:4544)

[Wa-66]
C. T. C. Wall, Classification problems in differential topology. V. On certain $ 6$-manifolds, Invent. Math. 1, 2, (1966), 355-374.

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57R55, 57R65

Retrieve articles in all Journals with MSC (2000): 57R55, 57R65

Additional Information:

Ahmet Beyaz
Affiliation: Department of Mathematics, University of California, Irvine, California 92697 -- and -- Department of Mathematics, Middle East Technical University, Ankara 06531, Turkey
Email: abeyaz@math.uci.edu, beyaz@metu.edu.tr

DOI: 10.1090/S0002-9947-08-04462-0
PII: S 0002-9947(08)04462-0
Keywords: $6$-manifold, $4$-manifold, homotopy complex projective space, surgery, plumbing
Received by editor(s): September 19, 2006
Posted: March 12, 2008
Additional Notes: The author would like to thank his thesis advisor, Ronald J. Stern, for his support during the Ph.D. study.
Copyright of article: Copyright 2008, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia