Transactions of the American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support My Subscriptions Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Small exotic rational surfaces without 1- and 3-handles

Author(s): Kouichi Yasui
Journal: Trans. Amer. Math. Soc. 362 (2010), 5893-5907.
MSC (2010): Primary 57R55; Secondary 57R65, 57N13
Posted: June 2, 2010
MathSciNet review: 2661501
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: We give new rational blowdown constructions of exotic $\mathbf{CP}^2\char93 n\overline{\mathbf{C}\mathbf{P}^2}$ $(5\leq n\leq 9)$ without using elliptic fibrations. We also show that our 4-manifolds admit handle decompositions without 1- and 3-handles, for $7\leq n\leq 9$ . A strategy for rational blowdown constructions of exotic $\mathbf{CP}^2\char93 n\overline{\mathbf{C}\mathbf{P}^2}$ $(1\leq n\leq 4)$ is also proposed.

References:

1.: S. Akbulut, The Dolgachev Surface, arXiv:0805.1524.
2.: S. Akbulut and K. Yasui, Corks, plugs and exotic structures, J. Gökova Geom. Topol. GGT2 (2008), 40-82. MR 2466001 (2009k:57036)
3.: A. Akhmedov and B. D. Park, Exotic Smooth Structures on Small -Manifolds, Invent. Math. 173 (2008), 209-223. MR 2403396 (2009g:57046)
4.: A. Akhmedov and B. D. Park, Exotic Smooth Structures on Small -Manifolds with Odd Signatures, arXiv:math/0701829
5.: R. Fintushel and R. J. Stern, Rational blowdowns of smooth -manifolds, J. Differential Geom. 46 (1997), no. 2, 181-235. MR 1484044 (98j:57047)
6.: R. Fintushel and R. J. Stern, Double node neighborhoods and families of simply connected -manifolds with $b\sp +=1$ , J. Amer. Math. Soc. 19 (2006), no. 1, 171-180. MR 2169045 (2006h:57024)
7.: R. Fintushel and R. J. Stern, Six Lectures on Four -Manifolds, IAS/Park City Math. Ser., Vol. 15, Amer. Math. Soc., Providence, RI, 2009. MR 2503498
8.: R. E. Gompf and A. I. Stipsicz, -Manifolds and Kirby calculus, Graduate Studies in Mathematics, 20. American Mathematical Society, 1999. MR 1707327 (2000h:57038)
9.: P. Ozsváth and Z. Szabó, On Park's exotic smooth four-manifolds, Geometry and topology of manifolds, Fields Inst. Commun., vol. 47, Amer. Math. Soc. Providence, RI, 2005, 253-260. MR 2189937 (2006i:57060)
10.: J. Park, Seiberg-Witten invariants of generalised rational blow-downs, Bull. Austral. Math. Soc. 56 (1997), no. 3, 363-384. MR 1490654 (99b:57067)
11.: J. Park, Simply connected symplectic -manifolds with $b\sp +\sb 2=1$ and $c\sp 2\sb 1=2$ , Invent. Math. 159 (2005), no. 3, 657-667. MR 2125736 (2006c:57024)
12.: J. Park, A. I. Stipsicz and Z. Szabó, Exotic smooth structures on $\mathbb{CP}\sp 2\char93 5\overline{\mathbb{CP}\sp 2}$ , Math. Res. Lett. 12 (2005), no. 5-6, 701-712. MR 2189231 (2006i:57059)
13.: R. J. Stern, Will we ever classify simply-connected smooth -manifolds?, Floer Homology, Gauge Theory, and Low-dimensional Topology, (D. Ellwood, et al., eds.), CMI/AMS publication, 2006, 225-239. MR 2249255 (2007f:57060)
14.: A. I. Stipsicz and Z. Szabó, An exotic smooth structure on $\mathbb{CP}\sp 2\char93 6\overline{\mathbb{CP}\sp 2}$ , Geom. Topol. 9 (2005), 813-832. MR 2140993 (2006f:57030)
15.: K. Yasui, Exotic rational elliptic surfaces without -handles, Algebr. Geom. Topol. 8 (2008), 971-996. MR 2443105 (2009m:57048)
16.: K. Yasui, Elliptic surfaces without -handles, Journal of Topology 1 (2008), 857-878. MR 2461858 (2009i:57065)

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 57R55, 57R65, 57N13

Retrieve articles in all Journals with MSC (2010): 57R55, 57R65, 57N13

Additional Information:

Kouichi Yasui
Affiliation: Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043, Japan
Address at time of publication: Department of Mathematics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Email: kyasui@cr.math.sci.osaka-u.ac.jp, kyasui@math.kyoto-u.ac.jp

DOI: 10.1090/S0002-9947-2010-05205-5
PII: S 0002-9947(2010)05205-5
Keywords: Handle decomposition, rational blowdown, $1$-handle, small exotic $4$-manifold.
Received by editor(s): August 6, 2008
Posted: June 2, 2010
Additional Notes: The author was partially supported by JSPS Research Fellowships for Young Scientists, and by GCOE, Kyoto University.
Copyright of article: Copyright 2010, American Mathematical Society

	Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
\|