Premium
Constructing infinitely many smooth structures on small 4‐manifolds
Author(s) -
Akhmedov Anar,
Baykur R. İnanç,
Park B. Doug
Publication year - 2008
Publication title -
journal of topology
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.447
H-Index - 31
eISSN - 1753-8424
pISSN - 1753-8416
DOI - 10.1112/jtopol/jtn004
Subject(s) - mathematics , symplectic geometry , diffeomorphism , homeomorphism (graph theory) , scheme (mathematics) , euler characteristic , pairwise comparison , pure mathematics , euler's formula , simply connected space , topology (electrical circuits) , combinatorics , mathematical analysis , statistics
The purpose of this article is two‐fold. First we outline a general construction scheme for producing simply connected minimal symplectic ‐manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic ‐manifolds homeomorphic but not diffeomorphic to ℂ ℙ 2 # ( 2 k + 1 )ℂ ℙ ¯ 2for k = 1, …, 4, or to 3 ℂ ℙ 2 # ( 2 l + 3 )ℂ ℙ ¯ 2for l = 1, …, 6. Secondly, for each of these homeomorphism types, we show how to produce an infinite family of pairwise nondiffeomorphic nonsymplectic 4‐manifolds belonging to it. In particular, we prove that there are infinitely many exotic irreducible nonsymplectic smooth structures on, ℂ ℙ 2 # 3ℂ ℙ ¯ 2 , 3 ℂ ℙ 2 # 5ℂ ℙ ¯ 2and 3 ℂ ℙ 2 # 7ℂ ℙ ¯ 2 .