R. Inanc Baykur

Constructing infinitely many smooth structures on small 4-manifolds

The purpose of this article is twofold. First we outline a general construction scheme for producing simply-connected minimal symplectic 4-manifolds with small Euler characteristics. Using this scheme, we illustrate how to obtain irreducible symplectic 4-manifolds homeomorphic but not diffeomorphic to

Simply connected minimal symplectic 4-manifolds with signature less than - 1

\CP#(2k+1)\CPb

Round handles, logarithmic transforms and smooth 4-manifolds

for

Kähler decomposition of 4–manifolds

k = 1,...,4

Multisections of Lefschetz fibrations and topology of symplectic 4–manifolds

, or to

INDECOMPOSABLE SURFACE BUNDLES OVER SURFACES

3\CP# (2l+3)\CPb

Positive factorizations of mapping classes

for

Lefschetz fibrations and Torelli groups

l =1,...,6

Simplified broken Lefschetz fibrations and trisections of 4-manifolds

. 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

Knotted surfaces in 4-manifolds and stabilizations

\CP#3\CPb