Definite four-manifolds with exotic smooth structures

Abstract In this paper, we study smooth structures on closed, oriented four-manifolds with fundamental group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>⁢</m:mo> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> </m:math> \mathbb{Z}/2\mathbb{Z} and definite intersection form. We construct infinitely many irreducible, oriented, closed, definite, smooth four-manifolds with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>π</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>⁢</m:mo> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> </m:mrow> </m:math> \pi_{1}=\mathbb{Z}/2\mathbb{Z} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>b</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> b_{2}=1 , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>b</m:mi> <m:mn>2</m:mn> </m:msub> <m:mo>=</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> b_{2}=2 . As an application, we prove that, when the second Betti number <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>b</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> b_{2} of a definite four-manifold with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>π</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>=</m:mo> <m:mrow> <m:mrow> <m:mi mathvariant="double-struck">Z</m:mi> <m:mo>/</m:mo> <m:mn>2</m:mn> </m:mrow> <m:mo>⁢</m:mo> <m:mi mathvariant="double-struck">Z</m:mi> </m:mrow> </m:mrow> </m:math> \pi_{1}=\mathbb{Z}/2\mathbb{Z} is positive and it admits a smooth structure, then it admits infinitely many smooth structures.