Dolgachev surface

In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by. They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds, no two of which are diffeomorphic.

Properties
The blowup $$X_0$$ of the projective plane in 9 points can be realized as an elliptic fibration all of whose fibers are irreducible. A Dolgachev surface $$X_q$$ is given by applying logarithmic transformations of orders 2 and q to two smooth fibers for some $$q\ge 3$$.

The Dolgachev surfaces are simply connected, and the bilinear form on the second cohomology group is odd of signature $$(1,9)$$ (so it is the unimodular lattice $$I_{1,9}$$). The geometric genus $$p_g$$ is 0 and the Kodaira dimension is 1.

found the first examples of homeomorphic but not diffeomorphic 4-manifolds $$X_0$$ and $$X_3$$. More generally the surfaces $$X_q$$ and $$X_r$$ are always homeomorphic, but are not  diffeomorphic unless $$q=r$$.

showed that the Dolgachev surface $$X_3$$ has a handlebody decomposition without 1- and 3-handles.