In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by Igor Dolgachev (1981). They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds, no two of which are diffeomorphic.
The blowup X 0 {\displaystyle 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 {\displaystyle X_{q}} is given by applying logarithmic transformations of orders 2 and q to two smooth fibers for some q ≥ ≥ --> 3 {\displaystyle q\geq 3} .
The Dolgachev surfaces are simply connected, and the bilinear form on the second cohomology group is odd of signature ( 1 , 9 ) {\displaystyle (1,9)} (so it is the unimodular lattice I 1 , 9 {\displaystyle I_{1,9}} ). The geometric genus p g {\displaystyle p_{g}} is 0 and the Kodaira dimension is 1.
Simon Donaldson (1987) found the first examples of homeomorphic but not diffeomorphic 4-manifolds X 0 {\displaystyle X_{0}} and X 3 {\displaystyle X_{3}} . More generally the surfaces X q {\displaystyle X_{q}} and X r {\displaystyle X_{r}} are always homeomorphic, but are not diffeomorphic unless q = r {\displaystyle q=r} .
Selman Akbulut (2012) showed that the Dolgachev surface X 3 {\displaystyle X_{3}} has a handlebody decomposition without 1- and 3-handles.
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