Summary: THE DOLGACHEV SURFACE
Disproving Harer-Kas-Kirby Conjecture
SELMAN AKBULUT
Abstract. We prove that the Dolgachev surface E(1)2,3 admits a handle-
body decomposition without 1- and 3- handles, and we draw the explicit
picture of this handlebody. We also locate a "cork" inside of E(1)2,3, so that
E(1)2,3 is obtained from E(1) by twisting along this cork.
0. Introduction
It is a curious question whether an exotic copy of a smooth simply connected
4-manifold admits a handle decomposition without 1- and 3- handles?. Clearly
if exotic S4
and CP2
exist, their handle decomposition must contain either 1-
or 3-handles. Hence It is a particularly interesting problem to find smallest
exotic manifolds with this property. Twenty two years ago Harer, Kas and
Kirby conjectured that the Dolgachev surface E(1)2,3, which is an exotic copy of
CP2
#9 ¯CP
2
, must contain 1- and 3-handles [HKK]. Recently, Yasui constructed

Source: Akbulut, Selman - Department of Mathematics, Michigan State University

Collections: Mathematics