Presents an exposition of Kirby calculus, or handle body theory on 4-manifolds. This book includes such topics as branched coverings and the geography of complex surfaces, elliptic and Lefschetz
Part I of the text presents the basics of the theory at thesecond-year graduate level and offers an overview of currentresearch. Part II is devoted to an exposition of Kirby calculus, orhandlebody theory on 4-manifolds. It is both elementary andcomprehensive. Part III offers in depth a broad range of topics fromcurrent 4-manifold research. Topics include branched coverings and thegeography of complex surfaces, elliptic and Lefschetz fibrations,(h)-cobordisms, symplectic 4-manifolds, and Steinsurfaces.
Some ambiguity exists in the literature on the precise use ofthe term "Kirby moves". Different presentations of "Kirby calculus"have a different set of moves and these are sometimes called Kirbymoves. Kirby's original formulation involved two kinds of move, the"blow-up" and the "handle slide"; Fenn and Rourke exhibited anequivalent construction in terms of a single move, the Fenn--Rourkemove, that appears in many expositions and extensions of the Kirbycalculus. Rolfsen's book, Knots and Links, from which manytopologists have learned the Kirby calculus, describes a set of twomoves: 1) delete or add a component with surgery coefficientinfinity 2) twist along an unknotted component and modify surgerycoefficients appropriately (this is called the Rolfsen twist). Thisallows an extension of the Kirby calculus to rationalsurgeries.
In ,the Kirby calculus in is a method for modifying in the using a finite setof moves, the Kirby moves. It is named for . Usingfour dimensional , he proved that if M and N are , resulting from onframed links L and J respectively, then they are if and only if L and J arerelated by a sequence of Kirby moves. According to the any 3-manifold isobtained by such surgery on some link in the 3-sphere.
|4-Manifolds and Kirby Calculus (Graduate Studies in Mathematics).pdf||10.18 MB|