# 4-Manifolds and Kirby Calculus (Graduate Studies in by András I. Stipsicz, Robert E. Gompf

By András I. Stipsicz, Robert E. Gompf

The earlier 20 years have introduced explosive progress in 4-manifold concept. Many books are presently showing that strategy the subject from viewpoints resembling gauge thought or algebraic geometry. This quantity, even if, bargains an exposition from a topological viewpoint. It bridges the space to different disciplines and provides classical yet vital topological thoughts that experience no longer formerly seemed within the literature. half I of the textual content offers the fundamentals of the idea on the second-year graduate point and provides an summary of present examine. half II is dedicated to an exposition of Kirby calculus, or handlebody idea on 4-manifolds. it truly is either basic and finished. half III bargains extensive a vast diversity of themes from present 4-manifold examine. issues comprise branched coverings and the geography of complicated surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. purposes are featured, and there are over three hundred illustrations and various routines with options within the ebook.

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**Extra info for 4-Manifolds and Kirby Calculus (Graduate Studies in Mathematics, Volume 20)**

**Sample text**

Consequently a closed electron trajectory Γ has to satisfy condition (34): e c A dl = 2πn, Γ n = 0, 1, 2, ... M. Kosevich Equation (36) is equivalent to the semiclassical quantization of the magnetic ﬂux (17). This relation is the physical basis of the Aharonov–Bohm effect [17] and explains the quantization of electron orbits in a singular ﬁeld of the vector-potential [29]. The latter phenomenon has the following realization. An inﬁnitely thin straight-line solenoid creates the magnetic ﬂux localized along the z-axis and the vector-potential A in the surroundings.

2 Topology, Quasiperiodic Functions,and the Transport Phenomena 35 Fig. 4. The picture from [2] representing the “thin spatial net” and the corresponding directions of B on the unit sphere where the nonclosed electron trajectories exist Case 2 (open periodic trajectories, Fig. 3) where ∗ indicates some dimensionless constants of the order of 1. We can see that conductivity reveals the strong anisotropy in the plane orthogonal to B in the second case, and the mean direction of the electron trajectory in p-space (not in x) can be measured experimentally as the zero eigen-direction of σ ik for B → ∞.

2). Also the Fermi surfaces SF will then be periodic in p-space with periods a1 , a2 , a3 . Remark. From a topological point of view, we consider the Brillouin zone as the compact three-dimensional torus T3 . The corresponding Fermi surfaces will then also be compact surfaces of ﬁnite size embedded in T3 . Identified sides Identified sides Identified sides Fig. 2. 1) p˙ = [v gr (p) × B] = [∇ε(p) × B] , c c where ε(p) is the dependence of energy on the quasimomentum (dispersion relation) and v gr (p) = ∇ε(p) is the group velocity at the state p.