# zbMATH — the first resource for mathematics

The Seiberg-Witten equations and applications to the topology of smooth four-manifolds. (English) Zbl 0846.57001
Mathematical Notes (Princeton). 44. Princeton, NJ: Princeton Univ. Press. vi, 128 p. (1996).
The author gives an introduction to the Seiberg-Witten equations. These equations are analogues of Donaldson’s instanton equations and they lead to new invariants of smooth 4-manifolds. Experts believe that these new invariants contain the same information as the Donaldson invariants. The main difference between the theories is the gauge group which is the (commutative) circle group for the Seiberg-Witten theory whereas Donaldson used SU(2)-principal bundles. Moreover, the Seiberg-Witten moduli space is compact which on one hand simplifies the theory but on the other hand makes some constructions impossible (like finding the original 4-manifold at the “end” of a 5-dimensional moduli space). The text is written for non-experts and is kept on an introductory level. In later chapters it emphasizes the computations of the invariants on Kähler surfaces. Taubes’ results for symplectic manifolds are not covered, neither is the proof of the 10/8 conjecture.
Reviewer: P.Teichner (Mainz)

##### MSC:
 57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes 57N13 Topology of the Euclidean $$4$$-space, $$4$$-manifolds (MSC2010) 57R15 Specialized structures on manifolds (spin manifolds, framed manifolds, etc.)
##### Keywords:
4-manifolds; gauge theory; Kähler surfaces; Dirac operator