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Selected applications of geometry to low-dimensional topology. Marker lectures in the mathematical sciences held at the Pennsylvania State University, February, 2-5, 1987. (English) Zbl 0691.57001

University Lecture Series, 1. Providence, RI (USA): American Mathematical Society (AMS). x, 79 p. $ 33.00 (1989).
This book describes some areas of recent activity in low-dimensional geometric topology and is based on a series of lectures given by the first author. The word “geometry” is interpreted broadly; in particular geometries in the sense of Thurston play no role, although the hyperbolization theorem for atoroidal Haken 3-manifolds is mentioned in several places. The first chapter is introductory, defining manifold, smooth manifold, tangent vector and Riemannian metric and giving a 2-page survey of the classification problem for manifolds in various dimensions. Chapter 2 begins by stating a transversality theorem and defining via differential topology the degree of a smooth map between oriental manifolds of the same dimension, and then concentrates on computing the Euler characteristics of a closed orientable surface, using the Hopf index formula and Morse theory to prove the Gauss-Bonnet theorem. The next chapter moves from vector fields on surfaces to codimension-1 foliations on closed orientable 3-manifolds. After giving several examples, including the Reeb foliation of a solid torus, two theorems on codimension-1 foliations with no compact leaves are proven. The first (due to Sullivan) is that a manifold with such a foliation admits a Riemannian metric for which all the leaves are minimal hypersurfaces, and the second (due to Novikov) asserts that if \(M\) is a 3-manifold with such a foliation then \(\pi_ 2(M)=0\). A more general version of Novikov’s theorem is then described and related to the torus decomposition of irreducible 3-manifolds and the work of Gabai on foliations without Reeb components on Haken 3-manifolds. The chapter concludes with an argument of the second author which shows that if \(T_g\) is a closed orientable surface of genus \(>1\) then \(T_g\times [0,1]\) has no smooth codimension-1 foliation such that the boundary components are leaves and all the other leaves are homeomorphic to \(\mathbb{R}^2\).
The second half of the book is on aspects of 4-manifold theory. Chapter 4 begins with a description of the Kummer surface and other nonsingular hypersurfaces in \(\mathbb{CP}^3\) and states the topological classification of 1-connected closed 4-manifolds in terms of their intersection pairings and Kirby-Siebenmann invariants. The rest of the chapter gives a detailed outline of the high-dimensional smooth h- cobordism theorem and Whitney’s lemma, leading up to a discussion of Casson handles and the celebrated Disk Theorem of the first author. (As Whitehead’s result that the intersection form of a 1-connected closed 4-manifold determines its homotopy type is mentioned, it seems an oversight that no reference is made to Wall’s result that homotopy equivalent 1-connected closed 4-manifolds are h-cobordant, for these results are both needed to relate the classification theorem to the h-cobordism theorem). The next chapter considers smooth 4-manifolds and the theorem of Donaldson that if the intersection form of a 1-connected closed smooth 4-manifold is definite then it is diagonalizable over \(\mathbb{Z}\). After discussing vector bundles, connections, curvature and the Yang-Mills functional the strategy of Donaldson’s theorem is outlined, and a simpler related result due to Fintushel and Stern is proven, using Yang-Mills theory of SO(3)-bundles. The final chapter brings together the results of the preceding two chapters to construct an exotic smooth structure on \(\mathbb{R}^4\), and outlines briefly a method for constructing infinitely many such exotic structures.
The book is well written and should be of interest in particular to anyone seeking an informal introduction to current work on the topology and differential topology of 4-manifolds.

MSC:

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
57N10 Topology of general \(3\)-manifolds (MSC2010)
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
57R80 \(h\)- and \(s\)-cobordism
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