Representing 3-manifolds by filling Dehn surfaces (Q2797846)

Closed 3-manifolds can be represented in various ways; among the most popular ones there are Heegaard splittings, surgery on links in \(S^3\), and branched spines (leading also to various notions of complexity of a 3-manifold). The present book is an in-depth study of the representation of closed 3-manifolds by \textit{filling Dehn surfaces}. A Dehn surface in a 3-manifold \(M\) is a compact immersed surface whose singularities are double and triple points (lines of double points meeting in triple points), codified by a Johansson diagram which indicates how the preimages of the double and triple points get identified (and allows to reconstruct \(M\) in the case of a filling Dehn surface). A Dehn surface is \textit{filling} if it defines a cellular decomposition of \(M\) (the triple points form the 0-skeleton, the double lines the 1-skeleton and the surface the 2-skeleton).NEWLINENEWLINEHistorically, the term Dehn surface has been introduced by Papakyriakopoulos 1957 in his proof of Dehn's Lemma; [\textit{W. Haken}, Lect. Notes Math. 375, 97--107 (1974; Zbl 0289.55003)] observed that every homotopy 3-sphere has a filling Dehn sphere and proposed filling Dehn surfaces and their Johansson diagrams as a method for representing homotopy 3-spheres. \textit{R. Fenn} and \textit{C. Rourke} [Lect. Notes Math. 722, 31--36 (1979; Zbl 0406.57002)] showed that every 3-manifold has a quasi-filling Dehn sphere (i.e., the complement is a disjoint union of open 3-balls), and finally \textit{J. M. Montesinos-Amilibia} [in: Contribuciones matemáticas. Libro-homenaje al Profesor D. Joaquín Arregui Fernández. Madrid: Editorial Complutense. 239--247 (2000; Zbl 0981.57009)] showed that every closed orientable 3-manifold has a filling Dehn sphere.NEWLINENEWLINEThe present book is an introduction to Dehn surfaces in 3-manifolds as well as to various related topics and results. The main result is a Reidemeister-type theorem (``Reidemeister moves'') from the 2006 PhD-thesis of the first author (in Spanish), stating that null-homotopic (e.g. if the 3-manifold is irreducible) filling Dehn spheres of the same 3-manifold are related by \textit{f}-moves (finger and saddle moves), slightly modifying moves considered by \textit{T. Homma} and \textit{T. Nagase} [Yokohama Math. J. 33, 103--119 (1985; Zbl 0595.57015)]. A second main theorem states that two filling Johansson diagrams in the 2-sphere represent the same 3-manifold if and only if suitably defined duplicates are \textit{f}-equivalent (Johansson, in an attempt from 1938 to prove Dehn's Lemma, had shown that each abstract diagram is realized for some 3-manifold).NEWLINENEWLINEThe main chapters of the present book are the following: Filling Dehn surfaces (surgery on Dehn surfaces, Montesinos theorem); Johansson diagrams; fundamental group of a Dehn sphere (from its Johansson diagram, isomorphic to the fundamental group of the underlying 3-manifold); filling homotopies (Haken moves, diagram moves, Amendola moves); proof of the main theorem; the triple point spectrum (Montesinos complexity, the minimal number of triple points for Dehn surfaces of every fixed genus \(g\)); 2-knots, 1-knots and open problems. (Since not all Haken moves are local, in fact, some do not preserve the property of being filling of a Dehn surface, \textit{G. Amendola} [Algebr. Geom. Topol. 9, No. 2, 903--933 (2009; Zbl 1176.57016)] and [J. Knot Theory Ramifications 19, No. 12, 1549--1569 (2010; Zbl 1221.57033)] has considered a set of moves which preserve the filling-property and used this to define a new 3-manifold invariant; the \textit{f}-moves of the present book are the Haken moves which preserve the filling-property).NEWLINENEWLINEThe methods used in the book are classical, that is mainly geometric-combinatorial; differential topology, regular homotopies and isotopies, triangulations, subdivisions and simplicial collapsings, and shellability are illustrated by many (more than 600) nice and carefully elaborated figures (the proof of a Key Lemma for the main theorem, relegated to a first appendix, ends with a sequence of 41 pages of figures without any text, and both the second and third appendix with 8 pages of figures).NEWLINENEWLINEConcluding, this an interesting and carefully written book (also quite technical in parts), with many hints to the relevant literature and related topics. As the authors note, filling Dehn spheres is a barely explored research topic: there are many open questions related to them, the associated complexities or invariants are hard to compute, and there are not many examples and applications so far. The main examples discussed in the book are \(S^3\), \(S^2 \times S^1\) and the lens space \(L(3,1)\) (but e.g. no infinite families of 3-manifolds), so also an enumeration of 3-manifolds of small complexity seems to be difficult at present (in analogy with the much considered Matveev complexity using branched spines). Interesting would be also a comparison of the present methods with other types of representations of 3-manifolds, and some application to the resolution of problems outside the technical range of the methods themselves (e.g., none of the classical methods for 3-manifolds seem strong enough to prove the Poincaré conjecture which motivated Haken, and it will be interesting to see if and when such a proof by classical geometric-combinatorial methods will emerge, and by which approach).