By Sergei Matveev
From the stories of the first edition:
"This e-book offers a finished and special account of other themes in algorithmic third-dimensional topology, culminating with the popularity process for Haken manifolds and together with the up to date ends up in desktop enumeration of 3-manifolds. Originating from lecture notes of varied classes given by means of the writer over a decade, the publication is meant to mix the pedagogical technique of a graduate textbook (without routines) with the completeness and reliability of a examine monograph…
All the cloth, with few exceptions, is gifted from the unusual standpoint of precise polyhedra and exact spines of 3-manifolds. This selection contributes to maintain the extent of the exposition rather straight forward.
In end, the reviewer subscribes to the citation from the again hide: "the ebook fills a spot within the latest literature and should develop into a typical reference for algorithmic three-d topology either for graduate scholars and researchers".
Zentralblatt f?r Mathematik 2004
For this 2nd version, new effects, new proofs, and commentaries for a greater orientation of the reader were additional. particularly, in bankruptcy 7 a number of new sections relating functions of the pc application "3-Manifold Recognizer" were integrated.
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Extra info for Algorithmic topology and classification of 3-manifolds
Namely, in a neighborhood of each edge e of K we attach a beam which runs along e and joins the blow-ups of its vertices. Denote by U the polyhedron obtained in this way. It consists of the union H of balls and beams (which is a handlebody) and of remnants of the triangles of K (which are 2-cells). Our next step consists in collapsing each ball Bv to a Bing membrane contained in it by penetrating inside Bv through two holes and exhausting the 3-dimensional material of Bv . We certainly can choose the holes in such a way that they intersect neither Dv nor the beams.
Then both of them collapse to F with a thin solid tube running along ∂F . See Fig. 37 for the genus 1 Fig. 36. Handle sliding Fig. 37. A common spine of the solid torus and solid Klein bottle 40 1 Simple and Special Polyhedra ˜ is a solid Klein bottle. To get a common case when H is a solid torus and H ˜ we collapse the tube onto a simple subpolyhedron. 19. If K is a simplicial complex, then W (K) is well deﬁned up to (T, U, L)-equivalence. Proof. We will show step by step that arbitrary choices made by constructing W (K) (see items 1–4 above) do not aﬀect its (T, U, L)-type.
Suppose that special polyhedra Pa and Pb are obtained from P by creating loops at a and b, respectively. Then one can transform Pa into Pb by moves T ±1 , U . Proof. We ﬁrst consider the case when a and b lie in the same edge of P . Then we may assume that they coincide. 26. Suppose that Pa , Pb are obtained by two of them. Then one can transform Pa into Pb by a composition of moves β −1 , α, T −1 , and β as shown in Fig. 41. 28 to split β ±1 and α into compositions of moves T ±1 , U . If a and b lie in edges of P having a common vertex, the transformation of Pa to Pb is carried out by β −1 and β: we create a U -turn instead of the loop at a, and then replace it by a loop at b.
Algorithmic topology and classification of 3-manifolds by Sergei Matveev