By Sergei Matveev
From the studies of the first variation: "This ebook presents a finished and certain account of other themes in algorithmic three-d topology, culminating with the popularity method for Haken manifolds and together with the updated leads to machine enumeration of 3-manifolds. Originating from lecture notes of varied classes given through the writer over a decade, the booklet is meant to mix the pedagogical process of a graduate textbook (without workouts) with the completeness and reliability of a examine monograph… the entire fabric, with few exceptions, is gifted from the atypical perspective of exact polyhedra and specific spines of 3-manifolds. This selection contributes to maintain the extent of the exposition rather basic. In end, the reviewer subscribes to the citation from the again disguise: "the e-book fills a niche within the present literature and should develop into a typical reference for algorithmic three-dimensional topology either for graduate scholars and researchers". Zentralblatt f?r Mathematik 2004 For this second variation, new effects, new proofs, and commentaries for a greater orientation of the reader were additional. particularly, in bankruptcy 7 a number of new sections pertaining to functions of the pc software "3-Manifold Recognizer" were integrated.
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Extra resources for Algorithmic Topology and Classification of 3-Manifolds (Algorithms and Computation in Mathematics)
X ∼ Y if π1 (X) = π1 (Y ). This relation is fairy rough. For example, taking the one-point union with S 2 preserves the fundamental group but increases the Euler characteristic by 1. 22) is essentially an addition of S 2 . (II) The same fundamental group and Euler characteristic. X ∼ Y if π1 (X) = π1 (Y ) and χ(X) = χ(Y ). (III) Homotopy equivalence. Recall that two topological spaces X and Y are homotopy equivalent if there exist maps f : X → Y and g: Y → X such 32 1 Simple and Special Polyhedra that the maps f g : X → X and gf : Y → Y are homotopic to the identity.
34 1 Simple and Special Polyhedra (V) 3-Deformation equivalence. This relation is obtained by restricting the dimension of simple homotopy equivalence. 3. Two polyhedra X, Y of dimension ≤ 2 are 3-deformation 3d equivalent (we write 3d-equivalent or X 2 ∼ Y 2 ), if there is a sequence of elementary moves of dimension no greater than 3 taking X–Y . Clearly, two 2-dimensional polyhedra X, Y which are 3-deformation equivalent are simple homotopy equivalent, and there is a 3-dimensional polyhedron Z such that Z X and Z Y .
13, all of them give equivalent blow-ups. (2) One may choose another simple spine P1 of H 3 . 27, there is a sequence of moves T ±1 , L±1 transforming P1 into P . The same sequence can be used to relate the corresponding blow-ups. Attached cells do not prevent us from making the moves. 13 for shifting the curve away. (3) Suppose that H 3 is orientable. 17 that any two homotopy equivalences ϕ, ϕ : |K (1) | → H 3 diﬀer by a homeomorphism H 3 → H 3 . This means that the corresponding blow-ups are equivalent.