By Sergei Matveev

From the reports of the first edition:

"This booklet presents a entire and unique account of alternative issues in algorithmic three-dimensional topology, culminating with the popularity strategy for Haken manifolds and together with the updated ends up in computing device 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 strategy of a graduate textbook (without workouts) with the completeness and reliability of a study monograph…

All the cloth, with few exceptions, is gifted from the ordinary viewpoint of detailed polyhedra and distinct spines of 3-manifolds. This selection contributes to maintain the extent of the exposition relatively easy.

In end, the reviewer subscribes to the citation from the again disguise: "the publication fills a niche within the latest literature and may turn into a regular reference for algorithmic third-dimensional topology either for graduate scholars and researchers".

Zentralblatt f?r Mathematik 2004

For this 2^{nd} variation, new effects, new proofs, and commentaries for a greater orientation of the reader were further. specifically, in bankruptcy 7 a number of new sections bearing on purposes of the pc software "3-Manifold Recognizer" were incorporated.

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**Extra resources for Algorithmic topology and classification of 3-manifolds**

**Example text**

All the remaining L can be expressed through T ±1 and U ±1 . 25. This fairly deep result will be used in Sect. 4. Recall that a special polyhedron is unthickenable, if it cannot be embedded into a 3-manifold. 20, a special polyhedron is unthickenable if and only if the boundary curve of at least one 2-component has a nontrivial normal bundle. 8 we know that if two special polyhedra P, Q are 3d-equivalent, then one can pass from P to Q by a sequence of moves T ±1 , U ±1 . It turns out that if Q is unthickenable, then one can get rid of move −1.

By construction, X2 . Z X1 and Z Now both the T -move and the U -move may be regarded as moves that change the attaching map for a disc by a homotopy. The same is true for the lune move. 10, these moves can be realized by 3-deformations. 11. Two simple polyhedra P1 , P2 are (T, U )-equivalent (notaT,U tion: P1 ∼ P2 ) if one can pass from P1 to P2 by a ﬁnite sequence of moves T ±1 , U ±1 . If in addition moves L±1 are allowed, then we say that the polyhedra are (T, U, L)-equivalent. 12. We do not make use of the bubble move.

This relation was introduced by Whitehead. A basic reference for this material is Milnor’s paper [95]. Originally, Whitehead worked with simplicial complexes, but later found the theory easier to express in terms of CW complexes. 1) and inverse transformations called elementary polyhedral expansions. , an elementary collapse or expansion) has a dimension that by deﬁnition equals the dimension of the cell that disappears, respectively, appears during the move. s Two polyhedra X, Y are said to be simple homotopy equivalent (X ∼Y ) if there is a sequence of elementary expansions and collapses taking X to Y .