Algorithmic topology and classification of 3-manifolds by Sergei Matveev

By Sergei Matveev

From the experiences of the first edition:

"This publication presents a accomplished and particular account of alternative issues in algorithmic three-d topology, culminating with the popularity strategy for Haken manifolds and together with the up to date ends up in machine enumeration of 3-manifolds. Originating from lecture notes of varied classes given by way of the writer over a decade, the booklet is meant to mix the pedagogical process of a graduate textbook (without routines) with the completeness and reliability of a study monograph…

All the fabric, with few exceptions, is gifted from the ordinary standpoint of unique polyhedra and exact spines of 3-manifolds. This selection contributes to maintain the extent of the exposition relatively trouble-free.

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

Zentralblatt f?r Mathematik 2004

For this 2nd variation, new effects, new proofs, and commentaries for a greater orientation of the reader were additional. specifically, in bankruptcy 7 numerous new sections referring to functions of the pc application "3-Manifold Recognizer" were integrated.

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Extra info for Algorithmic topology and classification of 3-manifolds

Example text

See Fig. 29. 30. Let P1 and P2 be special spines of the same manifold M with at least two true vertices each. Then they are T -equivalent. Proof. 27, one can pass from P1 to P2 by moves T ±1 , L, which preserve the property of a spine of having only disc 2-components, and obius 2-components. Instead by moves L−1 , which may create annular and M¨ of doing L−1 , we erect an arch with a membrane and only then carry out the move. This can be realized by the moves L and T −1 , see Fig. 30. Fig. 29. Creating an arch with a membrane 30 1 Simple and Special Polyhedra Fig.

7. The elementary move U on a simple polyhedron P consists in removing a proper butterfly E ⊂ P and replacing it by EU , see Fig. 33. Fig. 33. The U -move 36 1 Simple and Special Polyhedra Notice that U increases the number of true vertices in a polyhedron by one, and that the EU does not embed into R3 . 8. Let P, Q be special polyhedra. Then P ∼Q if and only if one can transform P into Q by a finite sequence of moves T ±1 , U ±1 . 10. 9. Let 2-dimensional polyhedra X1 , X2 be obtained from a polyhedron Y by attaching discs D12 , D22 with homotopic attaching maps f1 , f2 : S 1 → Y .

An open ball V ⊂ M \ P is called proper (with respect to P ), if Cl(V ) \ V ⊂ P . It is worth mentioning that in general Cl(V ) may be not a 3-ball. this can happen when V approaches a 2-cell of P from two sides. In particular, if V is the complement to a spine of a closed 3-manifold M , then Cl(V ) = M . 13. Let an open 3-ball V in a 3-manifold M be proper with respect to a simple subpolyhedron P ⊂ M . Then closure Cl(V ) is a compact submanifold of M whose boundary is contained in P . Proof. Let x be a point of Cl(V ) \ V .

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