Algorithmic topology and classification of 3-manifolds by Sergei Matveev

By Sergei Matveev

From the stories of the first edition:

"This publication offers a entire and designated account of other themes in algorithmic third-dimensional topology, culminating with the popularity technique 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 through the writer over a decade, the e-book is meant to mix the pedagogical procedure 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 bizarre perspective of specified polyhedra and specified spines of 3-manifolds. This selection contributes to maintain the extent of the exposition rather ordinary.

In end, the reviewer subscribes to the citation from the again disguise: "the ebook fills a niche within the current literature and may develop into a regular 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 numerous new sections bearing on functions of the pc application "3-Manifold Recognizer" were incorporated.

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Two steps that realize the endpoint-through-vertex move realized by moves T ±1 on P . The realization for the first case is explicitly presented in Fig. 18: We bypass the true vertex by taking two steps T and T −1 . For a realization of the second case see Fig. 19. 2 2-Cell Replacement Lemma In this section we prove that, under certain conditions, replacement of one 2-cell of a simple subpolyhedron of a 3-manifold by another can be realized by a sequence of moves T ±1 , L±1. 12. Let P be a simple polyhedron in a 3-manifold M .

Xn | r1 , . . , rn is a balanced presentation of the trivial group. Then it can be reduced to the empty presentation by operations (i)–(iv) on the relators, and two new operations: (v) Introduce a new generator xn+1 and a new relator rn+1 that coincides with xn+1 . (vi) The inverse of (v). The referee pointed out that the weaker conjecture (now known as the Andrews–Curtis conjecture) can be formulated in an equivalent geometric form. 3 Special Polyhedra Which are not Spines 35 Conjecture (AC).

Indeed, we have a butterfly: The sheets B, C, D form a disc while the sheet A passes the point twice and thus produces two wings. Note that a regular neighborhood of the loop in the modified polyhedron obius band in the union of wings A and C. Assuming that P1 contains a M¨ obius band P1 embeds into a 3-manifold M , we get an embedding of the M¨ with the disc D attached along its core circle. But this is impossible, since the normal bundle of the disc, hence its restriction to the boundary, is trivial.

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