By Sergei Matveev
From the studies of the first edition:
"This ebook offers a accomplished and specific account of alternative issues in algorithmic three-d topology, culminating with the popularity technique for Haken manifolds and together with the updated ends up in laptop enumeration of 3-manifolds. Originating from lecture notes of assorted classes given by means of the writer over a decade, the publication is meant to mix the pedagogical technique of a graduate textbook (without workouts) with the completeness and reliability of a learn monograph…
All the cloth, with few exceptions, is gifted from the strange viewpoint of targeted polyhedra and specified spines of 3-manifolds. This selection contributes to maintain the extent of the exposition rather undemanding.
In end, the reviewer subscribes to the citation from the again hide: "the e-book fills a niche within the latest literature and should turn into a customary reference for algorithmic three-d 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. 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 info for Algorithmic topology and classification of 3-manifolds
Sample text
Then E1 ∪b W is dominated by E2 ∪c W . Proof. 38, we embed W × I into W × I and thus extend the given embedding i: E1 × I → E2 × I to an embedding i1 : (E1 ∪b W ) × I → i1 ((E1 ∪b W ) × I), (E2 ∪c W ) × I. To construct a collapse (E2 ∪c W ) × I we collapse W × I to i1 (W × I), and then apply the given collapse E2 × I i(E1 × I). 40 is a powerful tool for constructing new pairs of special polyP2 , one hedra such that one is dominated by the other. Given a pair P1 may attach, step by step, additional wings to P1 and P2 , each time getting a new pair.
13. (2-cell shifting) Let P be a simple polyhedron and f, g : S 1 → P two homotopic curves in general position. Then the simple polyhedra Q1 = P ∪f D2 and Q2 = P ∪g D2 are (T, U, L)-equivalent. Proof. 16. The only difference is that there is no 3-manifold, where D1 , D2 , and the trace of the homotopy between the curves could bound a proper 3-ball. 3 Special Polyhedra Which are not Spines 37 Let ft : S 1 → P be a homotopy between f and g. Define a map F : S 1 × I → P × I by the rule F (x, t) = (ft (x), t).
Creating a loop transform it to produce a new special polyhedron P1 that does not embed into a 3-manifold. Choose a point on a triple line and modify a neighborhood of this point by reattaching a sheet incident to this line such that there appears a new loop as shown in Fig. 32. 8. 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.