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Detecting and visualizing 3-dimensional surgery. (English) Zbl 1440.57018

Summary: Topological surgery in dimension 3 is intrinsically connected with the classification of 3-manifolds and with patterns of natural phenomena. In this, mostly expository, paper, we present two different approaches for understanding and visualizing the process of 3-dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how 3-dimensional surgery alters the fundamental group of the initial manifold and present ways to calculate the fundamental group of the resulting manifold. We also point out how the fundamental group can detect the topological complexity of non-trivial embeddings that produce knotting. The second approach can only be applied for standard embeddings. For such cases, we give new visualizations of 3-dimensional surgery as rotations of the decompactified 2-sphere. Each rotation produces a different decomposition of the 3-sphere which corresponds to a different visualization of the 4-dimensional process of 3-dimensional surgery.

MSC:

57K30 General topology of 3-manifolds
57M05 Fundamental group, presentations, free differential calculus
57R65 Surgery and handlebodies
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
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