Detecting and visualizing 3-dimensional surgery

Abstract

Topological surgery in dimension [Formula: see text] is intrinsically connected with the classification of [Formula: see text]-manifolds and with patterns of natural phenomena. In this, mostly expository, paper, we present two different approaches for understanding and visualizing the process of [Formula: see text]-dimensional surgery. In the first approach, we view the process in terms of its effect on the fundamental group. Namely, we present how [Formula: see text]-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 [Formula: see text]-dimensional surgery as rotations of the decompactified [Formula: see text]-sphere. Each rotation produces a different decomposition of the [Formula: see text]-sphere which corresponds to a different visualization of the [Formula: see text]-dimensional process of [Formula: see text]-dimensional surgery.