Yongheng Zhang

Statement

Hatcher's Algebraic Topology introduces deformation retraction and mapping cylinders through thin and thick letters of the English alphabet, before discussing the general notion of homotopy equivalence. A theorem then states that two spaces are homotopy equivalent if and only if the mapping cylinder of a map (homotopy equivalence) between the two spaces contains both spaces as deformation retracts. While deformation retraction to the target space is straightforward, coming up with one to the domain space is a nontrivial exercise, and can take quite some effort. 3D sculptures could help in this endeavor, by providing concrete shapes to touch, to look at, and to think about.

Artworks

Frequently referred to as Holland in the past, the Netherlands contains the former in its northwest. This 3D design is a mapping cylinder of a homotopy equivalence from the thin letter H to the thick letter N, recording a time evolution of deforming H to the northwest of N. The reader can piece together stages of homotopy in Chapter 0 of Hatcher's Algebraic Topology, or in Chapter I of G. W. Whitehead's Elements of Homotopy Theory, to get a strong deformation retraction of this mapping cylinder back onto H, or to come up with one on their own.