Ryan Buchanan investigates
Non-trivial spaces
Question: What happens when we apply the topological operations (suspension, loop-space, etc.) to highly non-trivial spaces?
Follow-up: What are some of the "nicest" (well-behaved) non-trivial spaces?
Example: the disjoint union of a sphere with a single point completely changes the mapping space
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In general, a "trivial" object is one that has very few special properties. This makes them easier to define than the non-trivial ones. For instance, a sphere is defined as all the points equidistant from an origin in n dimensions. This is a very "nice" space and very easy to define. A still nice space, but one that requires more work is a ball of radius r centered at an origin. This contains all of the points of a sphere, but it is not "hollow:" it contains all of the points in between the boundary and the origin as well. Both of these spaces are "contractible;" thus, they can be smoothly compactified into points. However, spaces with singular regions do not behave this way.
Discussion
2026.01.03 Andrius: Ryan, I think you are provoking a very deep question: What is trivial and non-trivial in math? This would help us organize all of math. For mathematicians, "trivial" means "obvious", and in Wondrous Wisdom, obvious is what can't be hidden, the negation of Whether, and as such is truth. We could investigate this together if you like.
2026.01.02 Andrius: Ryan, a sphere in 2-dimensions is a circle. But a circle is not contractible! We can loop around a circle an arbitrary number of times forwards or backwards. Thus the fundamental group of a circle is {$\mathbb{Z}$}.
- And a sphere in 1-dimensions consists of two points, 1 and -1. The only paths are trivial so it is contractible. But it has two components!