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?
Examples
- the disjoint union of a sphere with a single point completely changes the mapping space
- Spinors are important, starting with representations of {$SU(2)$}. See Andrius's investigation What do spinors say about Bott periodicity?
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.
2026.02.10 Ryan: So, some of the simplest non-trivial spaces seem to be trivial in a different sense: from what I can gather, any based space (which is non-trivial in the sense that it is not just a "space," and also do remember that every space admits an infinity-groupoid presentation) comes with a natural stratification that I have dubbed the "trivial" one. I guess you could also call it the canonical or pointed stratification.
I discuss this in a recent note, if you're interested, but it's really nothing special.
The interesting thing (magic 🌟) starts to appear when you move to simplicial complexes/{$\Delta$}-complexes, because then you can start to do homology and stuff. The stratification is given by considering each n-face of the simplex (i.e., each subcomplex of dimension n) as a stratum.
Physically, this is really interesting, because you can begin to think of n-forms over each stratum, and then there are further stratifications given by the polytope-structure.
Discussion
2026.01.13 Andrius: Ryan, what do you mean by stratification?
- Stratification here just means a way to break the space down into layers or subsets based on some property. For a based space, it's based on the basepoint structure. For the simplicial complex, it's just the dimension of the faces.
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!