Property Suggestion: Homotopically Trivial

[mathmaster13]

Edit: The suggestion of #818 is a much more comprehensive version of this. But I also understand that #818 had problems in its day because of needing experts to verify the algebraic topology stuff. Still, maybe it can happen?

Property Suggestion

A space is said to be homotopically trivial (also known as weakly contractible) provided it is weak homotopy equivalent to The Singleton.

Equivalently, it is a space whose homotopy groups are all trivial (including the zeroth one, so that the space must be path-connected). It is difficult to find examples of spaces where "path-connected" and "connected" differ, but the definition is usually stated as "path components" because the set of path components is the zeroth homotopy set.

Rationale

The property is in this textbook. For CW-complexes, the Whitehead theorem (also in this book) says that two CW-complexes are weak homotopy equivalent if and only if they are homotopy equivalent. Thus this property is only different from Contractible for spaces that aren't so nice, which is exactly the kind of space I'd argue should be in here. Sure enough, the majority of our simply connected non-contractible spaces (which are not your everyday space!) fit this bill, and should be recognized as showing the failure of the Whitehead theorem in general.

Further developments with the concept should be in the sequel to that book, though I have no access to it myself. The concept is on Wikipedia, here and here.

It may be worthwhile to add a "has a CW structure" property. As Felix pointed out, though, very few spaces in the database are CW-complexes (and most spaces can be automatically shown to be or to not be such—only 16 or 17 would need a quite easy manual input). So I'm okay with not adding it in.

Relationship to other properties

Weakly contractible -> Simply connected
Contractible -> Weakly contractible

Locally 1-Euclidean + Simply Connected + nonempty -> Weakly contractible
I have reservations about the empty set being called simply connected, but that's how it is right now.

[felixpernegger]

Edit: The suggestion of #818 is a much more comprehensive version of this. But I also understand that #818 had problems in its day because of needing experts to verify the algebraic topology stuff. Still, maybe it can happen?

The issues there were nothing remotely unsolvable. I'm pretty confident this can be added. The algebraic topology isn't terribly advanced at all.

Property Suggestion

A space is said to be homotopically trivial (also known as weakly contractible) provided it is weak homotopy equivalent to The Singleton.

Equivalently, it is a space whose homotopy groups are all trivial (including the zeroth one, so that the space must be path-connected). It is difficult to find examples of spaces where "path-connected" and "connected" differ, but the definition is usually stated as "path components" because the set of path components is the zeroth homotopy set.

I'm a bit unsure about this one / how common this actually is. A quick google search didnt give much except https://math.stackexchange.com/questions/2652789/non-contractible-space-with-trivial-homotopy-groups

In any case, do you have a theorem in mind when a space is weakly contractible but not contractible? Otherwise this property would mostly just lie between contractible and simply connected.

It may be worthwhile to add a "has a CW structure" property. Then we can say:

Homotopically Trivial + Has a CW structure implies Contractible. (The Whitehead theorem, an important one.) And other basic properties: Has a CW structure implies Paracompact. Has a CW structure implies Perfectly Normal. Has a CW structure implies Locally Path Connected. (Maybe even locally injectively path connected?).

The concept of having a CW structure is in the textbooks I gave but also in Hatcher's Algebraic Topology, which is of course freely available to legally download.

Relationship to other properties

I thought about this before and generally think this is a good idea. I dont think this is a common property at all, but given how common CW complexes are, this is probably enough to justify it anyway. According to a very short Wikipedia entry, such spaces are called cellular spaces. (though Hausdorff is assumed there)

[GeoffreySangston]

In my opinion (and I think this is the common opinion, but could be wrong), the term 'homotopically trivial' should be reserved for contractible, not weakly contractible. This is because the homotopy type of a space being trivial exactly means contractible; I did find a few resources about finite topology which use homotopically trivial in your way, so perhaps I shouldn't be so confident.

I think the finite topology literature has some examples of weakly contractible and non-contractible spaces. Funnily enough, page 4 of the following SMALLEST HOMOTOPICALLY TRIVIAL NON-CONTRACTIBLE SPACES has an example.

Edit: I previously claimed that the so-called complete feather from MANIFOLDS: HAUSDORFFNESS VERSUS HOMOGENEITY is an example of a weakly contractible but non-contractible space, but I was confused! Sorry! I haven't looked at this paper in a while, but reading the proof now I think that their definition of contractible is the standard one. This example is not relevant to this page, because it's a contractible but not strongly contractible space (strongly contractible requires the point being contracted at to be fixed during the homotopy).

Another Edit: Thorgott found a bachelors thesis with a fantastic argument using standard techniques that simply connected implies weakly contractible for Locally 1-Euclidean) spaces, so I'll record here that this would be a good theorem to add if weakly contractible ever gets added.

[mathmaster13]

I mostly think about finite topology, so I am biased :) but I support weakly contractible. The thesis is really cool and I'd love to have the result included!

The reason I suggested weakly contractible is not exactly for its use in theorems. In "ordinary" work, you have the Whitehead theorem, which says that weakly contractible CW-complexes are contractible. (In finite topology it's a different story.)

I opened this issue because the Whitehead theorem is important and (when we add more algebraic topology in general) deserves counterexamples on the topology counterexample database. Although I do now wonder why it's important—I'll ask around. That being said, I'm sure anyone who does serious algebraic topology has seen it; it's not rare.

What's even better, we have Whitehead counterexamples in abundance. Of all the nonempty spaces pi-base knows to be simply connected but not contractible, exactly ONE of them, $S^2$, is not homotopically trivial. The rest are S100, the Warsaw circle, and many long lines—all not CW-complexes, because this is a great site for weird topologies!