Weakly contractible definition + trivial theorems

[felixpernegger]

Updating #1761.
Note this still misses whitehead theorem.

This PR has high priority!

[prabau]

If $f:X\to Y$ is a weak homotopy equivalence between two spaces (i.e., induces isomorphisms between all homotopy groups). does there necessarily exist an "inverse" i.e. a weak homotopy equivalence going the other way $Y\to X$?
I.e., is the relation of "weak homotopy equivalence" between spaces symmetric? (it's obviously reflexive and transitive).

[prabau]

The definition is kind of ok, although it could be slightly more precise. I'll suggest an edit.

[prabau]

Need to mention a specific definition/page where this is defined, in Hatcher for example, or some other AT book.

[prabau]

No need to add https://en.wikipedia.org/wiki/Homotopy_group in the refs: section, as it's only an accessory concept. It's good that it's linked to directly from the text.

On the other hand, need to add a direct link to something for "weakly homotopy equivalent".

[felixpernegger]

If f : X → Y is a weak homotopy equivalence between two spaces (i.e., induces isomorphisms between all homotopy groups). does there necessarily exist an "inverse" i.e. a weak homotopy equivalence going the other way Y → X ? I.e., is the relation of "weak homotopy equivalence" between spaces symmetric? (it's obviously reflexive and transitive).

according to this no
https://math.stackexchange.com/questions/80217/existence-of-weak-homotopy-equivalence-not-a-symmetric-relation
(very good question)

[felixpernegger]

No need to add https://en.wikipedia.org/wiki/Homotopy_group in the refs: section, as it's only an accessory concept. It's good that it's linked to directly from the text.

On the other hand, need to add a direct link to something for "weakly homotopy equivalent".

I thought we usually link it in the refs section as well even if we link directly in text?
In any case, there is also https://en.wikipedia.org/wiki/Weak_equivalence_(homotopy_theory) (which I actually dont like so much, but we should link it anyways)

[felixpernegger]

Need to mention a specific definition/page where this is defined, in Hatcher for example, or some other AT book.

I for the sake of it cant find a textbook containing either of the names (even though the term appears a lot)
A paper with over 70 citations by a reputable author defining the term is for example
https://zbmath.org/0979.55010
maybe we can use this?

[prabau]

If f : X → Y is a weak homotopy equivalence between two spaces (i.e., induces isomorphisms between all homotopy groups). does there necessarily exist an "inverse" i.e. a weak homotopy equivalence going the other way Y → X ? I.e., is the relation of "weak homotopy equivalence" between spaces symmetric? (it's obviously reflexive and transitive).

according to this no https://math.stackexchange.com/questions/80217/existence-of-weak-homotopy-equivalence-not-a-symmetric-relation (very good question)

Good to know.

Of course, for the case of a weakly contractible space, it does not matter: if there is a weak homotopy equivalence from $X$ to a singleton (by the unique map possible), any map from the singleton to $X$ will also be a weak homotopy equivalence, and vice versa. And it's equivalent to all homotopy groups being trivial.