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.

[prabau]

The Lück,-Meintrup paper does not seem easily accessible. We should replace it with something else. I'll keep looking.

We need one ref using the name "weakly contractible" (maybe even Hatcher), and another one using "homotopically trivial".

[felixpernegger]

We need one ref using the name "weakly contractible" (maybe even Hatcher), and another one using "homotopically trivial".

Hatcher doesnt have this. I disagree we need sources for both

[felixpernegger]

The Lück,-Meintrup paper does not seem easily accessible. We should replace it with something else. I'll keep looking.

Typing it to google scholar immediately gives a pdf, so it is easily accessible

https://scholar.google.com/scholar?hl=de&as_sdt=0%2C5&q=On+the+Universal+Space+for+Group+Actions+with+Compact+Isotropy&btnG=