5
$\begingroup$

Defintition: Two topological spaces $X$ and $Y$ are said to be homotopy equivalent if there exist continuous maps $f: X \to Y$ and $g: Y \to X$, such that the composition $f \circ g$ is homotopic to $\text{id}_Y$ and $g\circ f$ is homotopic to $\text{id}_X$.

Definition: A continuous map $f: X \to Y$ is called a weak homotopy equivalence if it induces isomorphisms $\pi_n (X, x_0) \to \pi_n(Y,f(x_0))$ for all $n \geq 0$ and all choices of basepoint $x_0$.

One can think of homotopy equivalent spaces as spaces, which can be deformed continuously into one another. However, I have absolutely no idea how to think about weak homotopy equivalences. How can one intuitively think about them?

Example: Warsaw circle; any map between the Warsaw circle and a point induces a weak homotopy equivalence. However these two spaces are not homotopy equivalent (link)

How can one intuitively see why we have weak homotopy equivalence, but not homotopy equivalence? Which spaces have (intuitively) the same weak homotopy type?

What are the properties that weak homotopy equivalences preserve? What are the properties that weak homotopy equivalence does not preserve, but homotopy ewuivalence preserves? Are there concrete examples of applications of weak homotopy equivalences?

Also, why bother defining weak homotopy equivalence? How do we benefit from it and where is it useful? Especially, what is the motivation for it outside of CW complexes (since for them, it is equivalent to homotopy equivalence by Whitehead's theorem).

$\endgroup$
4
  • $\begingroup$ Every space X is weakly equivalent to a CW complex, but not homotopy equivalent to one, iirc this is constructed inductively using the homotopy groups of X. I believe weak equivalence removes a lot of pathologies like warsaw circle. I don't think topologists work with non CW complexes (might be wrong). I think it's kind of like among projection resolutions, every quasi isomorphism is a homotopy equivalence and every complex has a projective resolution, both this case and for space are examples of cofibrant replacement (or fibrant replacement, forgot which). $\endgroup$ Jun 5, 2022 at 11:52
  • 3
    $\begingroup$ @TychusFindlay: Topologists work with non CW complexes all the time. Spaces that are homotopy equivalent but not homeomorphic to CW complexes are of particular importance. But there are plenty of spaces not even homotopy equivalent to CW complexes that topologists encounter, ponder, and study. $\endgroup$
    – Lee Mosher
    Jun 5, 2022 at 13:22
  • $\begingroup$ While I do not know how to answer your question about "benefit", here is a post which explains somewhat why mathematicians are satisfied with weak homotopy equivalence as being a useful concept, although the explanation goes quite far beyond topology per se. $\endgroup$
    – Lee Mosher
    Jun 5, 2022 at 14:37
  • $\begingroup$ @LeeMosher Thanks for the response, I said that because when I was at uni I was told topologists work in the "homotopy category" of spaces, and simplicial sets are another model for this (unstable) homotopy category in addition to CW complexes. Simplicial sets are more restrictive than CW complexes on the point set level, so I got the impression topologists don't care about non CW complexes. $\endgroup$ Jun 6, 2022 at 8:03

2 Answers 2

7
$\begingroup$

Let’s take it for granted that we are interested in spaces up to homotopy equivalence. But how can we prove that two given spaces $X,Y$ are actually homotopy equivalent? One option is of cause to provide an actual homotopy equivalence, but this might be actually quite hard to do (we have to specify a lot of maps).

There are a lot of functors associating to a space $X$ some algebraic object (usually a group, for example singular homology, or homotopy groups) in a homotopy invariant way in the sense that homotopy equivalent spaces have isomorphic associated groups. Since algebraic objects are easier to handle than topological ones (for example a map is injective iff the kernel is zero) we might ask ourselves, if we can reverse this process: Given a map $X\rightarrow Y$ of spaces, which induces an isomorphism of the associated algebraic objects, is it maybe already an homotopy equivalence?

The notion of a weak equivalence of spaces is precisely of this form. And indeed Whiteheads tells you that for reasonable good spaces (CW-complexes) this actually works. However there are obstructions for general spaces, as the warsaw circle and actual circle demonstrate.

$\endgroup$
2
  • 1
    $\begingroup$ This answer sidesteps the main question, which asks for motivation for weak homotopy equivalence outside of CW complexes. $\endgroup$
    – Lee Mosher
    Jun 5, 2022 at 13:19
  • 1
    $\begingroup$ I don’t think so. Weak homotopy equivalences are easier to handle than homotopy equivalences and the fact that both notions coincides on CW-complexes is not a drawback but a win. Also: are CW-complexes precisely the objects for which whe=he? For me Whitehead motivates CW-complexes, not the other way around. $\endgroup$ Jun 5, 2022 at 13:26
0
$\begingroup$

You are asking a lot of questions in one. Please allow me to answer with a few "motherhood" statements that may help. What we now call topology (it was then called "Analysis Situs") arose during the late 19th and early 20th centuries when it became apparent that a very abstract take on geometry could have fruitful consequences. However, the nature of that abstraction was not God-given: general topological spaces (which can be axiomatised and understood in many different ways), metric spaces of various kinds, manifolds, algebraic varieties etc., etc. all seemed like useful objects to study and so they were and are. I believe there was some shared background idea that later became known as Weyl's program leading mathematicians to think about mappings between these objects that preserve certain properties.

So, all that said, my point is that the relationship between topology and its many branches remains the work of Man. Topology is full of definitions that exist to make the proofs go through; the best way to understand the reason for those definitions is to study the proofs. Probably the best motivation for the concept "weak homotopy equivalence" is to study the proofs that use the concept.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .