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I recently read something along the lines of "General topology deals with the nice properties of pathological spaces, and algebraic topology with the pathological properties of nice spaces".

In a sense, this is consistent with my experience of both subjects, at least the part concerning which spaces we're talking about : in general topology often one thinks of the worst possible cases (e.g. totally disconnected spaces, non separated spaces, etc.), but in algebraic topology those are usually kept out : if someone says "but the space has to be connected", in algebraic topology we'll gladly add this as a hypothesis without a second thought; often we deal only with CW-complexes, or "connected, locally path-connected, semi-locally simply connected spaces" which are very far from the beasts one is concerned about in general topology.

Of course this is very schematic and I'm just generalizing stuff (though if you think this is a bad generalization, please tell me so !), especially about general topology.

However it's easy to come up with natural statements concerning nice spaces in general topology, we can easily add hypotheses such as path-connectedness etc.

But can we go in the other direction ? Is there some "algebraic topology of pathological spaces" ?

Of course it would have to look very different from what algebraic topology usually is (at least the one I know) : for instance the fundamental group of a totally disconnected space is quite useless.

But I can't see a reason why, a priori, one couldn't associate interesting algebraic invariants to pathological spaces, perhaps they wouldn't be groups, but some other thing. For a more formal question (though still vague) : is there a reason why there aren't nice functors from $\mathbf{Top}$ to "algebraic" categories ? (I'm adding the "category theory" tag because there might be an answer concerning the structure of $\mathbf{Top}$ that would hinder the existence of such functors)

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  • $\begingroup$ What do you means by "nice functors"? All the existing stuff works for any topological space. $\endgroup$ – user2520938 Aug 5 '18 at 16:08
  • $\begingroup$ There are definitely folk who are attempting to extend the ideas of algebraic topology to more general spaces (this paper and the [28]th citation therein are relevant), but I am not sure how successful the theory has been thus far. $\endgroup$ – Xander Henderson Aug 5 '18 at 16:10
  • $\begingroup$ @user2520938 : my question is very vague, and of course all of them work, but they're not useful for general topological spaces, e.g. the fundamental group of a zero dimensional space is always trivial, the homology groups of such a space are also useless, etc. $\endgroup$ – Max Aug 5 '18 at 16:12
  • $\begingroup$ @XanderHenderson thank you for the reference, I'll take a look ! $\endgroup$ – Max Aug 5 '18 at 16:14
  • $\begingroup$ Look at the work of Jeremy Brazas. $\endgroup$ – user98602 Aug 5 '18 at 16:21
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Actually, there is an active theory of algebraic topology for "pathological" spaces that has come a long way in the past two decades: Wild (algebraic/geometric) Topology. In fact, this has become a small field in its own right with a lot of recent momentum.

Descriptions of fundamental groups do become more complicated because in a wild space there may be shrinking sequences of non-trivial loops, which allow you to form various kinds of infinite products in $\pi_1$. Hence wild algebraic topology requires more than just the usual tools from algebraic topology but also is deeply connected to linear order theory, continuum theory, descriptive set theory, and topological algebra.

Here is an example of an astonishing result from this field, which addresses your interest in detecting homotopy type:

Homotopy Classification of 1-Dimensional Peano Continua (K. Eda): Two 1-dimensional Peano continua (e.g. Hawaiian earring, Sierpinski carpet/triangle, Menger cuber) are homotopy equivalent if and only if their fundamental groups are isomorphic.

The combined work of Greg Conner and Curtis Kent announced last year proves the same thing is true for planar Peano continua.

Once you realize how complicated these groups are due to the kinds of infinite products that can occur (although a word calculus of sorts does exist), it is absolutely remarkable that such theorems are true...almost scandalous. Results like the one above are very hard to prove. Eda's result required a lot of ingenuity and machinery that is being used and extended in current work.

Here is a little more pre-2000 history:

1950s - 1960s: There were a few scattered papers by some prominent mathematicians, e.g. Barrat/Milnor, H.B. Griffiths, Curtis/Fort.

1970s: Shape theory was developed to extend homotopy theoretic methods to provide invariants for more general spaces. The idea of space theory is to understand objects as (or at least approximated by) inverse limits of the usual "nice" spaces, applying your invariant to the nice approximating spaces, and call the inverse system of algebraic objects a "pro-invariant" and the inverse limit a "shape-invariant." The book Shape Theory by Segal and Mardesic is, I think, the best book on this topic. However, shape invariants only sometimes help with understanding homotopy type and traditional algebraic invariants of wild spaces.

1980s: Not much happened except for Morgan and Morrison fixing H.B. Griffiths description of the fundamental group of the Hawaiian earring.

1990s: Katsuya Eda, whose background was in logic, discovered that the Fundamental group of the Hawaiian earring behaves like a non-abelian version of the famous Specker group $\prod_{\mathbb{N}}\mathbb{Z}$. Eda was the first to make the key connection to order theory and describe the Hawaiian earring group as a group of reduced linear words $w$ (like a free group) where $w$ has countably many letters and each letter of your alphabet can only appear finitely many times in $w$. This work made the Hawaiian earring group practical to use; it is the key to many recent advancements.

Since Eda's work there has been a great deal done and there is now a huge amount of literature on the subject.

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  • $\begingroup$ Very interesting, thank you ! (Someone in the comments also mentioned I should look at your work, if you like you can also expand on this on your answer !) $\endgroup$ – Max Aug 8 '18 at 17:37
  • $\begingroup$ @Max I occasionally blog about these things to make them more accessible. Although I am currently focused more on the algebraic notions of infinite words and products, one thing also to consider is the possibility of putting topologies on the usual algebraic invariants (in a homotopy invariant way) that remember local properties forgotten by weak homotopy type. $\endgroup$ – Jeremy Brazas Aug 8 '18 at 17:42
  • $\begingroup$ @Max A lot of my own work in the past years has been on topological versions of fundamental groups and generalized covering spaces to extend the group $\leftrightarrow$ topology symbiotic relationship to a topological group $\leftrightarrow$ wild topology relationship. The natural quotient topology comes with issues so I introduced this one. $\endgroup$ – Jeremy Brazas Aug 8 '18 at 17:56
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In a sense this is a philosophical question. As you said, algebraic topology usually focusses on "nice spaces" like polyhedra, CW-complexes or manifolds. For one thing, this has historical reasons. For example, homology theory started with simplicial complexes and only over the time turned towards general spaces. On the other hand, if one is interested in effective computations of , say, singular homology groups, one must restrict to spaces making this possible. And again we come to polyhedra, CW-complexes and manifolds.

Many people have thought hard how to shift the border towards more general spaces, and obtained results like Cech (co)homology and Steenrod homology. See for example my answer to https://math.stackexchange.com/q/2807820.

In my opinion the general context of such approaches is shape theory. See the references in my above-mentioned answer and for example

Mardešic, Sibe, and Jack Segal. Shape theory: the inverse system approach. Vol. 26. Elsevier, 1982.

See also https://en.wikipedia.org/wiki/Shape_theory_(mathematics).

The general philosophy to study non-nice spaces $X$ is to approximate them by nice spaces. There are two "dual" methods: To study $X$ via maps from nice spaces to $X$ or to study $X$ via maps from $X$ to nice spaces. The first approach can be denoted as the singular approach; it ends with CW-substitutes for $X$ (roughly speaking, $X$ and a CW-substitute $X'$ admit the "same" maps living on nice spaces). The second approach leads to approximate $X$ by inverse systems of nice spaces which is the essence of shape theory. I do not say that one approach is superior to the other, these are just different points of view. However, I recommend to have a look at the above references to see that shape theory produced quite a number of interesting results and algebraic invariants.

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One reason is that most algebraic invariants are homotopy invariant, meaning the algebraic invariant takes the form of a functor $F:\mathbf{Top} \to \mathcal A$ and that any weak homotopy equivalence $f:X\to Y$ is mapped to an isomorphism $F(f)$ in $\mathcal A$.

But in $\mathbf{Top}$, every space $X$ can be associated with a a CW-complex $X'$ together with a weak equivalence $X' \to X$. (Explicitly $X'$ can be taken to be the geometric realization of the simplicial set $\mathrm{Sing}\,(X)$ of singular simplices.) So by only studying the value of the invariants on CW-complexes, one gets the invariant for any topological space, granted that one can compute such an $X'$ as before for any $X$.

Of course, it just gives a hint on why things are as they are and not on why people did not try go push algebraic topology to non-homotopy invariant functors. Maybe just because it is not consider algebraic topology anymore: this field is supposed to study algebraic objects computed from the general look/shape of a space, and homotopy-deforming is not supposed to change this look/shape. Or maybe we have gone to far into generality when defining topological spaces, and pathological spaces are non-welcomed artefacts: remember that topological spaces is a tentative modelization of the "real" spaces we have around (in physics and other related fields for example); there is no evidence that topological spaces are the right way to go, and some other notions appeared since, like locales (some kind of generalized sober spaces). This last paragraph is only wild guesses on my behalf and it does not necessarily reflect any general opinion.

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    $\begingroup$ Nice answer, but I question if your logic is circular. There are plenty of bona fide homotopy invariants that are not invariant under weak equivalence (e.g. many cohomology theories). The functorial CW replacement deforms $Top$ into a subcategory of, say, compactly generated weak Hausdoff spaces. Clearly there are many spaces which are completely destroyed by this functor. For example it sends the pseudocircle to $S^1$. I'm not sure that restricting to this latter category is what the op is looking to do. $\endgroup$ – Tyrone Aug 6 '18 at 9:35
  • $\begingroup$ @Tyrone What is your definition of homotopy invariant then? I guess it is that homotopic morphisms map to the same morphism between the invariants. But then what is your definition of homotopy? A continuous map $X\times I \to Y$ or a continuous map $X\to Y^I$ ($I$ is the interval here): bear in mind that these two notions are a priori not equivalent when $X$ is not cofibrant. Why favor the cylinder definition over the path one then? The only non biaised version of homotopy invariant functor is to map weak equivalences to isomorphism in my opinion. $\endgroup$ – Pece Aug 6 '18 at 10:06
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    $\begingroup$ So you see my objection? If we only consider what you would call "non biased" functors, then we are forced to define all our invariants on some convenient category, rather than the whole of $Top$. In this case how can we possibly expect them to capture any information about anything outside of the CW category? I think that the reasoning in your comment is more in line with what the op is looking for with his question "Is there a reason why there aren't nice functors from $Top$ to "algebraic" categories ?" than your original post. $\endgroup$ – Tyrone Aug 6 '18 at 10:24
  • $\begingroup$ @Tyrone I'm not sure what is your objection. You can still define your functor in all of $\mathbf{Top}$, for example singular homology is defined without restricting the topological spaces you input. But you should not expect to have more information about a space than the information contained in the invariant of its cofibrant replacement! Maybe I was not clear enough in my post: I was trying to convey why algebraic topologist are fine with requiring their spaces to be nice when needed. Short answer: because all spaces are such up to homotopy. $\endgroup$ – Pece Aug 6 '18 at 10:43
  • $\begingroup$ I understand your point, but as the first comment of Tyrone suggests, going to a weakly equivalent CW complex makes you lose lots of information it seems (there aren't that many CW complexes compared to topological spaces). I also understand your point about generality; but I don't understand the beginning of this paragraph : why couldn't we imagine assigning algebraic objects that aren't homotopy invariant ? I mean why should we expect homotopy to preserve "shapes" ? (E.g. intuitively I wouldn't say $\mathbb{R}^2\setminus\{0\}$ and $S^1$ have the same "shape") $\endgroup$ – Max Aug 6 '18 at 11:25

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