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Absolute neighborhood retracts (ANRs) are topological spaces $X$ which, whenever $i\colon X\to Y$ is an embedding into a normal topological space $Y$, there exists a neighborhood $U$ of $i(X)$ in $Y$ and a retraction of $U$ onto $i(X)$. They were invented by Borsuk in 1932 (Über eine Klasse von lokal zusammenhängenden Räumen, Fundamenta Mathematicae 19 (1), p. 220-242) and have been the object of a lot of developments from 1930 to the 60s (Hu's monograph on the subject dates from 1965), being a central subject in combinatorial topology.

The discovery that these spaces had good topological (local connectedness), homological (finiteness in the compact case) and even homotopical properties must have been a strong impetus for the developement of the theory. Also, they probably played some role in the discovery of the homotopy extension property (it is easy to extend homotopies whose source is a normal space and target an ANR) and of cofibrations.

I have the impression that this more or less gradually stopped being so in the 70s: a basic MathScinet search does not refer that many recent papers, although they seem to be used as an important tool in some recent works (a colleague pointed to me those of Steve Ferry).

My question (which does not want to be subjective nor argumentative) is the following: what is the importance of this notion in modern developments of algebraic topology?

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I study model categories, and the notion of cofibration is central to that field. So for me, NDRs, ANRs, etc are interesting because of their categorical properties (e.g. how they behave with respect to pushouts). At least once in my work I've added a hypothesis like "assume the cofibrations satisfy..." and then said this was motivated by analogy to certain maps in $Top$. A model category satisfying this hypothesis can be much easier to work with and then you just check if the examples of interest also satisfy that property. I know that's vague, but hopefully it gives some idea. –  David White Mar 4 '13 at 15:45
ANRs are a key tool in this paper in geometric group theory: ams.org/journals/jams/1991-04-03/S0894-0347-1991-1096169-1/… –  Ian Agol Mar 4 '13 at 19:09
I remember, maybe around 1980, Frank Quinn at the beginning of a lecture at the Cornell Topology Festival mentioned ANRs and then stopped, looked around at the audience, and said in a slightly exaggerated southern accent, "Oh, I forgot, y'all don't know about ANRs up here, do you?" –  Tom Goodwillie Mar 4 '13 at 19:10
I see ANR's all the time in modern geometric topology literature. Pretty much every book uses this notion extensively (except those about low dimensions), e.g. search for the term ANR in Steve Ferry's notes math.rutgers.edu/~sferry/ps/geotop.pdf, or in "Ends of complexes" by Hughes-Ranicki maths.ed.ac.uk/~aar/books/ends.pdf. –  Igor Belegradek Mar 4 '13 at 19:21
I have read the title as "The role of the agence nationale de la recherche in modern topology". –  Jonathan Chiche Mar 4 '13 at 20:15
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up vote 22 down vote accepted

Another reason you might not see the word ANR these days is that compact finite-dimensional spaces are ANRs if and only if they are locally contractible. Thus, "finite-dimensional and local contractible" can replace ANR in the statement of a theorem (and might help the result appeal to a wider audience).

In comparison geometry, for instance, the existence of a contractibility function takes the place of the ANR condition.

Borsuk conjectured that compact ANRs should have the homotopy types of finite simplicial complexes. Chapman and West proved that they even have preferred simple-homotopy types. This is part of the "topological invariance of torsion" package and is quite a striking result. Every compact, finite-dimensional, locally contractible space has a preferred finite combinatorial structure that is well-defined up to (even local!) simple-homotopy moves.

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Welcome to MO, Steve! –  Ryan Budney Mar 5 '13 at 4:54
Thanks a lot! I think it's fair to accept your answer. –  ACL Mar 7 '13 at 13:14
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I think the answer has more to do with the psychology of mathematicans as a culture than with actual mathematical facts.

I was not alive during the period where ANRs were mentioned in the topology literature but I've read quite a few early topology papers and also noticed before the 60's people couldn't seem to not mention them, and afterwards they were almost never mentioned.

I think this is mostly due to the more formal side of algebraic topology, with model categories. With the terminology cofibration one could largely avoid talking about ANRs and regular neighborhoods. You of course could continue to talk about those things but if you're attempting to write something short and concise with as few confusing side-roads as possible, you would omit it.

So fairly quickly people realized they didn't need to talk about ANRs. I think this kind of thing happens fairly often in mathematics, especially when the definition of a concept maybe slightly misses the mark of what you're aiming for, or if it isn't quite as general as you really need. Terminology like this cycles in and out of mathematics fairly frequently.

You could frame this in terms of the long-term survivability of a mathematical concept -- math verbiage evolution. The flaw in ANRs is they did not anticipate that point-set foundations would become less of a focus of topology, that the field would move on and become more scaleable.

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Does "more scaleable" mean more axiomatic? But I don't know any nontrivial model category where all objects, or all (co)fibrant objects are ANRs. (One problem is that the cone over a non-compact space is non-metrizable.) So I don't see how any talk about ANRs (or even regular neighborhoods) could be made implicit by cofibrations. If your point is that using ANRs looks dated in some AT textbooks, it's a question of presentation. The essential feature of ANRs is, of course, that they include topological manifolds and are similar enough to them, but more easily manageable. –  Sergey Melikhov Mar 5 '13 at 3:17
More scaleable meaning applicable in situations where you're not dealing with topological spaces -- applicable in a wider-variety of contexts. It isn't a question of being "dated" or not, it's a question of breadth of applicability. –  Ryan Budney Mar 5 '13 at 3:48
Also, by "topology literature" and "early topology papers" you probably mean algebraic topology? Things like Freedman's proof of the topological 4D Poincare Conjecture, Quinn's proof of the Annulus Conjecture, and Edwards and Cannon's proof of Milnor's Double Suspension Conjecture are very much about ANRs. For the record, these include 2 Annals papers from 1979, and a 1986 Fields medal; a further 1975 Annals paper mentions "ANRs" in its title (it's the main ingredient of West's proof that ANRs have finite types). –  Sergey Melikhov Mar 5 '13 at 4:03
Breadth of applicability is very good; I'm all for model categories (and homotopy type theory). I just don't see what this all has to do with ANRs (and topological manifolds). As you explain, ANRs are not really needed to do homotopy theory; on the other hand, model categories haven't yet helped anyone to do ANRs (and hence topological manifolds), AFAIK. –  Sergey Melikhov Mar 5 '13 at 4:15
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ANRs are (and have always been) irrelevant as long as homotopy-invariant properties of spaces homotopy equivalent to CW-complexes are concerned. But modern algebraic topologists do not seem to be really interested in (or anyway have real tools to deal with) more general spaces AFAIK. (Of course, "general nonsense" like simplicial model categories works for general spaces, but if you are using any invariants like homotopy groups or singular (co)homology theories to get substantinal results that do not mention those invariants, you'll probably need theorems such as Whitehead's - which means restricting to spaces homotopic to CW-complexes.)

Shape theory did go beyond spaces homotopic to CW-complexes. But being an ANR is not a shape-invariant property. It is an invariant of local shape (which Ferry, Quinn, Hughes and their collaborators do touch upon in their works) and indeed Quinn once wrote an expository paper on "Local algebraic topology". I don't think these "local" developments have ever been of interest for (mainstream) algebraic topology, but they have very good applications in geometric topology so are usually associated with the latter.

This area of geometric topology, where ANRs and topological manifolds natuarally belong, has been steadily falling out of fashion with younger generations (since the 80s I would say, not 60s), apparently because it's tough enough, but not nearly as attractive for an outsider as knots, say. That might as well be a problem of the generations rather than a "flaw" in ANRs.

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How about ANR homology manifolds?
See http://www.maths.ed.ac.uk/~aar/homology/tophom.pdf for an important article on the subject.

If I understand correctly, people expect (or know?) that these ANR homology manifold have transitive homeomorphism groups. The possible local models are indexed by the integers, and the value 0 corresponds to $\mathbb R^n$, i.e., to the notion of topological manifold.

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In this context, I think ANR acts as a placeholder niceness assumption. Yes, it is expected but not known that they are homogeneous. And, yes, one expects that their local homeomorphism type is determined by two integers, the dimension and Quinn invariant (in $1+8\mathbb Z$, 1 corresponds to $\mathbb R^n$). The substitute Bing-Borsuk conjecture is that homogeneous ANRs are built from these charts. You could take that as saying that general ANRs are not interesting. –  Ben Wieland Mar 5 '13 at 5:13
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