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In the course of filling in my map of non-pathological topology, I'd like to understand the class of ANRs (Absolute Neighborhood Retracts) as a sort of "neighborhood" of the class of CW complexes. This seems warranted by some of the nice properties of ANRs:

  • Every ANR has the homotopy type of a CW complex.

  • Every ANR is locally contractible, and as a partial converse, any locally contractible finite-dimensional metric space is an ANR.

But there are also important infinite-dimensional examples of ANR's:

  • The Hilbert cube is an ANR.

  • Many function spaces are ANRs.

This leaves me with some

Questions:

  1. What is a good example of a finite-dimensional ANR which is not a CW complex?

  2. Are (finite-dimensional) ANRs an appropriate setting to study either (a) fractals (wikipedia seems to define a fractal to be a subset of Euclidean space whose topological and Hausdorff dimensions differ) or (b) the limit sets of dynamical systems on CW complexes? I think my sense is that both (a) and (b) are generally wilder than ANRs, but I'm not really sure -- perhaps there's some overlap but no strict containments?

  3. Do ANRs admit some kind of "generalized cell structure" like CW complexes do? Or is there some other sense in which ANRs can be "classified"? Is there at least a "classification" of what ANRs can look like locally?

Less precisely, my feeling is that when somebody says "Let $X$ be a CW complex", I sort of know what they mean. But when somebody says "Let $X$ be an ANR", I don't -- I don't know what to think of as a "typical example", nor do I know what kinds of "typical pathologies" to watch out for. It would be nice if there were a book out there entirely devoted to the topology of ANRs but surprisingly I haven't been able to find one . I found a book Theory of Retracts by Sze-Tsen Hu, but I haven't yet found an example in it of a finite-dimensional ANR which is not a CW complex.

EDIT:

  • Another reference is Borsuk's The Theory of Retracts. This contains more examples in the later chapters, though I'm still struggling to piece together a coherent picture of the diversity of ANRs.

  • An important piece of context regarding (1): according to Thm V.10.1 of Borsuk, the (compact) finite-dimensional ANRs coincide with the retracts of (finite) polyhedra. Thus in finite dimensions, the question is: How wild can an idempotent on a polyhedron be?.

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    $\begingroup$ A fractal is typically not locally contractible. The Cantor set is not locally contractible. Also, direct from definition: an open neighborhood of the Cantor set in the real line has countably many components, hence has collapsed some. (But the Koch snowflake is a fractal embedding of the circle.) . . . I believe that Bing's Dogbone space is an ANR. But it's very close to a CW complex: its product with $\mathbb R$ is $\mathbb R^4$. $\endgroup$
    – Ben Wieland
    Commented Jun 28, 2019 at 20:21
  • $\begingroup$ @BenWieland Point well taken on fractals. And wow, the dogbone space seems complicated! I was hoping there might be a simpler example, especially because being an ANR is a local property (and the dogbone space seems like it was cooked up to have interesting global properties). $\endgroup$
    – Tim Campion
    Commented Jun 29, 2019 at 16:33
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    $\begingroup$ Bing contracts infinitely many tame arcs. What if we take a single wild arc in $\mathbb R^3$ and contract it to a point? Is the quotient space an ENR but not a CW complex? I think that's what "Product of Euclidean Spaces Modulo an Arc" is about. Probably start with its references. $\endgroup$
    – Ben Wieland
    Commented Jun 29, 2019 at 17:21
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    $\begingroup$ The simplest example I know is the subset of the plane which is the union of a sequence of segments $s_n$ of length $1/n$ all meeting at their common end-point. It is an ANR but not homeomorphic to a CW complex. $\endgroup$
    – Misha
    Commented Jun 29, 2019 at 18:22
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    $\begingroup$ It is not representative (too simple). $\endgroup$
    – Misha
    Commented Jun 29, 2019 at 18:44

3 Answers 3

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I find Cauty's result a good crutch to think about ANRs: A metrizable space is an ANR if and only if every open subset has the homotopy type of a CW complex.

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  • $\begingroup$ The more I think about it, the more I think this is a great answer -- I'm somewhat used to thinking of "having the homotopy type of a CW complex" as a reasonable "niceness" property for a space, and this kind of says that being an ANR is probably "what I really want" when using this property. I should be careful though -- I think I've been told that some pretty basic infinite CW complexes like $\mathbb R^\infty = \cup_n \mathbb R^n$ are not metrizable. $\endgroup$
    – Tim Campion
    Commented Feb 20, 2021 at 15:44
  • $\begingroup$ Having the homotopy type of a CW complex can hardly be verified in non-artificial examples; being a retract of a relatively open subset of a convex set in a normed (or Frechet) space can. Concerning metrizability, note that the usual definition of ANR spaces requires metrizability - at least, most of the results mentioned in the replies rely on this definition. Of course, there are various generalizations, but if you mean one of those you should specify them expllicitly. $\endgroup$
    – Martin Väth
    Commented Feb 20, 2021 at 17:26
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Do you know about ANR homology manifolds, and the Quinn invariant?

I like to think of the class of ANR homology manifolds as some kind of "idempotent completion" of the class of topological manifolds.

See e.g. the corollary on page 3 of

J. Bryant, S. Ferry, W. Mio, S. Weinberger, Topology of Homology Manifolds, Annals of Mathematics 143 No. 3 (1996) pp 435–467, doi:10.2307/2118532 (free .ps version)

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    $\begingroup$ I was also thinking about these, but is it known for sure they're not CW complexes (though it would be hard to see how they would be)? Incidentally, not sure if it helps with the original question, but Bryant and Ferry now have homogeneous versions of these, so ANRs can, in a sense, be weird in a uniform way. $\endgroup$
    – Greg Friedman
    Commented Jul 4, 2019 at 6:59
  • $\begingroup$ Thanks, this is really cool! It still all seems a bit mysterious; I wish there were some more easy-to-describe examples. $\endgroup$
    – Tim Campion
    Commented Jul 4, 2019 at 21:38
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    $\begingroup$ @GregFriedman They aren't CW cxs because the Quinn invariant is local. One manifold patch means that the whole (connected) thing is a manifold. In a finite dimensional CW complex has an open subset of a top dimensional cell is a manifold. $\endgroup$
    – Ben Wieland
    Commented Jul 6, 2019 at 16:42
  • $\begingroup$ @BenWieland Good point. $\endgroup$
    – Greg Friedman
    Commented Jul 7, 2019 at 6:26
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For Euclidean neighborhood retracts, there is the nice characterization of being locally connected (and locally compact). Unfortunately, in the infinite-dimensional case, locally connectedness is necessary but only "almost" sufficient, though there is the nice theorem that being an ANR is a local property.

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