This is a very general question. I am wondering what homology or cohomology theories have been developed for the analysis of spaces which are not locally connected. Particularly, theories that extend singular or simplicial homology, or Cech cohomology, which is basically all that I am familiar with.

For the area I am interested in, it would be fine to assume that the underlying spaces are locally compact, indeed even compact.

Are there any such theories that have proven more useful over the years? Particularly in finite or one dimensional cases. The only thing I have really heard of in this direction is shape theory.

  • $\begingroup$ "Theories that extend singular or simplicial homology, or Cech cohomology" - is it correct that you want to extend theories living on all locally connnected spaces? Or on polyhedra as "simplicial homology" suggests? $\endgroup$ – Paul Frost Jun 3 '18 at 7:53
  • $\begingroup$ Either! Maybe something that can deal well with inverse limits of polyhedra, which is an extremely broad class of spaces, at least in nice cases. $\endgroup$ – John Samples Jun 4 '18 at 11:36

The general strategy is to start with a theory which is defined for "locally nice spaces". But what is the meaning of "locally nice"? In a sense this is a philosophical question. I wouldn't say that locally connected spaces deserve to be considered as locally nice. At best they are locally nice in dimension $0$, but pathologies can occur also in higher dimensions. In fact, for each $n$ there are spaces which are $LC^n$ (=locally $n$-connected) but not locally $LC^{n+1}$. The standard interpretation of "$X$ is locally nice" seems to be that $X$ is a polyhedron or a CW-complex or an ANR. For (finite) polyhedra and CW-complexes you can compute the homology groups, at least for ordinary homology theories.

All spaces can be approximated by inverse systems of locally nice spaces. In general this will be a more sophisticated approximation than an "inverse limit" approximation, but for compact spaces the latter will do. Approximation is the essential point of shape theory. A nice essay on the "philosophical background" can be found in an old book review by R. Geoghegan https://projecteuclid.org/download/pdf_1/euclid.bams/1183551594 .

This leaves the question which homology theories "behave well" for general spaces. Generally speaking, these are theories which are continuous in a suitable sense, i.e. allow to express the homology groups of a space $X$ by the homology groups of an inverse system of locally nice spaces approximating $X$. Such theories are

  • Cech homology for compact pairs and a certain class of coefficient groups (unfortunately it does not satisfy the exactness axiom for integer cofficients). Spaces are approximated by the nerves of open covers.

  • Steenrod homology (aka Steenrod-Sitnikov homology) for compact metric pairs.

  • Cech cohomology for compact pairs.

  • Alexander Spanier cohomology (which agrees with Cech cohomology on compact pairs)

On finite polyhedra they agree with the standard (singular) theories.

A particular nice feature of these theories is that they support Alexander duality theorems. We have e.g. for a compact $X \subset S^n$: The Steenrod homology group $H^S_k(X)$ is canonically isomorphic to the ordinary cohomology group $H^{n-k-1}(S^n \backslash X)$; the Cech cohomology group $H_C^k(X)$ is canonically isomorphic to the ordinary homology group with compact support $H_{n-k-1}(S^n \backslash X)$.

Concerning Cech homology for integer coefficients see also What is Cech homology?. If $G$ is an abelian group such that Cech homology with coefficients in $G$ is exact, its restriction to compact metric pairs agrees with Steenrod homology with coefficients in $G$.

For ordinary homology theories (i.e. theories satisfying all Eilenberg-Steenrod axioms including the dimension axiom) it is well-known that on finite CW-complexes they are up to isomorphism uniquely determined by their coefficient group. There exist various explicit constructions of ordinary homology theories which extend the "standard homology theory" from finite CW-complexes to more general spaces. Here is a paper by E.G. Sklyarenko http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=inta&paperid=125&option_lang=eng (from 1989, but by no means outdated) where you can find a survey on such approaches. In my personal opinion it is essential for a "well-behaved" homology theory that you have a chance to compute homology groups - and this can only be done using theories compatible with approximating general spaces by CW-complexes or polyhedra. Steenrod homology would therefore be my favorite choice for compact metric spaces. However, we must be aware that it is focussed on the global structure of spaces and ignores their local features. If you want to use homology groups to detect any local characteristics where two spaces look different, then you must choose another theory. Maybe singular homology is adequate for such cases, but there is no general method to effectively compute something.

Generalized (co)homology theories - which are not required to satisfy the dimension axiom - give you some more variety. Examples are K-theory (complex and real), (co)bordism (again complex and real) and stable cohomotopy. However, all these theories have their focus on manifolds or CW-complexes. Extensions to more general spaces exist, but not very much seems to be known. An interesting exception is (complex) K-homology which is the homology theory associated to the BU-spectrum. In the 1970's Brown, Douglas and Fillmore https://www.encyclopediaofmath.org/index.php/Brown-Douglas-Fillmore_theory , https://www.jstor.org/stable/1970999?seq=1#page_scan_tab_contents studied - far from algebraic topology - questions in the context of operator algebras but came upon a homology theory for compact metric spaces which turned out to be an extension of the K-homology theory living on finite CW-complexes. This gave rise to the concept of generalized Steenrod homology theories (see Kaminker and Schochet https://www.jstor.org/stable/1997453?seq=1#page_scan_tab_contents and Kahn, Kaminker and Schochet https://projecteuclid.org/euclid.mmj/1029001885).

Generalized Steenrod homology theories have similar features as ordinary Steenrod homology, in particular they are not adequate to give a useful classification of tree-like continua.

  • $\begingroup$ Thanks! But lc is ultra nice in my setting, e.g. dendroids, tree-like continua and general acyclic curves. Are any of these theories better behaved than others in the one dimensional setting? $\endgroup$ – John Samples Jun 14 '18 at 23:50
  • $\begingroup$ @JohnSamples You are certainly right that being "nice" depends on the setting. I don't know very much about dendroids, tree-like continua etc. As far as I understand, a tree-like continuum can be defined as a continuum homeomorphic to the inverse limit of an inverse sequence of trees. This implies that its Steenrod homology groups are isomorphic to those of a point, so that in some sense they are not very interesting. So what are your options for choosing "good" homology theories? I shall add some remarks to my answer since comments are too limited in space. $\endgroup$ – Paul Frost Jun 16 '18 at 15:47
  • $\begingroup$ Ok, thanks! Btw, you might find it interesting to know that many tree-like continua are also circle-like, which has the same definition with onto bonding maps to the circle. As one might expect, the maps are inessential, i.e. have trivial 'topological winding.'. It may be that ordinary homology is inherently bad at distinguishing these sorts of spaces. What are the most common not-quite-ordinary homology theories that might be good at detecting geometric information in low dimensions? Also, computations would be second to identification or obstruction results to me. $\endgroup$ – John Samples Jun 17 '18 at 2:42
  • $\begingroup$ @JohnSamples If your aim is to find homological invariants of (low dimensional) continua which allow to classify them in a non-trivial way, I would be pessimistic. Intuitively I would think that you need some "custom-built"' invariants beyond homology. But of course, my perception may be wrong. I add more remarks to my answer. By the way, all inverse limits of inverse sequences of finite CW-complexes with inessential bondings have Steenrod homology groups isomorphic to those of a point. $\endgroup$ – Paul Frost Jun 17 '18 at 13:02

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