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The wikipedia page on Borel-Moore homology claims to give several definitions of it, all of which are supposed to coincide for those spaces $X$ which are homotopy equivalent to a finite CW complex and which admit a sufficiently nice embedding into a smooth manifold.

My question concerns one of the definitions on that page: it is asserted that if $X$ is a space and $\bar X$ is some compactification, then there is a definition of Borel-Moore homology given by $$ H_*^{\text{BM}}(X) := H_*(\bar X,\bar X \setminus X) . $$ The entry additionally requires that the pair $(\bar X,X)$ should be a CW pair for this to work.

It then goes on to assert that $\bar X = X^+$, the one-point compactification, will suffice for this.

Frankly, I don't see how $(X^+,X)$ will form a CW pair in most instances. For example suppose $X = \Bbb Z$ is the set of integers which is considered as a discrete space. Then in this instance $(X^+,X)$ certainly fails to be a CW pair.

Another example: $X = S^1 \setminus \ast$ with one point compactification $S^1$. Then $(S^1,S^1 \setminus \ast)$ is not a CW pair either.

My Questions:

(1) For what class of spaces $X$ does $H_*(X^+,+)$ coincide with definition of Borel-Moore homology given by locally finite chains?

(2) Does the wikipedia entry contain a mistaken assumption? Perhaps we do not need to assume that $(X^+,X)$ can be given the structure of a CW pair?

Note: if $X = \Bbb Z$ and if we use the above as a definition of Borel-Moore homology then $ H_0^{\text{BM}}(X) $ is a free abelian group whose generators are given by the underlying set of $\pi_0(X) = \Bbb Z$. This is clearly the wrong answer: it should be the countably infinite cartesian product of copies of the integers indexed over $X$ instead (using, say, the definition of Borel-Moore homology given by locally finite chains).

Another Question:

(3) Is there a definition the above kind (using compactifications) which will work (i.e., coincide with the locally finite chain definition) for $X = \Bbb Z$?

(I suspect not, since ordinary singular homology in degree zero is always free abelian.)

Incidentally, later in the page it lists the main variance property: Borel-Moore homology is supposed to be covariant with respect to proper maps. The page gives a proof using the above definition. But since the above definition doesn't work in general, I don't see how this is really supposed to be a proof.

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Maybe there is some language confusion here, but doesn't "finite CW-complex" mean a CW-complex with finitely many cells (that is, both finite-dimensional and finite type)? So how would $X=\mathbb{Z}$ (with the discrete topology) fit in that context? –  Alex Suciu Jul 29 '13 at 23:50
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What is your exact question concerning BM-homology? –  André Henriques Jul 29 '13 at 23:50
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@JohnKlein, if you are looking for a good treatment of locally finite singular or cellular homology, then obviously this Wikipedia page is not a very good source. I recommend instead taking a look at Chapters 3, 4 and Appendix A of Ranicki's Ends of Complexes (maths.ed.ac.uk/~aar/books/ends.pdf). –  Igor Khavkine Jul 30 '13 at 0:42
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Thanks Igor. I'm familiar with (Hughes' and) Ranicki's work. But they don't give a comparison of the various definitions. They are content to give two definitions: one the locally finite singular theory and the other a cellular version. –  John Klein Jul 30 '13 at 0:43
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@Ricardo and Scott: I did what you suggested. –  John Klein Jul 30 '13 at 1:07

1 Answer 1

up vote 6 down vote accepted

In Chriss & Ginzburg, it's asserted that the Borel-Moore homology is given by $H_*(\bar X, \bar X\setminus X)$ if $\bar X$ is the one-point compactification or any compactification such that the inclusion is cellular. They very pointedly do not assume that the inclusion of $X$ into $X^+$ is cellular. They reference Bredon but don't cite a particular chapter.

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Ben: I guess you mean that $\bar X \setminus X \to \bar X$ should be cellular. That's a reasonable condition, provided a cell structure on $\bar X$ is given. It seems then that the wikipedia entry has an error then. –  John Klein Jul 30 '13 at 12:53
    
Ben: I just checked Chriss & Ginzburg. It is clear now that the wikipedia entry has a mistake. –  John Klein Jul 30 '13 at 13:26
    
I took the liberty to correct the wikipedia page. –  Ricardo Andrade Jul 30 '13 at 22:56
    
Thanks Ricardo. –  John Klein Jul 31 '13 at 1:44
    
Another reference (with Google Books preview available) is "Representation Theories and Algebraic Geometry" books.google.pl/books?id=Q0Nyon7-qDMC , pp. 129-130 –  Michał Kukieła Aug 16 '13 at 15:50

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