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Bott & Tu in Differential forms in Algebraic Topology write in Remark 5.17, pg.52

The two Poincare duals of a compact orientated submanifold correspond to two homology theories - closed and compact homology. Closed homology has now fallen into disuse, while compact homology is known these days as the homology of singular chains...in general Poincare duality sets up an ismorphism between closed homology and de Rham cohomology, and between compact homology and compact de Rham cohomology.

What is closed homology, and why did it fall into disuse? Did a better alternative turn up?

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    $\begingroup$ The book title should be Differential Forms in Algebraic Topology. $\endgroup$ – AaronS Jul 8 '18 at 13:20
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    $\begingroup$ I think it's just Borel-Moore homology. And it didn't fall into disuse... $\endgroup$ – user40276 Jul 8 '18 at 14:18
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I think that closed homology is another name for Borel-Moore homology, which certainly has the isomorphism you suggest: https://en.m.wikipedia.org/wiki/Borel-Moore_homology

It still is used quite frequently.

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    $\begingroup$ Yep. Another common name for it is "homology with closed supports". $\endgroup$ – Dan Petersen Jul 8 '18 at 14:36
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    $\begingroup$ Provided the space $X$ is tame at infinity, the Borel-Moore homology is also isomorphic to the relative homology $H_*(\overline X,\{\infty\})$, where $\overline X=X\cup\{\infty\}$ denotes the one-point compactification of $X$. $\endgroup$ – André Henriques Jul 8 '18 at 21:33

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