Talk:Intersection homology

I think intersection homology would be a more standard page title here. Charles Matthews 15:34, 6 Sep 2004 (UTC)

I agree! It should say "Intersection Homology". That is also what Goresky and MacPherson use. (129.206.26.185 00:11, 13 January 2007 (UTC))

History
Somebody should write about the history of IC. References? Mhym 20:08, 13 April 2006 (UTC)

Disputed
The statement that IC_X \otimes IC_X is the Verdier dualizing object D_X is false; there is a canonical map IC_X \otimes IC_X --> D_X which induces a dual pairing, but it is almost never an isomorphism. It's not immediately clear how to fix the original author's intended development without a major rewrite. TCBraden 18:46, 2 May 2006 (UTC)

Q coefficients
I think that the Poincare pairing between i-dimensional and i-codimensional cycles is defined and perfect over the integers (up to boundaries). With intersection homology you lose perfection over the integers, but retain it over every field. All the astonishing applications of IH in representation theory definitely require rational (or bigger) coefficients, at least so far, but is there some reason to fret about it in this article, in its current version? Even Deligne's definition makes sense over Z. Changbao 23:13, 3 April 2007 (UTC)


 * The pairing over the integers is not perfect if the homology groups have torsion. R.e.b. 15:16, 4 April 2007 (UTC)

Mistake in definition
In the Deligne definition with the iterated pushforwards and truncations, the locally constant sheaf with which you begin (here it's been taken constant) has to be shifted by n. Cf. Goresky Macpherson Inventiones II for instance (and also otherwise you have a contradiction with the calculation of the IC of the cone over an elliptic curve later down the page) 2001:861:388A:CAE0:5DB3:FF0E:1C56:3372 (talk) 15:56, 5 March 2024 (UTC)