Talk:Borel–Moore homology

Results

The basic results in http://mysite.science.uottawa.ca/jlema072/Borel-Moore.pdf should be discussed. — Preceding unsigned comment added by 161.98.8.4 (talk) 22:12, 9 August 2017 (UTC)

Computations

• Complement of hyperplane arrangement in $\mathbb {P} ^{2}$. The cohomology of the closed subset can be computed using mixed hodge theory fairly easily (check out https://www3.nd.edu/~lnicolae/hodge_normcross.pdf). This computation should be partly done on the hodge structure page
• Take the complement of a K3 surface by a generic degree 1 hyperplane section. This requires more tools, such as logarithmic cohomology. Username6330 (talk) 03:18, 10 October 2017 (UTC)

Fundamental classes and cycle classes

There is a cryptic statement in the introduction: « a closed oriented submanifold defines a class in Borel–Moore homology, but not in ordinary homology unless the submanifold is compact ».

1 "closed manifold" usually means "compact without boundary". Thus if one means that the submanifold must be closed as a subset of the ambient manifold, one should change the statement.

2 For each oriented manifold there is "fundamental class" in BM homology, a class in the top-dimensional rank-1 homology group of a manifold, which restricts to an orientation class at each point. And we can define a "cycle class" for a submanifold other than the ambient manifold itself, by pushing forward a fundamental class of the submanifold, but this is not clear from the formulation here. For instance a proper submanifold of euclidean space will have its fundamental class pushforward to 0, as the Borel-Moore homology of euclidean space is 0 outside top dimension; so it takes a bit of justification to convince someone that this is useful, and there is none in the article.

Thus i would change the statement to « an oriented manifold without boundary has a fundamental class in Borel-Moore but not in ordinary homology, and the fundamental class of a submanifold which is closed as a subset can be pushed forward to give a cycle class in the Borel-Moore homology of its ambient manifold -intuitively this is the cycle defined by the not-necessarily-compact submanifold, while only compact (nonzero) cycles exist in ordinary homology ». If you find this too long-winded you can delete the part from the dash on. Plm203 (talk) 18:01, 25 July 2023 (UTC)