Introduction to Springer theory

(d) Hm−∗(M,M \X), where M is a smooth oriented manifold of dimension m and there exists an embedding X ↪→M as before. (d’) Hm−∗(X) if X itself is a smooth oriented manifold of dimension m. (e) H∗(D•(X), d). Assume there exists an embedding X ↪→M to a smooth manifold M of dimension m. Then we define D•(X) be a chain complex of distributions supported on X. (A distributions of degree k is a continuous linear function φ : Ωm−k c → R defined on the space of (m− k)-forms with compact support.) Now we define HBM ∗ (X), the Borel-Moore homology on X, as any of these notions. Remark. From now on, we follow the notation of [CG], and write H∗ for H BR ∗ and H ord ∗ for H∗.