I am reading the paper of Fulton and Lazarsfeld on the connectivity of degeneracy loci of morphisms of vector bundles, but there is a comment in the article that I don't quite understand.

Let $G$ be a smooth complex projective variety of dimension $r$ and let $Y \subseteq G$ be a closed algebraic subset. How do I prove that there is an injection (I assume it will be an isomorphism?) in singular cohomology $$ H^1(G,Y;\mathbb Q) \overset{\simeq ?}{\longrightarrow} H_{2r-1}(G-Y;\mathbb Q) $$ I actually don't know what they meant by Lefschetz duality since I am using Hatcher to learn algebraic topology, and in that book they mention that the isomorphism $H^k(M;R) \simeq H_{r-k}(M, \partial M;R)$ (where $M$ is compact and orientable) is sometimes called "Lefschetz duality" (page 254, Theorem 3.43) but I don't know what Fulton & Lazarsfeld mean by using Lefschetz duality in this context. The theorem in Hatcher works also with boundaries of smooth compact submanifolds of codimension $1$ but my closed algebraic subset $Y$ is not a smooth $(r-1)$-manifold (in fact, it never has codimension $1$ in $G$ as a real manifold).

Another way to obtain this injection, according to the paper, is to write down the exact sequence of low degree terms of the Zeeman spectral sequence. I couldn't find it anywhere (the paper of Zeeman explains the spectral sequence, but I don't have enough experience with spectral sequences to write it down myself), so it would be nice to at least see it and wonder if I can work out the injection.