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This isn't a complete answer, but for example, if you know that $H_T^*(X;\mathbb{Z})$ injects into the cohomology of the fixed point set $X^T$, then (for $G=GL_n(\mathbb{C})$) the canonical map is injective. See Theorem 2.10 and Corollary 2.11 in T. Holm and R. Sjamaar, "Torsion and abelianization in equivariant cohomology." Transform. Groups 13 (2008), no. 3-4, 585–615.
Oh sorry, I read too quickly! The Berkeley Math Dept has an eraser cleaner which presumably was bought outside Asia, but it looks like it is 50 years old, and I've never seen another.
Thanks for your comment, Peter. It's good to know the parts are independently readable. That should make my goals of learning some parts of it, and attracting as many grad student participants as possible, both attainable!
