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Abstract: Let G be a topological group and let G act on a space X. What
can one deduce about the G-fixed point space X^G and the orbit
space X/G from information about X? I will explain classical
cohomological results, P.A. Smith theory and the Connor conjecture,
that give quite remarkable answers to these questions. I will use
modern proofs of these results to describe equivariant cohomology
theory, which means different things to different people. The proof
of the Connor conjecture will lead us directly to the idea that
cohomology should be graded on the real representation ring of G
rather than just on the integers. In turn, this will lead us to a
remarkable relationship between Mackey functors in algebra and
equivariant stable homotopy groups. All concepts will be defined.
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