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I have sometimes wondered about the following:

(1) Who was the first articulate that in dealing with $G$-equivariant cohomology theories ($G$ a finite group or a compact Lie group), it is best to work in an $RO(G)$-graded context?

(2) Who was the first to realize that the correct set up for equivariant stable homotopy was to work in a complete universe?

(3) At what time did these ideas first come to the surface? The seventies? The eighties? How much do they predate the Segal conjecture?

(Maybe I should thank Peter May in advance?)

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Suggestion: add the ho.history-and-overview tag – Yemon Choi Sep 3 '12 at 23:19
Thanks. I did it. – John Klein Sep 3 '12 at 23:41
I think Bredon was the father of equivariant cohomology. – Benjamin Steinberg Sep 4 '12 at 0:42
But did Bredon introduce the idea of grading by $RO(G)$? – John Klein Sep 4 '12 at 0:43
I also want to mention Kosniowski: Equivariant Cohomology and Stable Cohomotopy - this paper from 1974 also introduces RO(G)-graded cohomology theories and attributes the idea to Graeme Segal. – Lennart Meier Nov 5 '15 at 13:31
up vote 17 down vote accepted

Well if you insist John :) (1) The first explicit formulation I know of is in the nice paper: Klaus Wirthmuller. Equivariant homology and duality. Manuscripta Math. 11 (1974), 373–390. He writes: "The ideas developed here partly originate from suggestions made by T. tom Dieck, who introduced me to the subject." They were thinking about equivariant Poincar\'e duality and about equivariant cobordism, which make $RO(G)$ grading inevitable. (2) That may be tom Dieck and may be me. I'm honestly not sure. I was explicitly using universes nonequivariantly in 1972, with good reason. I think tom Dieck may have at least implicitly used complete universes in his work on cobordism. Certainly I was using complete $G$-universes by some time around or before 1974-75. The question is confused by the fact that tom Dieck was in Chicago lecturing on equivariant things that year. See Tammo tom Dieck. The Burnside ring and equivariant stable homotopy. Lecture notes by Michael C. Bix. Department of Mathematics, University of Chicago, Chicago, Ill., 1975. (3) The Segal conjecture in its simplest form dates from 1970. However, early work on it was not based on equivariant stable homotopy theory, let alone $RO(G)$ grading. As late as 1983, Frank Adams wrote a paper ``Graeme Segal's Burnside ring conjecture'' in which he barely mentioned equivariant cohomotopy, and then a bit skeptically. There is a 1982 paper "Classifying $G$-spaces and the Segal conjecture'' by Lewis, McClure, and myself that proves the equivalence of the nonequivariant and equivariant versions of the Segal conjecture, before Carlsson's proof.

It is worth emphasizing that, in a sense, $RO(G)$-grading has no philosophical justification. Logically, grading should be on the equivariant Picard group which is considerably larger (at least under space level equivalence) or on the stable equivalence classes of representation spheres (and their negatives), which is considerably smaller.

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I just saw the comment about Bredon. He introduced ordinary $\mathbf{Z}$-graded cohomology theories, in 1966 I believe. (I heard him talk about it that year). Most people outside core algebraic topology mean Borel cohomology, not Bredon cohomology, when they say "equivariant cohomology'', and Borel certainly came earlier. Of course, neither considered $RO(G)$-grading. – Peter May Sep 4 '12 at 1:11
Thanks Peter. You've given me a fairly thorough answer. – John Klein Sep 4 '12 at 1:28

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