Atiyah–Segal completion theorem explained

The Atiyah–Segal completion theorem is a theorem in mathematics about equivariant K-theory in homotopy theory. Let G be a compact Lie group and let X be a G-CW-complex. The theorem then states that the projection map

\pi\colon X x EG\to X

induces an isomorphism of prorings

\pi ^{*\colon} K ^{*(X)} _{\widehat{I} } \to K^*((X\times EG)/G).

Here, the induced map has as domain the completion of the G-equivariant K-theory of X with respect to I, where I denotes the augmentation ideal of the representation ring of G.

In the special case of X being a point, the theorem specializes to give an isomorphism

K ^{*(BG)\cong} R(G) _{\widehat{I} } between the K-theory of the classifying space of G and the completion of the representation ring.

The theorem can be interpreted as giving a comparison between the geometrical process of taking the homotopy quotient of a G-space, by making the action free before passing to the quotient, and the algebraic process of completing with respect to an ideal.^[1]

The theorem was first proved for finite groups by Michael Atiyah in 1961,^[2] and a proof of the general case was published by Atiyah together with Graeme Segal in 1969.^[3] Different proofs have since appeared generalizing the theorem to completion with respect to families of subgroups.^{[4] [5]} The corresponding statement for algebraic K-theory was proven by Alexander Merkurjev, holding in the case that the group is algebraic over the complex numbers.

See also

• Segal conjecture

Notes and References

1. Greenlees, J.P.C. . 1996 . An introduction to equivariant K-theory. . Equivariant homotopy and cohomology theory . CBMS Regional Conference Series . 91 . 143–152 . Published for the Conference Board of the Mathematical Sciences, Washington, DC.
2. Atiyah. M.F.. Michael Atiyah . 1961 . Characters and cohomology of finite groups . Publications Mathématiques de l'IHÉS. 9 . 1 . 23–64 . 10.1007/BF02698718. 54764252.
3. Atiyah. M.F.. Michael Atiyah . Segal. G.B.. Graeme Segal . 1969 . Equivariant K-theory and completion . . 3 . 1–2. 1–18 . 2008-06-19. 10.4310/jdg/1214428815. free .
4. Jackowski . S. . 1985 . Families of subgroups and completion . J. Pure Appl. Algebra . 37 . 2 . 167–179 . 10.1016/0022-4049(85)90094-5. free .
5. Adams . J.F. . Haeberly. J.P.. Jackowski. S. . May. J.P.. J. Peter May . 1988 . A generalization of the Atiyah-Segal Completion Theorem . Topology . 27 . 1 . 1–6 . 10.1016/0040-9383(88)90002-X. free.