Atiyah–Segal completion theorem

Atiyah–Segal completion theorem 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 {displaystyle pi colon Xtimes EGto X} induces an isomorphism of prorings {displaystyle pi ^{*}colon K_{G}^{*}(X)_{widehat {I,}}to K^{*}((Xtimes 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 a point, the theorem specializes to give an isomorphism {displaystyle 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 References ^ Greenlees, J.P.C. (1996). "An introduction to equivariant K-theory.". CBMS Regional Conference Series. Equivariant homotopy and cohomology theory. Vol. 91. Published for the Conference Board of the Mathematical Sciences, Washington, DC. pp. 143–152. ^ Atiyah, M.F. (1961). "Characters and cohomology of finite groups". Publications Mathématiques de l'IHÉS. 9 (1): 23–64. doi:10.1007/BF02698718. S2CID 54764252. ^ Atiyah, M.F.; Segal, G.B. (1969). "Equivariant K-theory and completion" (PDF). Journal of Differential Geometry. 3 (1–2): 1–18. doi:10.4310/jdg/1214428815. Retrieved 2008-06-19. ^ Jackowski, S. (1985). "Families of subgroups and completion". J. Pure Appl. Algebra. 37 (2): 167–179. doi:10.1016/0022-4049(85)90094-5. ^ Adams, J.F.; Haeberly, J.P.; Jackowski, S.; May, J.P. (1988). "A generalization of the Atiyah-Segal Completion Theorem". Topology. 27 (1): 1–6. doi:10.1016/0040-9383(88)90002-X.

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