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
- [math]\displaystyle{ \pi\colon X\times EG\to X }[/math]
induces an isomorphism of prorings
- [math]\displaystyle{ \pi^*\colon K_G^*(X)_{\widehat{I\,}} \to K^*((X\times EG)/G). }[/math]
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 [math]\displaystyle{ K^*(BG)\cong R(G)_{\widehat{I\,}} }[/math] 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
References
- ↑ Greenlees, J.P.C. (1996). "An introduction to equivariant K-theory.". Equivariant homotopy and cohomology theory. 91. Published for the Conference Board of the Mathematical Sciences, Washington, DC. pp. 143–152.
- ↑ "Characters and cohomology of finite groups". Publications Mathématiques de l'IHÉS 9 (1): 23–64. 1961. doi:10.1007/BF02698718. http://www.numdam.org/item/PMIHES_1961__9__23_0/.
- ↑ "Equivariant K-theory and completion". Journal of Differential Geometry 3 (1–2): 1–18. 1969. doi:10.4310/jdg/1214428815. http://www.maths.ed.ac.uk/~aar/papers/atiyahsegal1.pdf. 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. (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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