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In "$K$-homology of universal spaces and local cohomology of the representation ring" Greenlees proves that $$ K_i(BG_+) \simeq H_J^i(R(G)) $$ for $i=0,1$, where $J$ is the augmentation ideal and $H_J^i(R(G))$ refers to local cohomology. In fact Greenlees proves something more general using equivariant homotopy theory (for which the above is a special case), and this more general object is also studied in Chapter 18 of Greenlees and May's "Generalized Tate Cohomology"

Equivalently, one can use Anderson's universal coefficient theorem for $K$-theory (see here). Anderson shows that if $G$ is even a compact connected Lie group that $KU_1(BG) = 0$ and $$KU_0BG = \operatorname{lim}_n\operatorname{Hom}(R(G)/J^n,\mathbb{Z}).$$

There is a discussion of the relationship between the two results in Remark 2.8 of "Topological K-(co)homology of classifying spaces of discrete groups", by Joachim and Lück (which also contains more results along these lines).

In "$K$-homology of universal spaces and local cohomology of the representation ring" Greenlees proves that $$ K_i(BG_+) \simeq H_J^i(R(G)) $$ for $i=0,1$, where $J$ is the augmentation ideal and $H_J^i(R(G))$ refers to local cohomology. In fact Greenlees proves something more general using equivariant homotopy theory (for which the above is a special case), and this more general object is also studied in Chapter 18 of Greenlees and May's "Generalized Tate Cohomology"

Equivalently, one can use Anderson's universal coefficient theorem for $K$-theory (see here). Anderson shows that if $G$ is even a compact connected Lie group that $KU_1(BG) = 0$ and $$KU_0BG = \operatorname{lim}_n\operatorname{Hom}(R(G)/J^n,\mathbb{Z}).$$

There is a discussion of the relationship between the two results in Remark 2.8 of "Topological $K$-(co)homology of classifying spaces of discrete groups", by Joachim and Lück (which also contains more results along these lines).

In "$K$-homology of universal spaces and local cohomology of the representation ring" Greenlees proves that $$ K_i(BG_+) \simeq H_J^i(R(G)) $$ where $J$ is the augmentation ideal and $H_J^i(R(G))$ refers to local cohomology. In fact Greenlees proves something more general using equivariant homotopy theory (for which the above is a special case), and this more general object is also studied in Chapter 18 of Greenlees and May's "Generalized Tate Cohomology"

Equivalently, one can use Anderson's universal coefficient theorem for $K$-theory (see here). Anderson shows that if $G$ is even a compact connected Lie group that $KU_1(BG) = 0$ and $$KU_0BG = \operatorname{lim}_n\operatorname{Hom}(R(G)/J^n,\mathbb{Z}).$$

There is a discussion of the relationship between the two results in Remark 2.8 of "Topological K-(co)homology of classifying spaces of discrete groups", by Joachim and Lück.

In "$K$-homology of universal spaces and local cohomology of the representation ring" Greenlees proves that $$ K_i(BG_+) \simeq H_J^i(R(G)) $$ for $i=0,1$, where $J$ is the augmentation ideal and $H_J^i(R(G))$ refers to local cohomology. In fact Greenlees proves something more general using equivariant homotopy theory (for which the above is a special case), and this more general object is also studied in Chapter 18 of Greenlees and May's "Generalized Tate Cohomology"

Equivalently, one can use Anderson's universal coefficient theorem for $K$-theory (see here). Anderson shows that if $G$ is even a compact connected Lie group that $KU_1(BG) = 0$ and $$KU_0BG = \operatorname{lim}_n\operatorname{Hom}(R(G)/J^n,\mathbb{Z}).$$

There is a discussion of the relationship between the two results in Remark 2.8 of "Topological K-(co)homology of classifying spaces of discrete groups", by Joachim and Lück (which also contains more results along these lines).

In "$K$-homology of universal spaces and local cohomology of the representation ring" Greenlees proves that $$ K_i(BG_+) \simeq H_J^i(R(G)) $$ where $J$ is the augmentation ideal and $H_J^i(R(G))$ refers to local cohomology. In fact Greenlees proves something more general using equivariant homotopy theory (for which the above is a special case), and this more general object is also studied in Chapter 18 of Greenlees and May's "Generalized Tate Cohomology"

Equivalently,one can use Anderson's universal coefficeint theorem for $K$-theory (see here). Anderson shows that if $G$ is even a compact connected Lie group that $KU_1(BG) = 0$ and $$KU_0BG = \operatorname{lim}_n\operatorname{Hom}(R(G)/J^n,\mathbb{Z}).$$

There is a discussion of the relationship between the two results in Remark 2.8 of "Topological K-(co)homology of classifying spaces of discrete groups", by Joachim and Lück.

In "$K$-homology of universal spaces and local cohomology of the representation ring" Greenlees proves that $$ K_i(BG_+) \simeq H_J^i(R(G)) $$ where $J$ is the augmentation ideal and $H_J^i(R(G))$ refers to local cohomology. In fact Greenlees proves something more general using equivariant homotopy theory (for which the above is a special case), and this more general object is also studied in Chapter 18 of Greenlees and May's "Generalized Tate Cohomology"

Equivalently, one can use Anderson's universal coefficient theorem for $K$-theory (see here). Anderson shows that if $G$ is even a compact connected Lie group that $KU_1(BG) = 0$ and $$KU_0BG = \operatorname{lim}_n\operatorname{Hom}(R(G)/J^n,\mathbb{Z}).$$

There is a discussion of the relationship between the two results in Remark 2.8 of "Topological K-(co)homology of classifying spaces of discrete groups", by Joachim and Lück.