EQUIVARIANT CONNECTIVE K THEORY FOR COMPACT LIE GROUPS

J.P.C.GREENLEES

Abstract. We construct a complex oriented, multiplicative, G-equivariant analogue of connective K-theory for an arbitrary compact Lie group G. We describe its role in the theory of equivariant formal group laws and calculate its coefficient ring for a variety of groups.

Contents

1. Background. 1 2. Statement of results. 2 3. The construction. 5 4. Multiplicative structure. 7 5. The unitary groups. 9 6. Filtrations on the representation ring. 10 7. Coefficient rings. 11 8. Some specific examples. 13 9. Some terms in the representing spectrum. 14 10. The Noetherian condition. 16 11. Complex orientability. 17 References 20

1. Background.

There are sevaral ways of explaining the theoretical role of connective K theory. It can be viewed as an embodiment of the universal ring for multiplicative formal group laws, as a bundle theory with triviality conditions [13], a continuous relative of algebraic K theory, or as a refined home for certain indices. The usual construction is either by using an infinite loop space machine, or by using a brutal truncation of Atiyah’s periodic complex K theory.

In the equivariant world these different roles suggest a variety of different constructions. The view of a bundle theory with a triviality condition, and the usual constructions of equivariant algebraic K theory suggest the use of connective infinite loop space theory to give the brutal truncation K[0, ∞) of Atiyah-Segal equivariant complex K theory, which has coefficient ring K[0, ∞)_{∗}^{G} = R(G)[v], where R(G) is the complex representation ring of G. This has the serious disadvantage that it is not complex orientable if G ≠ 1. It also appears from Carlsson’s work on descent in algebraic K theory that this type of construction is also inadequate for some purposes in algebraic K theory.

In the present article we construct a G-equivariant analogue of connective K-theory for an arbitrary compact Lie group G. This is a complex oriented, ring valued theory from which Atiyah-Segal complex equivariant K theory can be obtained by inverting the Bott class. In fact, this theory has all of the desirable properties listed in [8] except that its coefficients need not be in even degree. Furthermore it agrees with the theory constructed there for the group of order p, and the construction here is simpler than that construction even in that special case. Apart from its value in understanding complex oriented cohomology theories and equivariant formal group laws, this theory can be used to define equivariant e-invariants with much smaller indeterminacy than periodic K theory.

The coefficient ring is closely related to the Rees ring (Section 2) of the representation ring for the augmentation ideal (and equal to it when G is a cyclic group). More generally, it is closely related to the universal ring for multiplicative equivariant formal group laws [9] (and equal to it when G is a product of two cyclic groups).

In retrospect, this type of coefficient ring seems very natural. Indeed, since ku is complex orientable we expect that ku has a representing G-spectrum with V th term \underline{ku}_{V} depending only on the dimension of a complex representation V. We then expect the first few terms to be analagous to the non-equivariant case, with

\underline{ku}_{0} = ℤ × BU, \underline{ku}_{2} = BU, \underline{ku}_{4} = BSU

where BU, BSU denote the classifying spaces for equivariant U or SU bundles (this is proved in Section 9). This gives calculations of some coefficient groups, since ku_{−n}^{G} = π_{0}^{G}(\underline{ku}_{n}). Since bundles over a point are representations, we obtain

ku_{0}^{G} = R(G), ku_{−2}^{G} = J, ku_{−4}^{G} = J_{2}

where J = ker(R(G) ⟶ ℤ) consists of bundles over a point with virtual dimension 0, and J_{2} consists of bundles over a point which are of virtual dimension 0 and of determinant 1.

This work has its roots in [3], and the author thanks R.R. Bruner for many valuable conversations. G. Carlsson informs me that he has also considered this construction.

2. Statement of results.

The main theorem is stated in terms of the representing G-spectrum ku for the cohomology theory, defined by the condition

k̃u_{G}^{∗}(X) = [X, ku]_{G}^{∗}

for based G-spaces X. Theorem 2.1. The G spectrum ku has the properties

(i) It is a highly structured commutative ring spectrum (an algebra over the equivariant sphere spectrum in the terminology of [4])

(ii) If H is any subgroup of G then if we view the G-equivariant spectrum ku as an H-spectrum we obtain the H-equivariant construction. In particular, ku is nonequivariantly the connective cover of the periodic K-theory spectrum K.

(iii) ku is a split ring spectrum, and hence ku_{G}^{∗} is an algebra over ku^{∗} = ℤ[v].

(iv) There is a ring map ku ⟶ K of G-spectra which is localization to invert v.

(v) ku_{G}^{∗} is a Noetherian ring. Its Krull dimension equal to two or the maximal rank of an abelian subgroup of G, whichever is greater.

(vi) There is a ring map MU ⟶ ku of G-spectra, so that ku is complex orientable (if

G is abelian this agrees with the classical notion of orientability by [2]). The G-spectrum is constructed in Section 3. This construction makes Parts (i), (ii) and

(iv) obvious. Part (iii) is proved in Section 4, Part (v) in Section 10 and Part (vi) in Section 11.

The connection with the non-equivariant theory is as good as possible in that the completion theorem and the local cohomology theorem hold. Since ku is a commutative algebra over the sphere spectrum with Noetherian coefficient ring, the usual proof given in [6, 7] applies. Let I = ker(ku_{G}^{∗} ⟶ ku^{∗}) be the augmentation ideal. Proposition 2.2. For any compact Lie group G,

(i) the completion theorem holds for ku in the sense that

ku^{∗}(BG) = ku_{G}^{∗}(EG) = (ku_{G}^{∗})_{I}^{∧}

(ii) the local cohomology theorem holds for ku in the sense that there is a spectral

sequence

H_{I}^{∗,∗}(ku_{∗}^{G}) ⇒ ku_{∗}^{G}(EG) = k̃u_{∗}(BG^{ad}). □

We have partial results about the coefficient ring. The most satisfactory of these are in terms of the ring homomorphism

p : ku_{G}^{∗} ⟶ K_{G}^{∗} = R(G)[v, v^{−1}]

comparing connective and periodic K-theory, where R(G) is the complex representation ring of G. A standard construction in commutative algebra is the Rees ring Rees(R, J) associated to an ideal J of a ring R. This is the graded subring of the graded ring R[v, v^{−1}] generated by R, v and v^{−1} · J, where v has degree 2. It is thus R in degree 0 and each positive even degree, and it is J^{n} in degree −2n. We apply this when R = R(G) and J is the augmentation ideal; in fact we need a slight modification. One may think of the Rees ring as a ring in which elements of J^{n} become divisible by v^{n}. In topology, the jth Chern class c_{j}(V) of an n-dimensional complex representation V naturally lies in degree −2j, and v^{j}c_{j}(V) = c_{j}^{R}(V), where

c_{j}^{R}(V) = ∑_{i=0}^{j}(−1)^{j} ( \begin{matrix}n − i \\ n − j \end{matrix} ) λ^{i}(V) ∈ R(G)

is the representation theory Chern class. Thus we would expect the representation theory Chern class c_{j}^{R}(V) of an n-dimensional representation to be divisible by v^{j} in ku_{G}^{∗}, even if it is not in J^{j}. Accordingly we define an algebraic model for ku_{G}^{∗} with this property. Definition 2.3. [3] The modified Rees ring ModRees(G) is the subring of R(G)[v, v^{−1}] generated by R(G), v and v^{−j}c_{j}^{R}(V) for all representations V. Remark 2.4. (i) If G is abelian, then ModRees(G) = Rees(R(G), J), but in general the inclusion

Rees(R(G), J) ⊆ ModRees(G)

is proper. (ii) The Rees ring only depends on R(G) as an augmented ring. However, the modified Rees ring also depends on the exterior powers, so we write it as a functor of G rather than the ring R(G). Example 2.5. The groups Q_{8} and D_{8} have isomorphic augmented representation rings, and hence also isomorphic Rees rings. Their modified Rees rings are not only different from the Rees ring, but also different from each other. Indeed, if V is the 2-dimensional simple representation of Q_{8} then c_{2}(V) = 2 − V, whilst if W is the 2-dimensional simple representation of D_{8} then c_{2}(V) = 1 − V + r where r is the one dimensional representation with kernel the rotation subgroup.

We are now ready to state results about the coefficient rings these are proved in Section 7. First we have a general statement if we invert v or tensor with the rational numbers. Proposition 2.6. (i) In positive degrees connective and periodic K theory agree:

ku_{i}^{G} = K_{i}^{G} if i ≥ 0.

(ii) Above degree −6 the coefficients are as follows

ku_{i}^{G} = \{ \begin{matrix}0 & if i is odd and ≥ 0 \\ R(G) & if i is even and ≥ 0 \\ 0 & if i = −1 \\ J & if i = −2 \\ 0 & if i = −3 \\ J_{2} & if i = −4 \\ 0 & if i = −5 \end{matrix}

(iii) Localized away from v connective and periodic K theory agree:

ku_{G}^{∗}[1/v] ≅ K_{G}^{∗} = R(G)[v, v^{−1}].

(iv) If G is finite, the map ku_{G}^{∗} ⟶ K_{G}^{∗} is a rational monomorphism and the image is

the rationalized modified Rees ring. Remark 2.7. The lower coefficients behave in a more complicated way: it follows from calculations for elementary abelian groups in [3], summarized in Section 8 that ku_{−6}^{G} can contain torsion (and is therefore not equal to the modified Rees ring in this degree) and ku_{−7}^{G} can be non-zero.

For groups of special forms we can give global descriptions. Proposition 2.8. (i) If G is topologically cyclic, ku_{G}^{∗} is the Rees ring

ku_{G}^{∗} = Rees(R(G), J).

(ii) If G is a product of two topologically cyclic groups then ku_{G}^{∗} is the representing ring for multiplicative equivariant formal group laws described in [9]:

ku_{G} = L_{G}^{m}.

(iii) If G = U(n) then ku_{G}^{∗} is the modified Rees ring

ku_{U(n)}^{∗} = ModRees(U(n)).

Note that the calculation of ku_{U(n)}^{∗} allows us to define ku-theory Chern classes for arbitrary representations by naturality. Corollary 2.9. For any compact Lie group G, the image of the map

ku_{G}^{∗} ⟶ K_{G}^{∗} = R(G)[v, v^{−1}]

contains ModRees(G). □

Because of the central place of Chern classes we prove Part (iii) in Section 5, and the rest of the proposition in Sections 6 and 7.

The calculations of [3] give calculations for a number of other groups.

The rest of the paper is devoted to proving the results we have just described. In Section 3 we describe the construction, and in Section 4 we show it has the desired multiplicative structure. Section 5 deals with G = U(n) and hence establishes the existence of Chern classes, and Sections 6 to 10 deal with properties of the coefficient rings. Finally, in Section 11 we return to establish complex orientability.

3. The construction.

The construction is inspired by the Hasse pullback square, and is easy to describe. Definition 3.1. The equivariant connective K theory spectrum ku is defined by the homotopy pullback square

\xymatrix{ku \ar[r] \ar[d] & K \ar[d] \\ F (EG_{+}, inf_{1}^{G}ku) \ar[r] & F (EG_{+}, K)}

of G-spectra. Here K is the usual Atiyah-Segal periodic K theory spectrum and inf_{1}^{G}ku is the inflation of the nonequivariant ku spectrum. The bottom horizontal is induced by the composite

inf_{1}^{G}ku ⟶ inf_{1}^{G}K ⟶ K

where the first map is inverting v and the second arises since K is split. The right hand vertical is induced by the unique map EG ⟶ ∗.

The rest of this article is devoted to proving the results of Section 2 by establishing properties of this spectrum. This should convince the reader that this is the correct generalization of the non-equivariant spectrum. Remark 3.2. (a) Part (ii) of Theorem 2.1 is immediate from the construction.

(b) If we use the Elmendorf-May highly structured inflation [5], then inf_{1}^{G}ku and inf_{1}^{G}K are commutative algebras over the sphere spectrum, and hence F (EG_{+}, inf_{1}^{G}ku) and F (EG_{+}, inf_{1}^{G}K) ≃ F (EG_{+}, K) are too. Since K is a commutative algebra over the sphere spectrum (for finite groups by this follows by infinite loop space theory, and for general compact Lie groups it is proved in [11]) we may take the pullback in the category of commutative algebras over the sphere spectrum. This proves Part (i) of Theorem 2.1.

(c) Part (iv) of Theorem 2.1 follows from the fact that ku^{∗}(BG)[1/v] = K^{∗}(BG) ([3, 1.1.1]) and hence that the bottom horizontal becomes an equivalence after inverting v.

(d) It is worth pausing to explain why this simple construction is plausible. Assuming ku exists, one expects there to be a homotopy pullback square

\xymatrix{ku \ar[r] \ar[d] & ku[1/v] ≃ K \ar[d] \\ ku_{v}^{∧} \ar[r] & ku_{v}^{∧}[1/v]}

of G-spectra. Now ku_{v}^{∧} ≃ holim_{←}_{k} ku/v^{k}, and ku/v^{k} can be constructed from ku/v by a finite number of cofibre sequences. Finally this should represent an additive formal group law, for which the identity is the only point of finite order. Thus ku/v = F (EG_{+}, ku/v), and hence

ku_{v}^{∧} ≃ F (EG_{+}, ku_{v}^{∧}) ≃ F (EG_{+}, ku).

(e) An analogous construction is used by Hopkins and Mahowald [10] in constructing a “connective” spectrum eo_{2} from Hopkins-Miller EO_{2}.

First we should record the fact that this is indeed a generalization of the one previously existing family of examples. Lemma 3.3. If G is of prime order, then the spectrum ku is equivalent to the one constructed in [8]. Remark 3.4. The present construction is considerably simpler than the previous one. Nonetheless, I would never have recognized the present construction as the right one without the previous construction. Proof: Suppose G is of prime order. We continue to use ku to denote the form constructed here, and we let ku′ denote the construction from [8]. Both constructions construct the equivariant form of ku as a homotopy pullback of a diagram of G-spectra. These diagrams do not appear to be directly comparable, but we have the diagram

\xymatrix{F (EG_{+}, inf_{1}^{G}ku) \ar[d]^{≃} \ar[r] & F (EG_{+}, inf_{1}^{G}K) \ar[d] & K \ar[l]^{u} \ar[d] \\ F (EG_{+}, inf_{1}^{G}ku) \ar[r] & F (EG_{+}, inf_{1}^{G}K) ∧ ẼG & K ∧ ẼG \ar[l]^{u′′} \\ F (EG_{+}, inf_{1}^{G}ku) \ar[u]_{≃} \ar[r] & F (EG_{+}, inf_{1}^{G}ku) ∧ ẼG \ar[u] & ku′ ∧ ẼG \ar[l]^{u′} \ar[u]}

Here ku is the pullback of the top fork and ku′ is the pullback of the bottom fork. We let ku′′ denote the pullback of the middle fork. The left hand verticals in the diagram are all equivalences, so we may concentrate on the two right hand squares. We have comparison maps

fibre(u) ⟶ fibre(u′′) ⟵ fibre(u′),

and it suffices to show these are equivalences.

Since EG_{+} is non-equivariantly S^{0}, the map u is a non-equivariant equivalence and hence the fibre of u is the fibre of u′′, so that ku ≃ ku′′. Now let K/ku′ denote the cofibre of the map ku′ ⟶ K. By the octahedral axiom, there is a cofibre sequence

F (ẼG, K/ku′) ∧ ẼG ⟶ fibre(u′) ⟶ fibre(u′′),

so it remains to show F (ẼG, K/ku′) ∧ ẼG is contractible. However it is non-equivariantly contractible by construction so it suffices to shown its homotopy groups are zero.

First note that π_{i}^{G}(K/ku′) = 0 for i ≥ 0. Now ẼG = lim_{→}_{k} S^{kV} where V is the reduced regular representation and

F (ẼG, K/ku′)ẼG ≃ F (ẼG, K/ku′) ≃ holim_{←}_{k} K/ku′ ∧ S^{−kV} .

By complex orientability

K/ku′ ∧ S^{−kV} ≃ Σ^{−2k}K/ku′

so that

π_{s}^{G}(K/ku′ ∧ S^{−kV}) = 0 provided s + 2k < 0,

and the Milnor exact sequence shows

π_{s}^{G}(F (ẼG, K/ku′)) = 0

for each s as required. □

4. Multiplicative structure.

In this section we show that ku is a split commutative ring spectrum up to homotopy. To show ku is a ring we need to describe a map μ : ku ∧ ku ⟶ ku. The Mayer-Vietoris sequence of a homotopy pullback square gives a long exact sequence

⋯ ⟶ [EG_{+}∧X, K]_{1}^{G} ⟶ [X, ku]_{0}^{G} ⟶ [X, K]_{0}^{G}⊕[EG_{+}∧X, inf_{1}^{G}ku]_{0}^{G} ⟶ [EG_{+}∧X, K]_{0}^{G} ⟶ ⋯

for maps into ku. The thing that makes this practical to use is that [EG_{+} ∧ X, K]_{1}^{G} = 0 in the cases we need to use. Lemma 4.1. If Z is a non-equivariant spectrum so that H_{∗}(Z) is a free ℤ-module on even degree generators, then the graded group [EG_{+} ∧ Z, K]_{∗}^{G} is zero in odd degrees. Proof: First note that

[EG_{+} ∧ Z, K]_{∗}^{G} = [BG_{+} ∧ Z, K]_{∗} = [Z, F (BG_{+}, K)]_{∗}.

Now use the Atiyah-Hirzebruch spectral sequence togegher with the fact that F (BG_{+}, K) has coefficients in even degrees. □

First we define a splitting map compatible with that of K. Lemma 4.2. There is a non-equivariant equivalence i : inf_{1}^{G}ku ⟶ ku so that

inf_{1}^{G}ku ⟶^{i} ku
↓  ↓
inf_{1}^{G}K ⟶^{j} K

commutes, where j is the splitting for K. Proof: Take X = inf_{1}^{G}ku in the Mayer-Vietoris sequence. To define i we must construct compatible maps i′ : inf_{1}^{G}ku ⟶ K and i′′ : inf_{1}^{G}ku ⟶ F (EG_{+}, inf_{1}^{G}ku). For i′ we use the composite

inf_{1}^{G}ku ⟶ inf_{1}^{G}K ⟶^{j} K,

and for i′′ we use the map induced by projection EG ⟶ ∗. The choice of i′ guarantees compatibility with j, and compatibility of (i′, i′′) follows. This constructs a non-equivariant equivalence i.

This specifies i uniquely by 4.1 since H_{∗}(ku) is a free ℤ-module on even degree generators. □

Since we know that ku is a strictly commutative ring spectrum, we do not need to reprove that it has this property up to homotopy. However, it is convenient for 4.4 and Section 11 to have the details of the following proof to hand. This construction up to homotopy is visibly is compatible with the highly structured version. Lemma 4.3. The G-spectrum ku is a commutative and associative ring spectrum up to homotopy. Proof : To show ku is a ring we must construct a map μ : ku ∧ ku ⟶ ku. Taking X = ku ∧ ku in the Mayer-Vietoris sequence, we see that we need to construct compatible maps μ′ : ku ∧ ku ⟶ K and μ′′ : ku ∧ ku ⟶ F (EG_{+}, inf_{1}^{G}ku).

Since ku[1/v] = K, we may take μ′ to be the composite

ku ∧ ku ⟶ K ∧ K ⟶ K

where the last map is the multiplication for K. Since ku∧ku∧EG_{+} ≃ inf_{1}^{G}(ku∧ku)∧EG_{+} we may take μ′′ to correspond to the product in non-equivariant ku. These are compatible since the non-equivariant map ku ⟶ K is a ring map, and K is split so that its equivariant and non-equivariant products are compatible. More precisely, noting that

inf_{1}^{G}(A ∧ B) = inf_{1}^{G}(A) ∧ inf_{1}^{G}(B)

by [12, 3.14(iii)], compatibility follows because the diagram

\xymatrix{ & EG_{+} ∧ ku ∧ ku \ar[rr] &  & EG_{+} ∧ K ∧ K \ar[dd] \\ EG_{+} ∧ inf_{1}^{G}(ku ∧ ku) \ar@{-->}[ur] \ar[rr] \ar[dd] &  & EG_{+} ∧ inf_{1}^{G}(K ∧ K) \ar@{-->}[ur] \ar[dd] \\  &  &  & EG_{+} ∧ K \\ EG_{+} ∧ inf_{1}^{G}ku \ar[rr] &  & EG_{+} ∧ inf_{1}^{G}K \ar@{-->}[ur]}

commutes. Applying 4.1 with Z = ku ∧ ku we see that

[ku ∧ ku, F (EG_{+}, K)]_{∗}^{G} = [ku ∧ ku ∧ BG_{+}, K]_{∗}

is in even degrees, and hence the map μ : ku∧ku ⟶ ku corresponding to (μ′, μ′′) is unique. Since both μ′ and μ′′ are commutative it follows similarly that μ is commutative. Exactly analogous arguments with three factors ku in the domain show that μ is also associative. □

Finally we show the two bits of structure are compatible. Lemma 4.4. The splitting i is a ring map. Proof: We must show that the diagram

\xymatrix{inf_{1}^{G}ku ∧ inf_{1}^{G}ku \ar[d]_{=} \ar[r]^{i∧i} & ku ∧ ku \ar[dd]^{μ} \\ inf_{1}^{G}(ku ∧ ku) \ar[d]_{inf_{1}^{G}(μ)} \\ inf_{1}^{G}ku \ar[r]^{i} & ku}

commutes. Once again, because [inf_{1}^{G}(ku ∧ ku) ∧ EG_{+}, K]_{∗}^{G} is in even degrees by 4.1, it suffices to check after composing with ku ⟶ K or ku ⟶ F (EG_{+}, inf_{1}^{G}ku) by 4.1 and this is clear. □

5. The unitary groups.

The unitary groups have a universal role since all compact Lie groups may be embedded in them, and it is therefore valuable to understand the unitary case early in the analysis. In particular, once we have proved 2.8 (iii):

ku_{U(n)}^{∗} = ModRees(U(n)),

we may define Chern classes by pullback from the unitary case Proof of 2.8 (iii): The defining pullback square for ku gives a Mayer-Vietoris sequence

⋯ ⟶ ku_{U(n)}^{∗} ⟶ K_{U(n)}^{∗} ⊕ ku^{∗}(BU(n)) ⟶^{⟨i,−j⟩} K^{∗}(BU(n)) ⟶ · · · .

For U(n) we may identify all the groups and maps explicitly. Indeed,

R(U(n)) = ℤ[c_{1}^{R}, c_{2}^{R}, …, c_{n}^{R}, (λ^{n})^{−1}],

and since ku and K are complex oriented, we have

K^{∗}(BU(n)) = K^{∗}[[c_{1}, c_{2}, …, c_{n}]] and ku^{∗}(BU(n)) = ku^{∗}[[c_{1}, c_{2}, …, c_{n}]].

The critical fact is that the map ⟨i, −j⟩ is surjective. Indeed, an element of K^{2n}(BU(n)) may be written as a formal sum of terms m_{I}/v^{n} · (c^{R})^{I} where I = (i_{1}, i_{2}, …) is a sequence of natural numbers almost all zero, m_{I} is an integer and

(c^{R})^{I} = ∏_{s}(c_{s}^{R})^{i_{s}} = ∏_{s} v^{s}c_{s}^{i_{s}}.

For all but finitely many terms the power (Σ_{s}i_{s}) − n of v is positive. Subtracting off the finitely many negative terms using im(i) we are left with an element of im(j).

It now follows that ku_{U(n)}^{∗} = ker(⟨i, −j⟩), and since both i and j are monomorphisms

ku_{U(n)}^{∗} = ku^{∗}[[c_{1}, c_{2}, …, c_{n}]] ∩ K^{∗}[c_{1}, c_{2}, …, c_{n}, (λ^{n})^{−1}],

which is the modified Rees ring as required. □

6. Filtrations on the representation ring.

Consider the representation ring R(G), and let J denote the augmentation ideal. We need to consider two filtrations: one associated to the Rees ring and the other to the modified Rees ring. The point of this section is to show that they define the same topology, so we can repeat the method used to prove 2.8 (iii) in Section 5.

The first filtration is the J-adic filtration

R(G) ⊇ J ⊇ J^{2} ⊇ J^{3} ⊇ ⋯

by powers of the augmentation ideal. However for the modified Rees ring it is more relevant to use certain ideals generated by Chern classes. We therefore define a second filtration recursively by J_{0} = 0 and take

J_{n} = J_{1}J_{n−1} + J_{2}J_{n−2} + ⋯ + J_{n−1}J_{1} + (c_{n}(V) | V is a representation )

to be the ideal generated by Chern classes of the appropriate degrees. Noting that c_{1}(V) = |V |−V, we see that J = J_{1} is generated by first Chern classes. Again we have the filtration

R(G) ⊇ J = J_{1} ⊇ J_{2} ⊇ J_{3} ⊇ · · · .

Note that by definition

J_{i}J_{j} ⊆ J_{i+j}

so that this is a multiplicative filtration. The reason for considering these filtrations is that

Rees(R(G))_{−2n} = J^{n}

and

ModRees(G)_{−2n} = J_{n}.

Lemma 6.1. The two topologies on R(G) coincide. Proof: We have already noted that J^{n} ⊆ J_{n}, so that the J-adic topology is finer than the lower topology.

To see that the lower topology is finer than the J-adic topology note that, since c_{n}(V) is non-equivariantly zero, it is clear that J_{i} ⊆ J. Now recall c_{.}(V ⊕ W) = c_{.}(V)c_{.}(W). If the largest dimension of a simple representation is d it follows that c_{n}(V) is decomposable whenever n > d, and therefore

J_{dk} ⊆ J^{k}

□

Now consider the I-adic topology on ku_{G}^{∗}, where I = ker(ku_{G}^{∗} ⟶ ku^{∗}) is the augmentation ideal. Since ku^{∗} is zero in negative degrees, it follows that I_{2n} = ku_{2n}^{G} for n < 0. On the other hand ku_{2n}^{G} = K_{2n}^{G} for n ≥ 0, so that I_{2n} = J if n ≥ 0.

Now let \overline{I}(s) be the image of I^{s} in K_{G}^{∗}. For each n this defines a topology on K_{2n}^{G}. Lemma 6.2. The \overline{I} topology on K_{2n}^{G} coincides with the J-adic topology. Proof: First note that \overline{I}(1)_{2n} = J if n ≥ 0. Next, by 2.8(iii) (proved in Section 5), the Chern classes c_{n}(V) ∈ K_{−2n}^{G} lift back to ku_{∗}^{G}, and hence

J_{n} ⊆ \overline{I}(1)_{−2n} ⊆ J.

It follows that \overline{I}(s + 1)_{2n} = J^{s+1} if n ≥ 0 and

J_{n}J^{s} ⊆ \overline{I}(s + 1)_{−2n} ⊆ J^{s}.

The result follows from 6.1. □

7. Coefficient rings.

The pullback square defining ku automatically gives a Mayer-Vietoris sequence for its homotopy groups. However we show that in fact it induces a pullback square of rings and hence [3] gives a plentiful source of calculations: in most cases the calculations of ku^{∗}(BG) in [3] are stated so that the ring ku_{G}^{∗} is given by omitting a visible completion. These will be recorded in Section 8; in the present section we apply the pullback theorem to prove the general statements about coefficient rings. Theorem 7.1. The defining pullback square induces a pullback square

\xymatrix{ku_{G}^{∗} \ar[r] \ar[d] & K_{G}^{∗} = R(G)[v, v^{−1}] \ar[d] \\ ku^{∗}(BG) \ar[r] & K_{G}^{∗}(BG) = R(G)_{J}^{∧}[v, v^{−1}]}

of commutative rings. Proof: We must show that for each n the map

ku^{2n}(BG) ⊕ R(G) ⟶ R(G)_{J}^{∧}

is surjective. Evidently it suffices to check that the image of ku^{2n}(BG) ⟶ R(G)_{J}^{∧} contains (J^{s})_{J}^{∧} for some s. This follows from 6.2. □

Proof of 2.6: Most of Parts (i) and (ii) follow from the results about spaces in the representing spectrum in Section 9, but we give direct proofs here. In fact they follow from corresponding statements about ku^{n}(BG). Indeed, ku^{n}(BG) = K^{n}(BG) for n ≤ 0, ku^{−1}(BG) = ku^{−3}(BG) = 0 and ku^{−2}(BG) = J_{J}^{∧}. To see this we let K/ku denote the cofibre of the map ku ⟶ K, and note that

(K/ku)^{s}(BG) = \{ \begin{matrix}0 & if s ≥ −1 \\ H^{0}(BG) & if s = −2 \\ H^{1}(BG) & if s = −3 \end{matrix}

All this can be viewed as an argument with the Atiyah-Hirzebruch spectral sequence. To see that ku^{5}(BG) = 0 we need to argue that the differentials killing K^{5}(BG) all originate in ku, or equivalently that none originate in H^{0}(BG) or H^{2}(BG). This is obvious for H^{0}(BG), and for H^{2}(BG) it follows from the fact that H^{2}(BG) is generated by Chern classes. The statement about ku_{−4}^{G} follows by identifying the map BU ⟶ BS^{1} as Bdet.

Part (iii) is immediate from Theorem 2.1 (iv), which was proved in Section 3.

For Part (iv) it suffices to show that { \atop (K/ku)}^{∗}(BG) ⊗ ℚ = 0. However

{ \atop (K/ku)}^{s}(BG) = lim_{←}_{m}[BG^{(m)}, K/ku]^{s} = lim_{←}_{m} lim_{→}_{n}[BG^{(m)}, (K/ku)[−n, −2]]^{s}

with both direct and inverse limits being achieved at finite stages. Since H̃^{∗}(BG) ⊗ ℚ = 0

and (K/ku)[−n, −2] is a finite Postnikov tower, it follows that { \atop (K/ku)}^{∗}(BG) ⊗ ℚ = 0 as required. □

Proof of 2.8: Consider the relation between ku_{G}^{∗} and the representing ring L_{G}^{m} for multiplicative equivariant formal group laws. First if G = C is topologically cyclic, ku^{∗}(BC) ⟶ K^{∗}(BC) is injective, so the defining pullback square identifies ku_{C}^{∗} as a subring of K_{C}^{∗}. From the formal group calculation of ku^{∗}(BC) we can directly see that the image of ku_{C}^{∗} is the Rees ring, which was shown to be L_{C}^{m} in [9, 4.5(iii)].

Now suppose A = C × D where C and D are topologically cyclic. We claim

ku_{A}^{∗} = L_{A}^{m}.

To prove this we use the fact that both rings can be constructed as pullbacks, and compare the forks. For ku_{A}^{∗} the pullback property is established in 7.1. For L_{A}^{m} we use the Hasse square associated to the ideal (v); since L_{A}^{m} is Noetherian by [9], we have the pullback square

L_{A}^{m} ⟶ L_{A}^{m}[1/v] = L_{A}^{sm}
↓  ↓
(L_{A}^{m})_{(v)}^{∧} ⟶ (L_{A}^{m})_{(v)}^{∧}[1/v].

The equality L_{A}^{m}[1/v] = K_{A}^{m} is [9, 4.5 (ii)]; this ring is the universal ring L_{A}^{sm} of strictly multiplicative A-equivariant formal group laws. It now remains to show that (L_{A}^{m})_{(v)}^{∧} = ku^{∗}(BA) and that the two forks are comparable. Both maps in the fork can be identified explicitly and seen to agree Now by decoupling [9, 5.5] we have the tensor decomposition

L_{A}^{m} = L_{C}^{m} ⊗_{L^{m}} L_{D}^{m},

which we may complete to obtain

(L_{A}^{m})_{(v)}^{∧} = (L_{C}^{m})_{(v)}^{∧} ˆ⊗_{L^{m}}(L_{D}^{m})_{(v)}^{∧}.

On the other hand, because of the Gysin presentation for ku^{∗}(BC) when it is topologically cyclic, we have the Künneth theorem

ku^{∗}(BA) = ku^{∗}(BC) ˆ⊗_{ku_{∗}}ku^{∗}(BD).

By the completion theorem ku^{∗}(BG) = (ku_{G}^{∗})_{I}^{∧}, so the result follows once observe that the I-adic and v-adic topologies on the Rees ring coincide.

Part (iii) was proved in Section 5. □

Remark 7.2. If G = A is abelian we will see in Section 11 that ku is complex orientable, and that it has a canonical orientation, whose associated equivariant formal group law is multiplicative. It follows that there is a universal map L_{A}^{m} ⟶ ku_{A}^{∗}, and it is this map that is an isomorphism when A is a product of two topologically cyclic groups.

8. Some specific examples.

In this section we give the coefficient rings ku_{∗}^{G} for a few groups not covered by the general theorems above. Using the pullback theorem 7.1, this is immediate from the calculations in [3]. Lemma 8.1. If ku^{∗}(BG) has no v-torsion and is generated by Chern classes of representations, then

ku_{∗}^{G} = ModRees(G).

Proof: By the pullback theorem, since the map ku^{∗}(BG) ⟶ K^{∗}(BG) is injective, ku_{G}^{∗} is just the intersection with K_{G}^{∗} = R(G)[v, v^{−1}]. Since this intersection contains ModRees(G) by 2.9, the lemma follows. □

Certainly this applies to cyclic groups, but here are two more complicated examples. Example 8.2. (i) (Non-abelian pq.) If q is a prime and p|q−1 we let G_{p,q} be the semidirect product with a normal subgroup C_{q} and quotient C_{p} acting by an automorphism of order p. By [3, 2.3.2] we have

ku_{∗}^{G_{p,q}} = ModRees(G_{p,q}).

(ii) (Quaternion 2-groups.) For n ≥ 1 we let Q_{2^{n+2}} denote the quaternion group of order 2^{n+2}. By [3, 2.4.5] we have

ku_{∗}^{Q_{2^{n+2}}} = ModRees(Q_{2^{n+2}}).

Example 8.3. (Dihedral 2-groups [3, 2.5.4,2.5.5].) We consider the dihedral group D_{2^{n+2}} for n ≥ 1. It is again shown in [3] that ku^{∗}(BG) is generated by Chern classes, so the image in periodic K theory is the modified Rees ring, but now there is some v-torsion. In fact there is an extension

0 ⟶ T ⟶ ku_{∗}^{D_{2^{n+2}}} ⟶ ModRees(D_{2^{n+2}}) ⟶ 0.

of modules over ModRees(D_{2^{n+2}}), where T = Σ^{−6}𝔽_{2}[a, b, d]/(ab + b^{2}) with a, b of degree −2 and d of degree −4. Since the dihedral group is a 2-group, the completion map is injective in periodic K theory, and since the relations given in [3] are defined before completion they also give a a presentation of ku_{∗}^{G}, which we do not reproduce here.

Example 8.4. (Elementary abelian 2-groups [3, Chapter 4].) Let V denote an elementary abelian 2-group of rank r. Since it is a 2-group, completion is injective in periodic K theory, and since the image of ku^{∗}(BV) in K^{∗}(BV) is generated by Chern classes the same is true before completion. Thus there is an exact sequence

0 ⟶ T ⟶ ku_{V}^{∗} ⟶ ModRees(V) ⟶ 0

of ModRees(V)-modules. The module T is annihilated by 2 or by V, and its rather complicated structure is described in detail in [3]. For the present we only mention that T is zero if V is of rank 1, and more generally

T = T_{2} ⊕ T_{3} ⊕ ⋯ ⊕ T_{r}

where T_{i} is non-zero in degree −i − 4, and non-zero in all degrees ≤ −i − 4 with the same parity. In particular ku_{−6}^{V} contains torsion if r ≥ 2 and ku_{−7}^{V} ≠ 0 if r ≥ 3. Example 8.5. (Alternating group A_{4} [3, 2.6.2].) Although ku^{∗}(BA_{4}) is not generated by Chern classes, its image in periodic K theory is still generated by Chern classes. We therefore have a short exact sequence

0 ⟶ T ⟶ ku_{∗}^{A_{4}} ⟶ ModRees(A_{4}) ⟶ 0

of ModRees(A_{4})-modules, and T is non-zero in degree −6. A presentation may be read off from [3, 2.6.3]. Choose a non-trivial 1-dimensional representation α, and let τ denote the 3-dimensional simple module. Now let y = c_{1}(α) and ν = c_{3}(τ), μ = c_{2}(τ), and let π be a new generator of degree −6. We then have the presentation

ku_{∗}^{A_{4}} = ku_{∗}[y, μ, ν, π]/I

where

I = ([3](y), yμ, yν, yπ, 2ν, vν, vπ − 2μ, vμ^{2} − 2π, μ^{3} − π^{2} − πν − ν^{2}).

Indeed, this follows from the v-adic Hasse square for the R(G)-module ku_{∗}[y, μ, ν, π]/I because one may check the v-adic and J-adic topologies coincide.

9. Some terms in the representing spectrum.

We connect the spectrum ku to geometry by describing some of the terms in the representing Ω-spectrum. In particular this gives an alternative proof of Parts (i) and (ii) of (except for the statement that ku_{−5}^{G} = 0).

To be clear, for a unitary representation V we consider the group U(V) of unitary maps of V and note that it admits an action of G by conjugation. Next, there is a map U(V) ⟶ U(W) if there is an equivariant isometry V ⟶ W. Hence, using a complete complex universe 𝒰 to organize the terms, we may form the direct limit

U = lim_{→}_{V ⊆𝒰} U(V).

The result is compatible with restriction of the group of equivariance. Similarly we may form SU(V) and SU.

Next, we may form the equivariant classifying spaces BU and BSU of equivariant unitary or special unitary bundles. There is a Grassmannian model in the unitary case as usual. Thus

BU_{n} = Gr_{n}(𝒰)

is the classifying G-space for n-dimensional unitary bundles and we may form BU = lim_{→}_{n} BU_{n}. As usual, we may construct BSU_{n} as the fibre of Bdet : BU_{n} ⟶ BU_{1}.

Lemma 9.1. The following are terms in the Ω-spectrum for the G-spectrum ku:

\underline{ku}_{−n} = \{ \begin{matrix}BU × ℤ & if n is even and ≥ 0 \\ U & if n is odd and ≥ −1 \\ BU & if n = −2 \\ SU & if n = −3 \\ BSU & if n = −4 \end{matrix}

where BU, BSU are equivariant classifying spaces, and U, SU are groups of equivariant unitary or special unitary maps. Proof: We may identify the terms \underline{ku}_{−n} in an Ω-spectrum for ku by the same pullback square as defined ku itself. In fact, we claim that any map X ⟶ Y with the property that for each subgroup H of G there is a complex H-representation V such that Ω^{V} X ⟶ Ω^{V} Y is an equivalence, a pullback

X  ⟶  Y
↓  ↓
map(EG_{+}, X) ⟶ map(EG_{+}, Y).

The result will follow by taking Y = \underline{K}_{−n}, and X to be the candidate for \underline{ku}_{−n}. The point being that in each case this candidate is known to be correct non-equivariantly, so that the three terms of the fork are correct.

To prove the claim, note that the fibres of the verticals are

map(ẼG, X) ⟶ map(ẼG, Y).

Now we argue as usual: by induction on the group order we may assume that this is an H equivariant equivalence for all proper subgroups. Now take

S^{∞W} = lim_{→}_{W^{G}=0} S^{W}

and note that it is H-contractible for all proper subgroups H, but has G-fixed point set S^{0}. Now consider the map

ẼG ⟶ S^{∞W}

which is an equivalence in G fixed points. Its mapping cone may be constructed using cells G/H_{+} for proper subgroups H, so it suffices to show

map(S^{∞W} , X) ⟶ map(S^{∞W} , Y)

is an equivalence, which follows from the hypothesis. □

10. The Noetherian condition.

In this section we prove Part (vi) of Theorem 2.1, stating that ku_{G}^{∗} is Noetherian. We use the short exact sequence

0 ⟶ T ⟶ ku_{G}^{∗} ⟶ Q ⟶ 0

of ku_{G}^{∗}-modules, where T is the v-power torsion and Q = im(ku_{G}^{∗} ⟶ K_{G}^{∗}).

In fact we choose a faithful representation of G in U(n) and show that ku_{G}^{∗} is finitely generated as a module over the Noetherian ring Ch = ku_{U(n)}^{∗} = R(U(n))[v, c_{1}, …, c_{n}]. It suffices to show that T and Q are finitely generated Ch-modules.

We let Ĉh = ku^{∗}(BU(n)) = R(U(n))[v][[c_{1}, …, c_{n}]] denote the completion of Ch. Proposition 10.1. The v-power torsion module T is a finitely generated Ch-module. Proof : From the defining pullback square, T is also equal to the v-power torsion in ku^{∗}(BG). Since ku^{∗}(BG) is a finitely generated Ĉh-module by Quillen’s argument, it follows that its submodule T is also finitely generated.

Since the chain of annihilators of v, v^{2}, v^{3}, … is an ascending chain of Ĉh-modules, it follows that v^{k}T = 0 for some k. Now if t_{1}, …, t_{N} is a generating set for T as an Ĉhmodule we claim that it is also a generating set as an Ch-module. Indeed, if t = ∑_{i} f_{i}t_{i} with f_{i} ∈ Ĉh we may write

f_{i} = ∑_{I} a_{i,I}c^{I}

with a_{i,I} ∈ R(U(n))[v] where the sum is over finite sequences I = (i_{1}, i_{2}, …) with i_{s} ≥ 0 and c^{I} = ∏_{s} c_{s}^{i_{s}}. Since |c^{I}| ⟶ −∞ as |I| ⟶ ∞ it follows that |a_{i,I}| ⟶ ∞ as |I| ⟶ ∞ and the power of v dividing a_{i,I} is greater than k for all but finitely many terms. Hence the power series f_{i} ∈ Ĉh can be replaced by polynomials f′_{i} ∈ Ch and still have

t = ∑_{i} f′_{i}t_{i}

as required. □

Proposition 10.2. The module Q is a finitely generated Ch-module. Proof: From the defining pullback square, Q may be calculated by

Q = Q̂ ∩ R(G)[v, v^{−1}],

where ∩R(G)[v, v^{−1}] indicates the inverse image under the natural map R(G) ⟶ R(G)_{J}^{∧}. Each Q_{2n} is a submodule of the Noetherian ring R(G) and hence finitely generated. The central point is the fact that Q̂ is rationally generated in the sense of the following lemma. Lemma 10.3.

Q̂_{2n} = (Q_{2n})_{J}^{∧}.

We pause to note that this is not a triviality: for example it is easy to find a non-zero sub-ℤ_{p}^{∧}-module T ≅ ℤ_{p}^{∧} of ℤ_{p}^{∧} ⊕ ℤ_{p}^{∧} with T ∩ (ℤ ⊕ ℤ) = 0. Proof: It suffices to prove the result completed at p for each integer prime p, and we work completed at p for the rest of the proof.

Suppose first that G is finite. Note that Q̂_{−2n} is a sub-ℤ_{p}^{∧}-module of Ĵ, and that it is necessarily free. The point is that Q̂_{−2n} and Ĵ have equal ranks and Ĵ is the completion of J. Now if we choose a ℤ_{p}^{∧}-basis of Q̂_{−2n} and express it in terms of the basis of J, we may put it into diagonal form. Multiplying by p-adic units, the diagonal entries may be taken to be powers of p. These integral basis elements lie in Q_{−2n} and their completion gives Q̂_{−2n} as required.

If G is not finite we argue similarly, except that J must be replaced by a direct summand of finite codimension to ensure equality of ranks. The summand is the kernel of the composite

K_{G}^{2n} ⟶ K^{2n}(BG) ⟶ (K/ku)^{2n}(BG),

where we note that (K/ku)^{2n}(BG) is a finitely generated abelian group. □

Returning to the proof of 10.1, we note that since Q̂_{≥0} = (R(G))_{J}^{∧}[v] and Q̂_{−2} = J_{J}^{∧} the corresponding statements, Q_{≥0} = R(G)[v] and Q_{−2} = J follow for Q. Accordingly Q_{≥−2n} generates a finitely generated ring for each n ≥ 0, and it remains to show that this is the whole of Q for n sufficiently large.

For each n we may consider the image P̂_{−2n} ⊆ Q̂_{−2n} of products of negative degree elements. We may now consider the diagram

0 ⟶ P_{−2n} ⟶ Q_{−2n} ⟶ \overline{Q}_{−2n} ⟶ 0
↓ ↓ ↓
0 ⟶ P̂_{−2n} ⟶ Q̂_{−2n} ⟶ \overline{Q̂}_{−2n} ⟶ 0

Since J-completion preserves sums and is exact on finitely generated modules, it follows that P̂_{−2n} is the completion of P_{−2n}, and hence that \overline{Q̂}_{−2n} is the completion of \overline{Q}_{−2n}.

Next note that J\overline{Q}_{−2n} = 0 since P_{−2n} contains

Q_{−2} · Q_{−2n+2} = J · Q_{−2n+2} ⊇ J · Q_{−2n}.

Hence \overline{Q}_{−2n} is J-complete and

\overline{Q}_{−2n} = \overline{Q̂}_{−2n}.

Since Q̂ is finitely generated it follows that Q̂_{−2n} = \overline{Q̂}_{−2n} = 0 for sufficiently large n as required. □

11. Complex orientability.

We must construct a ring map ν : MU ⟶ ku. The Mayer-Vietoris sequence of a homotopy pullback square gives a long exact sequence

⋯ ⟶ [EG_{+}∧X, K]_{1}^{G} ⟶ [X, ku]_{0}^{G} ⟶ [X, K]_{0}^{G}⊕[EG_{+}∧X, inf_{1}^{G}ku]_{0}^{G} ⟶ [EG_{+}∧X, K]_{0}^{G} ⟶ · · · .

Accordingly, to construct ν, we need to construct compatible maps ν′ : MU ⟶ K and ν′′ : MU ⟶ F (EG_{+}, inf_{1}^{G}ku).

For ν′′ we use the composite

MU ⟶ F (EG_{+}, MU) ≃ F (EG_{+}, inf_{1}^{G}MU) ⟶ F (EG_{+}, inf_{1}^{G}ku)

where the first is induced by EG ⟶ ∗, the second is the equivalence induced by the fact that MU is split, and the third is induced by the non-equivariant map MU ⟶ ku. These may all be chosen to be maps of S-algebras.

For ν′ we prove a lemma. Lemma 11.1. There is a ring G-map MU ⟶ K which is non-equivariantly the Todd genus.

We will turn to the proof when we have described how this implies the theorem.

The compatibility condition for the maps ν′ and ν′′ is described by their images in [MU ∧ EG_{+}, K]_{G}. By the following result it is practical to check this, since taking Z = MU^{∧i} in 4.1 we obtain. Lemma 11.2. The graded group [EG_{+} ∧ MU^{∧i}, K]_{∗}^{G} is zero in odd degrees. □

The compatibility condition is therefore given by the rectangle

\xymatrix{MU \ar[d] \ar[r] & F (EG_{+}, inf_{1}^{G}MU) \ar[r] & F (EG_{+}, inf_{1}^{G}ku) \ar[d] \\ K \ar[rr] &  & F (EG_{+}, inf_{1}^{G}K)}

or equivalently, by the rectangle

\xymatrix{MU ∧ EG_{+} \ar[d] \ar[r] & MU \ar[r] & ku \ar[d] \\ K ∧ EG_{+} \ar[rr] &  & K.}

Embedding this in the diagram

\xymatrix{inf_{1}^{G}MU ∧ EG_{+} \ar[rr] \ar@{-->}[dr]^{≃} \ar[dd] &  & inf_{1}^{G}MU \ar[dr] \ar[dd] \\  & MU ∧ EG_{+} \ar[rr] \ar[dd] &  & MU \ar[dd] \\ inf_{1}^{G}ku ∧ EG_{+} \ar[rr] \ar@{-->}[dr]^{≃} &  & inf_{1}^{G}ku \ar[dr] \\  & ku ∧ EG_{+} \ar[rr] &  & ku \ar[d] \\  &  &  & K}

we see that commutativity follows from the non-equivariant description of MU ⟶ ku and the compatibility of splittings for the pair (ku, K) for the pair (MU, K).

Uniqueness of the lift ν of (ν′, ν′′) follows since [EG_{+} ∧ MU, K]_{1}^{G} = 0. Similarly the fact that ν is a commutative and associative ring map follows from commutativity and associativity after smashing with EG_{+} or inverting v together with the fact that [EG_{+} ∧ MU^{∧i}, K]_{1}^{G} = 0 for i = 2, 3. □

Proof of 11.1: From the Atiyah-Segal completion theorem

K_{G}^{0}(BU(n)) = R(G × U(n))_{J(G,U(n))}^{∧}

where J(G, U(n)) is the ideal in R(G × U(n)) of elements restricting to zero on every subgroup H ⊆ G × U(n) intersecting 1 × U(n) trivially. In fact

J(G, U(n)) = (e^{R}(V ⊗ λ^{1}), e^{R}(V ⊗ λ^{2}), …, e^{R}(V ⊗ λ^{n}) | V is a representation of G),

where λ^{i} is the ith exterior power of the natural representation of U(n).

If V is an n-dimensional representation of G we have

MU(V) = Gr_{n}(V ⊕ 𝒰)^{γ_{n}}

where 𝒰 is a complete complex universe and γ_{n} is the natural n-plane bundle. If we identify Gr_{n}(V ⊕ 𝒰) with BU(n), the K theory Thom class of γ_{n} is e^{R}(V ⊗ λ^{1}), so

K_{G}^{∗}(MU(V)) = e^{R}(V ⊗ λ^{1}) · (R(G × U(n))_{J(G,U(n))}^{∧}.

Because of the multiplicative property, these Thom classes are compatible under suspension as V varies and hence give rise to an element of

K_{G}^{0}(MU) = lim_{←}_{V} K_{G}^{0}(MU(V)),

where the absence of a lim_{←}^{1}-term is due to the fact that K_{G}^{1}(MU(V)) = 0.

The fact that these give a ring map again follows from multiplicativity of Euler classes. The non-equivariant behaviour follows since the Euler classes are compatible under forgetting equivariance. □

Remark 11.3. It seems likely that ν can be chosen to be a map of commutative algebras over the sphere spectrum, but there are some questions of compatibility to be decides. To start with, the given map ν′′ can be chosen to be a highly structured ring map. Next, Joachim [11, Appendix A] constructs a highly structured Atiyah-Bott-Shapiro map MSpin^{c} ⟶ K, and there is a model for MU giving a highly structured ring map MU ⟶ MSpin^{c}. It remains to check that the two implied structures on F (EG_{+}, inf_{1}^{G}K) agree.

Finally we should discuss the equivariant formal group law [1] associated to ku when G = A is abelian. Lemma 11.4. The natural map

ku_{A}^{∗}(ℂP (𝒰)) ⟶ K_{A}^{∗}(ℂP (𝒰))

is injective in cohomological degree 2, and the canonical complex orientation y(ϵ) := e^{R}(z)/v lies in the image. Proof: The fact that e^{R}(z)/v is in the image is immediate by considering the diagram

ku_{A×U(1)}^{∗} ⟶ K_{A×U(1)}^{∗}
↓  ↓
ku_{A}^{∗}(ℂP (𝒰)) ⟶ K_{A}^{∗}(ℂP (𝒰))

The injectivity follows from 2.6(ii) and the Artin-Rees lemma. □

Lemma 11.5. The element y(ϵ) ∈ ku_{A}^{2}(ℂP (𝒰)) is a complex orientation and the associated universal group law is multiplicative. Proof: In the range where we need to calculate ku^{∗}(ℂP (𝒰)) and ku^{∗}(ℂP (𝒰) × ℂP (𝒰)) are embedded in the corresponding groups for periodic K theory; for ku^{∗}(ℂP (𝒰)) this is in 11.4, and for ku^{∗}(ℂP (𝒰)) we have the analagous diagram

ku_{A×U(1)×U(1)}^{∗}  ⟶  K_{A×U(1)×U(1)}^{∗}
↓  ↓
ku_{A}^{∗}(ℂP (𝒰) × ℂP (𝒰)) ⟶ K_{A}^{∗}(ℂP (𝒰) × ℂP (𝒰))
□

References

[1] M.M.Cole, J.P.C.Greenlees and I. Kriz “Equivariant formal group laws.” Proc. London Math. Soc 81 (2000) 355-386.

[2] M.M. Cole, J.P.C.Greenlees and I. Kriz “The topological universality of equivariant bordism.” Math. Z. (to appear)

[3] R.R.Bruner and J.P.C.Greenlees “Connective K theory of finite groups.” Mem. American Maths. Soc. (to appear), 129pp.

[4] A. D. Elmendorf, I.Kriz, M.Mandell and J. P. May “Rings modules and algebras in stable homotopy” AMS (1997)

[5] A. D. Elmendorf, and J. P. May “Algebras over equivariant sphere spectra” J. Pure and Applied Algebra 116 (1997) 139-149

[6] J.P.C.Greenlees “K-homology of universal spaces and local cohomology of the representation ring” Topology 32 (1993) 295-308.

[7] J.P.C.Greenlees “Augmentation ideals of equivariant cohomology rings.” Topology 37 (1998) 1313-1323

[8] J.P.C.Greenlees “Equivariant forms of connective K theory.” Topology 38 (1999) 1075-1092.

[9] J.P.C.Greenlees “Multiplicative equivariant formal group laws.” J. Pure and Applied Algebra (to appear) 20pp.

[10] M.J.Hopkins and M.E.Mahowald “From elliptic curves to homotopy theory.” Preprint (1998)

[11] M. Joachim “Higher coherences for equivariant K theory.” Preprint (2001) 27pp

[12] L.G.Lewis, J.P.May, M.Steinberger (with contributions by J.E.McClure) “Equivariant stable homotopy theory.” Lecture notes in maths. 1213 Springer-Verlag (1986)

[13] G.B.Segal “K-homology theory and algebraic K theory” Lecture notes in maths. 575, Springer-Verlag (1977) 113-127.

EQUIVARIANT CONNECTIVE K THEORY 21

Department of Pure Mathematics, Hicks Building, Sheffield S3 7RH. UK.

E-mail address: j.greenlees@sheffield.ac.uk