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CHERN CLASSES IN EQUIVARIANT BORDISM

STEFAN SCHWEDE

Abstract. We introduce Chern classes in U(m)-equivariant homotopical bordism that refine the Conner- Floyd-Chern classes in the MU- of BU(m). For products of unitary groups, our Chern classes form regular sequences that generate the augmentation ideal of the equivariant bordism rings. We also use the Chern classes to give a reformulation of the MU-completion theorem of Greenlees-May and La Vecchia.

Introduction Complex MU is arguably the most important cohomology theory in . It represents the bordism theory of stably almost complex , and it is the universal complex oriented cohomology theory; via Quillen’s celebrated theorem, MU is the entry gate for the theory of formal group laws into stable , and thus the cornerstone of chromatic stable homotopy theory. Tom Dieck’s homotopical equivariant bordism MUG [17], defined with the help of equivariant Thom spaces, strives to be the legitimate equivariant refinement of , for compact Lie groups G. The theory MUG is the universal equivariantly-complex oriented theory; and for abelian compact Lie ∗ groups, the coefficient ring MUG carries the universal G-equivariant formal group law [8]. Homotopical equivariant bordism receives a homomorphism from the geometrically defined equivariant bordism theory; due to the lack of equivariant transversality, this homomorphism is not an isomorphism for non-trivial ∗ groups. For general compact Lie groups G, the equivariant bordism rings MUG are still largely mysterious; the purpose of this paper is to shed new light on their structure for unitary groups, and for products of unitary groups. Chern classes are important characteristic classes for complex vector bundles that were originally introduced in integral singular cohomology. Conner and Floyd [4, Corollary 8.3] constructed Chern classes for complex vector bundles in complex cobordism; in the universal cases, these yield classes 2k ck ∈ MU (BU(m)) that are nowadays referred to as Conner-Floyd-Chern classes. Conner and Floyd’s construction works in much the same way for any complex oriented cohomology theory; in singular coho- mology, it reduces to the classical Chern classes. The purpose of this note is to define and study Chern ∗ classes in U(m)-equivariant homotopical equivariant bordism MUU(m) that map to the Conner-Floyd- Chern classes under tom Dieck’s bundling homomorphism [17, Proposition 1.2]. Our classes satisfy the analogous formal properties as their classical counterparts, including the equivariant refinement of the Whitney sum formula, see Theorem 1.3. We will use our Chern classes and the splitting of [15] to prove new structure results about the equivariant ∗ bordism rings MUU(m) for unitary groups, or more generally for products of unitary groups. To put this into context, we recall that in the special case when G is an abelian compact Lie group, a lot of structural results about equivariant homotopical bordism have been established to date. Indeed, for abelian compact ∗ Lie groups, the MUG is concentrated in even degrees and free as a module over the non- ∗ ∗ ∗ equivariant cobordism ring MU [3, Theorem 5.3], [11]; the bundling homomorphism MUG −→ MU (BG) ∗ ∗ is completion at the augmentation ideal of MUG [2, Theorem 1.1], [12]; and the coefficient ring MUG carries the universal G-equivariant formal group law [8]. For non-abelian compact Lie groups G, however, ∗ the equivariant bordism rings MUG are still largely mysterious; the evenness conjecture [7, Conjecture ∗ ∗ 1.2] asks whether in this context, MUG is also concentrated in even degrees and free over MU . At the time of this writing, the evenness conjecture is still wide open; in particular, it is presently not known for unitary groups U(m) with m ≥ 2. The main result of this note is the following:

Date: September 4, 2021; 2020 AMS Math. Subj. Class.: 55N91, 55P91, 55Q91. 1 2 STEFAN SCHWEDE

Theorem. Let m ≥ 1 be a natural number. (m) (m) (m) (i) The sequence of Chern classes cm , cm−1 . . . , c1 is a regular sequence that generates the augmenta- ∗ tion ideal of the graded- MUU(m). ∗ ∗ (ii) The completion of MUU(m) at the augmentation ideal is a graded MU -power series algebra in the above Chern classes. ∗ ∗ (iii) The bundling homomorphism MUU(m) −→ MU (BU(m)) extends to an isomorphism ∗ ∧ ∗ (MUU(m))I −→ MU (BU(m)) from the completion at the augmentation ideal. We prove this result as a special case of Theorem 2.3 below; the more general version applies to products of unitary groups. As we explain in Remark 3.6, the regularity of the Chern classes also implies that the Greenlees-May local spectral sequence converging to MU∗(BU(m)) degenerates because the relevant local homology groups vanish in positive degrees. As another application we use the Chern classes in equivariant bordism to to give a reformulation of work of Greenlees-May [7] and La Vecchia [9] on the completion theorem for MUG, see Theorem 3.5.

1. Equivariant MU-Chern classes In this section we introduce the Chern classes in U(m)-equivariant homotopical bordism, see Definition 1.1. We establish their basic properties in Theorem 1.3, including a Whitney sum formula and the fact that the bundling homomorphism takes our Chern classes to the Conner-Floyd-Chern classes in MU- cohomology. We begin by fixing our notation. For a compact Lie group G, we write MUG for the G-equivariant homotopical bordism introduced by tom Dieck [17]. For our purposes, it is highly relevant that the theories MUG for varying compact Lie groups G assemble into a global stable homotopy type, see [14, n G Example 6.1.53]. For an integer n, we write MUG = π−n(MU) for the G-equivariant coefficient group in cohomological degree n. Since MU comes with the structure of a global ring spectrum, it supports graded-commutative mul- ∗ tiplications on MUG, as well as biadditive, associative, unital and commutative external multiplication pairings k l k+l × : MUG × MUK −→ MUG×K for all pairs of compact Lie groups G and K. We write νk for the tautological representation of the unitary group U(k) on Ck; we denote its Euler class by 2k ek = eU(k),νk ∈ MUU(k) . A 0 We write U(k, m − k) for the block subgroup of U(m) consisting of matrices of the form ( 0 B ) for (A, B) ∈ U(m) ∗ ∗ U(k) × U(m − k). We write trU(k,m−k) : MUU(k,m−k) −→ MUU(m) for the degree zero transfer associated to the inclusion U(k, m − k) −→ U(m), see for example [14, Construction 3.2.11]. Definition 1.1. For 0 ≤ k ≤ m, the k-th in equivariant complex bordism is the class (m) U(m) 2k ck = trU(k,m−k)(ek × 1m−k) ∈ MUU(m) ,

0 (m) where 1m−k ∈ MUU(m−k) is the multiplicative unit. We also set ck = 0 for k > m.

In the extreme cases k = 0 and k = m, we recover familiar classes: since e0 is the multiplicative unit in ∗ (m) 0 the non-equivariant cobordism ring MU , the class c0 = 1m is the multiplicative unit in MUU(m). In (m) the other extreme, cm = em = eU(m),νm is the Euler class of the tautological U(m)-representation. As (m) we will show in Theorem 1.3 (ii), the classes ck are compatible in m under restriction to smaller unitary groups. (m) The Chern class ck is defined as a transfer; so identifying its restriction to a subgroup of U(m) involves a double coset formula. The following double coset formula will take care of all cases we need in this paper; it ought to be well-known to experts, but I do not know a reference. The case m = 2 is treated in [5, Example IV.9], and the case l = 1 is established in [16, Lemma 4.2]. The double coset space CHERN CLASSES IN EQUIVARIANT BORDISM 3

U(i, j)\U(m)/U(k, l) is discussed at various places in the literature, for example [13, Example 3], but I have not seen the resulting double coset formula spelled out explicitly. Proposition 1.2 (Double coset formula). Let i, j, k, l be positive natural numbers such that i + j = k + l. Then U(i+j) U(k+l) X U(i,j) ∗ U(k,l) resU(i,j) ◦ trU(k,l) = trU(d,i−d,k−d,j−k+d) ◦γd ◦ resU(d,k−d,i−d,l−i+d) , 0,k−j≤d≤i,k where γd ∈ U(m) is the permutation matrix of the shuffle permutation χd ∈ Σm given by m for 1 ≤ m ≤ d,  m − d + i for d + 1 ≤ m ≤ k, χd(m) = m − d + k for k + 1 ≤ m ≤ k + i − d, and  m for m > k + i − d.

G G Proof. We refer to [10, IV 6] or [14, Theorem 3.4.9] for the general double coset formula for resK ◦ trH for two closed subgroups H and K of a compact Lie group G; we need to specialize it to the situation at hand. We first consider a matrix A ∈ U(m) such that the center Z of U(i, j) is not contained in the U(i, j)-stabilizer A SA = U(i, j) ∩ U(k, l) of the coset A · U(k, l). Then SA ∩ Z is a proper subgroup of the center Z of U(i, j), which is isomorphic to U(1) × U(1). So SA ∩ Z has strictly smaller dimension than Z. Since the center of U(i, j) is contained in the normalizer of SA, we conclude that the group SA has an infinite Weyl group inside U(i, j). All summands in the double coset formula indexed by such points then involve infinite index transfers, and hence they vanish. So all non-trivial contributions to the double coset formula stem from double cosets U(i, j) · A · U(k, l)   such that S contains the center of U(i, j). In particular the matrix −Ei 0 then belongs to S . We A 0 Ej A write L = A · (Ck ⊕ 0l), a complex k-plane in Ck+l; we consider x ∈ Ci and y ∈ Cj such that (x, y) ∈ L.   Because −Ei 0 · L = L, we deduce that (−x, y) ∈ L. Since (x, y) and (−x, y) belong to L, so do 0 Ej the vectors (x, 0) and (y, 0). We have thus shown that the k-plane L = A · (Ck ⊕ 0l) is spanned by the intersections i j i j L ∩ (C ⊕ 0 ) and L ∩ (0 ⊕ C ) .

We organize the cosets with this property by the dimension of the first intersection: we define Md as the closed subspace of U(m)/U(k, l) consisting of those cosets A · U(k, l) such that i j i j dimC(L ∩ (C ⊕ 0 )) = d and dimC(L ∩ (0 ⊕ C )) = k − d .

If Md is non-empty, we must have 0, k − j ≤ d ≤ i, k. The group U(i, j) acts transitively on Md, and the coset γd · U(k, l) belongs to Md; so Md is the U(i, j)-orbit type of U(m)/U(k, l) for the conjugacy class of γd Sγd = U(i, j) ∩ U(k, l) = U(d, i − d, k − d, j − k + d) .

The corresponding orbit space U(i, j)\Md = U(i, j) · γd · U(k, l) is a single point inside the double coset space, so its internal is 1. This orbit type thus contributes the summand

U(i,j) ∗ U(k,l) trU(d,i−d,k−d,j−k+d) ◦γd ◦ resU(d,k−d,i−d,l−i+d) to the double coset formula.  In [4, Corollary 8.3], Conner and Floyd defined characteristic classes for complex vector bundles ξ : E −→ X in the non-equivariant MU-cohomology rings MU∗(X). In the universal cases, these yield 2k classes ck ∈ MU (BU(m)) that are nowadays referred to as Conner-Floyd-Chern classes. The next (m) theorem spells out the key properties of our Chern classes ck ; parts (i), (ii) and (iii) roughly say that all the familiar structural properties of the Conner-Floyd-Chern classes in MU∗(BU(m)) already hold for our Chern classes in U(m)-equivariant MU-theory. Part (iv) of the theorem refers to the bundling maps ∗ ∗ MUG −→ MU (BG) defined by tom Dieck in [17, Proposition 1.2]. 4 STEFAN SCHWEDE

Theorem 1.3. (i) For all 0 ≤ k ≤ m = i + j, the relation U(m) (m) X (i) (j) resU(i,j)(ck ) = cd × ck−d d=0,...,k 2k holds in the group MUU(i,j). (ii) The relation ( c(m−1) for 0 ≤ k ≤ m − 1, and resU(m) (c(m)) = k U(m−1) k 0 for k = m 2k holds in the group MUU(m−1). (iii) Let T m denote the diagonal maximal of U(m). Then the restriction homomorphism U(m) 2k 2k resT m : MUU(m) −→ MUT m (m) ∗ ∗ takes the class ck to the k-th elementary symmetric polynomial in classes p1(e1), . . . , pm(e1), where m pi : T −→ T = U(1) is the projection to the i-th factor. (iv) The bundling map ∗ ∗ MUU(m) −→ MU (BU(m)) (m) takes the class ck to the k-th Conner-Floyd-Chern class. U(m) U(m) Proof. (i) This property exploits the double coset formula for resU(i,j) ◦ trU(k,m−k) recorded in Proposition 1.2, which is the second in equation in the following list: U(m) (m) U(m) U(m) resU(i,j)(ck ) = resU(i,j)(trU(k,m−k)(ek × 1m−k)) X U(i,j) ∗ U(k,m−k) = trU(d,i−d,k−d,j−k+d)(γd (resU(d,k−d,i−d,j−k+d)(ek × 1m−k))) d=0,...,k X U(i,j) ∗ = trU(d,i−d,k−d,j−k+d)(γd (ed × ek−d × 1i−d × 1j−k+d)) d=0,...,k X U(i,j) = trU(d,i−d,k−d,j−k+d)(ed × 1i−d × ek−d × 1j−k+d) d=0,...,k X U(i) U(j) = trU(d,i−d)(ed × 1i−d) × trU(k−d,j−k+d)(ek−d × 1j−k+d) d=0,...,k X (i) (j) = cd × ck−d d=0,...,k Part (ii) for k < m follows from part (i) by restriction from U(m − 1, 1) to U(m − 1): U(m) (m) U(m−1,1) U(m) (m) resU(m−1)(ck ) = resU(m−1) (resU(m−1,1)(ck )) U(m−1,1) (m−1) (1) (m−1) (1) = resU(m−1) (ck−1 × c1 + ck × c0 ) (m−1) U(1) (1) (m−1) U(1) (1) (m−1) = ck−1 × res1 (c1 ) + ck × res1 (c0 ) = ck . (1) U(1) ∗ ∗ We have used that the class c1 = e1 is in the kernel of the augmentation res1 : MUU(1) −→ MU . (m) ∗ The class cm = eU(m),νm restricts to 0 in MUU(m−1) because the restriction of the the tautological U(m)-representation νm to U(m − 1) splits off a trivial 1-dimensional summand. (iii) An inductive argument based on property (i) shows that U(m) (m) U(m) (m) resT m (ck ) = resU(1,...,1)(ck ) X Y ∗ (1) Y ∗ (1) = pa(c1 ) · pb (c0 ) A⊂{1,...,m},|A|=k a∈A b6∈A X Y ∗ = pa(e1) . A⊂{1,...,m},|A|=k a∈A This is the desired relation. CHERN CLASSES IN EQUIVARIANT BORDISM 5

(iv) As before we let T m denote the diagonal maximal torus in U(m). The splitting principle holds for non-equivariant complex oriented cohomology theories, i.e., the right vertical map in the commutative square of graded rings is injective:

∗ ∗ MUU(m) / MU (BU(m))

U(m) (Bi)∗ resT m

 ∗ ∗  m ∗ ∗ ∗ MUT m / MU (BT ) MU [[p1(e1), . . . , pm(e1)]]

The k-th Conner-Floyd-Chern class is characterized as the unique element of MU2k(BU(m)) that maps ∗ ∗ to the k-th elementary symmetric polynomial in the classes p1(e1), . . . , pm(e1). Together with part (iii), this proves the claim. 

U(m) ∗ ∗ We alert the reader that the restriction homomorphism resT m : MUU(m) −→ MUT m is not injective for m ≥ 2, see Example 2.2. So in contrast to the non-equivariant situation for complex oriented cohomology ∗ theories, the Chern classes in MUU(m) are not characterized by their restrictions to the maximal torus.

2. Regularity results In this section we use the Chern classes to formulate new structural properties of the equivariant bordism ∗ ∗ ring MUU(m). In particular, we can say what MUU(m) looks like after dividing out some of the Chern classes, and after completing at the Chern classes. The following theorem states these facts more generally for U(m) × G instead of U(m); by induction on the number of factors, we can then deduce corresponding results for products of unitary groups, see Theorem 2.3. The results in this section make crucial use of the splitting theorem for global established in [15]. Theorem 2.1. For every compact Lie group G and all 0 ≤ k ≤ m, the sequence of Chern classes

(m) (m) (m) (cm × 1G, cm−1 × 1G, . . . , ck+1 × 1G) ∗ is a regular sequence in the graded-commutative ring MUU(m)×G that generates the kernel of the surjective restriction homomorphism

U(m)×G ∗ ∗ resU(k)×G : MUU(m)×G −→ MUU(k)×G .

(m) (m) (m) In particular, the sequence of Chern classes (cm , cm−1, . . . , c1 ) is a regular sequence that generates the ∗ augmentation ideal of the graded-commutative ring MUU(m). Proof. We argue by downward induction on k. The induction starts with k = m, where there is nothing to show. Now we assume the claim for some k ≤ m, and we deduce it for k − 1. The inductive hypothesis (m) (m) ∗ shows that cm ×1G, . . . , ck+1 ×1G is a regular sequence in the graded-commutative ring MUU(m)×G, and U(m)×G that the restriction homomorphism resU(k)×G factors through an isomorphism ∗ (m) (m) ∼ ∗ MUU(m)×G/(cm × 1G, . . . , ck+1 × 1G) = MUG×U(k) .

We exploit that the various equivariant bordism spectra MUG underlie a global spectrum, see [14, Example U(k)×G 6.1.53]; thus the restriction homomorphism resU(k−1)×G is surjective by Theorem 1.4 and Proposition 2.2 of [15]. Hence the standard long exact sequence unsplices into a short exact sequence of graded MU∗-modules:

resU(k)×G ∗−2k (ek×1G)·− ∗ U(k−1)×G ∗ 0 −→ MUU(k)×G −−−−−−−→ MUU(k)×G −−−−−−−−→ MUU(k−1)×G −→ 0 Because U(m)×G (m) (k) resU(k)×G (ck × 1G) = ck × 1G = ek × 1G , (m) ∗ (m) (m) (m) we conclude that ck × 1G is a non zero-divisor in MUU(m)×G/(cm × 1G, cm−1 × 1G, . . . , ck+1 × 1G), and (m) ∗ that additionally dividing out ck × 1G yields MUU(k−1)×G. This completes the inductive step.  6 STEFAN SCHWEDE

Example 2.2 (Mind the ordering of the Chern classes). We alert the reader that the order of the Chern classes in Theorem 2.1 is crucial for the regularity property, and that when taken in a different order, the Chern classes will typically not form a regular sequence. We illustrate this in the simplest non-trivial example of the group U(2). We let N denote normalizer of U(1, 1) inside U(2), a semidirect product of a cyclic group of order 2 permuting the two factors of U(1, 1). The class

U(2) 0 κ = 1 − trN (1) ∈ MUU(2) has infinite order because the U(2)-geometric fixed point map takes it to the multiplicative unit. In particular, the class κ is nonzero. The double coset formula [10, IV Corollary 6.7 (i)]

U(2) U(2) N resU(1,1)(trN (1)) = resU(1,1)(1) = 1

U(2) 0 0 implies that the class κ is in the kernel of the restriction homomorphism resU(1,1) : MUU(2) −→ MUU(1,1). Reciprocity for restriction and transfers then yields the relation (2) U(2) U(2) U(2) c1 · κ = trU(1,1)(e1 × 1) · κ = trU(1,1)((e1 × 1) · resU(1,1)(κ)) = 0 .

(2) ∗ (2) (2) The Chern class c1 is thus a zero-divisor in the ring MUU(2), and so the sequence (c1 , c2 ) is not regular. (2) Since we are at it, we also argue that the class κ is infinitely divisible by the Euler class e2 = c2 . So κ is also in the kernel of the completion map MU∗ −→ (MU∗ )∧ at the ideal (e ). To prove our U(2) U(2) (e2) 2 −4n claim we inductively define classes κn ∈ MUU(2) that satisfy

(2) κn · e2 = κn−1 and c1 · κn = 0 .

The construction starts with κ0 = κ. For the inductive step we suppose that κn−1 has been constructed (2) and satisfies c1 · κn−1 = 0. Then U(2) U(2) (2) e1 · resU(1)(κn−1) = resU(1)(c1 · κn−1) = 0 .

∗ U(2) Since e1 is a non zero-divisor in MUU(1), this implies that resU(1)(κn−1) = 0. Since the Euler class e2 U(2) generates the kernel of resU(1), there is a class κn such that κn · e2 = κn−1. Then

(2) (2) c1 · κn · e2 = c1 · κn−1 = 0 . ∗ (2) Because e2 is a non zero-divisor in MUU(2), this implies that c1 · κn = 0. This completes the inductive construction of the classes κn with the desired properties. ∗ ∗ We can now determine the completion of MUU(m) at the augmentation ideal as an MU -power series algebra on the Chern classes. We state this somewhat more generally for products of unitary groups, which we write as

U(m1, . . . , ml) = U(m1) × · · · × U(ml) ,

for natural numbers m1, . . . , ml ≥ 1. For 1 ≤ i ≤ l, we write pi : U(m1, . . . , ml) −→ U(mi) for the projection to the i-th factor, and we set c[i] = p∗(c(mi)) = 1 × c(mi) × 1 ∈ MU2k . k i k U(m1,...,mi−1) k U(mi+1,...,ml) U(m1,...,ml)

The following theorem was previously known for tori, i.e., for m1 = ··· = ml = 1.

Theorem 2.3. Let m1, . . . , ml ≥ 1 be positive integers. (i) The sequence of Chern classes (2.4) c[1] , . . . , c[1], c[2] , . . . , c[2], . . . , c[l] , . . . , c[l] m1 1 m2 1 ml 1 is a regular sequence that generates the augmentation ideal of the graded-commutative ring MU∗ . U(m1,...,ml) (ii) The completion of MU∗ at the augmentation ideal is a graded MU∗-power series algebra U(m1,...,ml) in the Chern classes (2.4). CHERN CLASSES IN EQUIVARIANT BORDISM 7

(iii) The bundling map MU∗ −→ MU∗(BU(m , . . . , m )) extends to an isomorphism U(m1,...,ml) 1 l (MU∗ )∧ −→ MU∗(BU(m , . . . , m )) U(m1,...,ml) I 1 l from the completion at the augmentation ideal. Proof. Part (i) follows from Theorem 2.1 by induction on the number l of factors. We prove parts (ii) and (iii) together. We must show that for every k ≥ 1, MU∗ /Ik is free as U(m1,...,ml) an MU∗-module on the monomials of degree less than k in the Chern classes (2.4). There is nothing to show for k = 1. The short exact sequence 0 −→ Ik/Ik+1 −→ MU∗ /Ik+1 −→ MU∗ /Ik −→ 0 U(m1,...,ml) U(m1,...,ml) and the inductive hypothesis reduce the claim to showing that Ik/Ik+1 is free as a MU∗-module on the monomials of degree exactly k in the Chern classes (2.4). Since the augmentation ideal I is generated by these Chern classes, the k-th power Ik is generated, as a module over MU∗ , by the monomials U(m1,...,ml) of degree k. So Ik/Ik+1 is generated by these monomials as a module over MU∗. The bundling map MU∗ −→ MU∗(BU(m , . . . , m )) is a homomorphism of augmented U(m1,...,mk) 1 k ∗ [i] MU -algebras, and it takes the Chern class ck to the inflation of the k-th Conner-Floyd-Chern class along the projection to the i-th factor. By the theory of complex orientations, the collection of these ∗ ∗ Conner-Floyd-Chern classes are MU -power series generators of MU (BU(m1, . . . , ml)); in particular, ∗ ∗ the images of the Chern class monomials are MU -linearly independent in MU (BU(m1, . . . , ml)). Hence k k+1 these classes are themselves linearly independent in I /I .  Remark 2.5. The previous regularity theorems are special cases of the following more general results that hold for every global MU-module E: (m) (m) • For every compact Lie group G, the sequence of Chern classes cm ×1G, . . . , c1 ×1G acts regularly ∗ ∗ on the graded MUU(m)×G-module EU(m)×G. • The restriction homomorphism U(m)×G ∗ ∗ resG : EU(m)×G −→ EG factors through an isomorphism ∗ (m) (m) ∼ ∗ EU(m)×G/(cm × 1G, . . . , c1 × 1G) = EG . • For all m , . . . , m ≥ 1, the sequence of Chern classes (2.4) acts regularly on the graded MU∗ - 1 l U(m1,...,ml) module E∗ . U(m1,...,ml) Remark 2.6 (Equivariant Stiefel-Whitney classes). The global complex bordism spectrum MU has a real analog MO, the global Thom ring spectrum defined in [14, Example 6.1.7]. For every compact Lie group G, the underlying G-homotopy type of MO is the real analog of tom Dieck’s equivariant bordism. By a theorem of Br¨ocker and Hook [1, Theorem 4.1], the G-equivariant homology theory represented by MO is stable equivariant bordism. Restricted to elementary abelian 2-groups, the equivariant homotopy groups of MO carry the universal global 2-torsion group law [8, Theorem D]. We can define Stiefel-Whitney classes in equivariant MO-theory in much the same way as for MU, by setting (m) O(m) R k wk = trO(k,m−k)(ek × 1m−k) ∈ MOO(m) . Here R k e = e R ∈ MO . k O(k),νk O(k) R k is the real Euler class of the tautological representation νk of the O(k) on R . I suspect that these Stiefel-Whitney classes will also have all the desirable properties, and lead to real analogs of Theorem 1.3, 2.1 and 2.3. However, the orthogonal analog of the double coset formula in Proposition 1.2 will have additional terms, because the relevant double coset space will have additional points whose stabilizers have finite Weyl groups. One has to carefully check that these extra terms don’t create any trouble, and I have not done that. What makes me optimistic is that for the splitting of [15], the extra transfer term in the real case (in contrast to the unitary and symplectic cases) ultimately did not interfere. 8 STEFAN SCHWEDE

We can also define symplectic versions of the Chern classes and Stiefel-Whitney class

(m) Sp(m) H 4k pk = trSp(k,m−k)(ek × 1m−k) ∈ MSpSp(m) . Here H 2k e = e H ∈ MSp . k Sp(k),νk Sp(k) H k is the symplectic Euler class of the tautological representation νk of the symplectic group Sp(k) on H .

3. The MU-completion theorem via Chern classes In this section we use the Chern classes to reformulate the work of Greenlees-May [7] and La Vecchia [9] on the completion theorem for MUG. We emphasize that the essential arguments of this section are all contained in [7] and [9]; the Chern class formalism just gives a way to arrange them in a slightly more conceptual and concise way. ∗ The references [7, 9] ask for a finitely generated ideal of MUG that is ’sufficiently large’ in the technical sense of [7, Definition 2.4]; our main innovation is the observation that the ideal generated by the Chern classes of any faithful G-representation exactly serves this purpose. Our discussion is self-contained, and it avoids explicit mentioning of ’sufficiently large’ ideals. Construction 3.1 (Chern classes of G-representations). We let V be a complex representation of a compact Lie group G. We let ρ : G −→ U(m) be any continuous homomorphism that classifies V , i.e., ∗ such that ρ (νm) is isomorphic to V ; here m = dimC(V ). For k ≥ 0, we set ∗ (m) 2k ck(G, V ) = ρ (ck ) ∈ MUG and refer to this as the k-th Chern class of the G-representation V . Then in particular, c0(G, V ) = 1, cm(G, V ) = eG(V ) is the Euler class, and ck(G, V ) = 0 for k > m. 2 Example 3.2. As an example, we consider the tautological representation ν2 of SU(2) on C . By the general properties of Chern classes we have c0(SU(2), ν2) = 1, c2(SU(2), ν2) = eSU(2)(ν2) is the Euler class, and ck(SU(2), ν2) = 0 for k ≥ 3. The first Chern class of ν2 can be rewritten by using a double coset formula as follows: U(2) (2) U(2) U(2) c1(SU(2), ν2) = resSU(2)(c1 ) = resSU(2)(trU(1,1)(e1 × 1)) SU(2) U(1,1) SU(2) = trT (resT (e1 × 1)) = trT (eT,χ) . λ 0 ∼ Here T = {( 0 λ¯ ): λ ∈ U(1)} is the diagonal maximal torus of SU(2), and χ : T = U(1) is the character that projects onto the upper left diagonal entry. Definition 3.3 (Chern-Koszul complex). The Chern-Koszul complex K(G, V ) of a faithful complex rep- resentation V of a compact Lie group G is the Koszul complex, in the of G-equivariant MUG- modules, of the sequence of Chern classes c1(G, V ), . . . , cm(G, V ), where m = dimC(V ). For the convenience of the reader, we recall the definition of the Chern-Koszul complex in more detail, l compare [7, Section 1]. For any equivariant homotopy class x ∈ MUG, we write MUG[1/x] for the localization at x, of MUG in the homotopy category of G-equivariant MUG-module spectra; in other words, MUG[1/x] is a homotopy colimit (mapping telescope) in the triangulated category of the sequence −·x l −·x 2l −·x 3l −·x MUG −−→ Σ MUG −−→ Σ MUG −−→ Σ MUG −−→ ....

We write K(x) for the fiber of the MUG −→ MUG[1/x]. Then the Chern-Koszul complex is

K(G, V ) = K(c1(G, V )) ∧MUG ... ∧MUG K(cm(G, V )) .

The smash product of the K(ci(G, V )) −→ MUG provides a morphism of G-equivariant MUG- module spectra V : K(G, V ) −→ MUG. By general principles, the complex K(G, V ), like any Koszul complex, only depends on the radical of the ideal generated by the classes c1(G, V ), . . . , cm(G, V ). But more is true: as a consequence of Theorem 3.5 below, the Koszul complex K(G, V ) is entirely independent, as a G-equivariant MUG-module, of the faithful representation V . CHERN CLASSES IN EQUIVARIANT BORDISM 9

Proposition 3.4. Let V be a faithful complex representation of a compact Lie group G.

(i) The morphism V : K(G, V ) −→ MUG is an equivalence of underlying non-equivariant spectra. (ii) For every non-trivial closed subgroup H of G, the H-geometric fixed point spectrum ΦH (K(G, V )) is trivial.

Proof. (i) We suppose that dimC(V ) = m. The Chern classes c1(G, V ), . . . , cm(G, V ) all belong to the ∗ ∗ augmentation ideal of MUG, so they restrict to 0 in MU{1}, and hence the underlying non-equivariant spectrum of MUG[1/ci(G, V )] is trivial for each i = 1, . . . , m. Hence the morphisms K(ci(G, V )) −→ MUG are underlying non-equivariant equivalences for i = 1, . . . , m. So also the morphism V is an underlying non-equivariant equivalence. (ii) We let H be a non-trivial closed subgroup of G. We set W = V − V H , the orthogonal complement of the H-fixed points. This is a complex H-representation with W H = 0; moreover, W is nonzero because

H acts faithfully on V and H 6= {1}. For k = dimC(W ) we then have H G eH (W ) = ck(H,W ) = ck(H,W ⊕ V ) = ck(H,V ) = resH (ck(G, V )) ; the second equation uses the fact that adding a trivial representations leaves Chern classes unchanged, by part (ii) of Theorem 1.3. H H ∗ ∗ Since W = 0, the geometric fixed point homomorphism Φ : MUH −→ ΦH (MU) sends the Euler G H G class eH (W ) = resH (ck(G, V )) to an invertible element. The Φ ◦ resH commutes with inverting H G elements. Since the class Φ (resH (ck(G, V ))) is already invertible, the localization morphism MUG −→ H G MUG[1/ck(G, V )] induces an equivalence on H-geometric fixed points. Since the functor Φ ◦ resH is exact, it annihilates the fiber K(ck(G, V )) of the localization MUG −→ MUG[1/ck(G, V )]. The functor H G Φ ◦ resH is also strong monoidal, in the sense of a natural equivalence of non-equivariant spectra H H H H Φ (X ∧MUG Y ) ' Φ (X) ∧Φ (MUG) Φ (Y ) , for all G-equivariant MUG-modules X and Y . Since K(G, V ) contains K(ck(G, V )) as a factor (with H respect to ∧MUG ), we conclude that the spectrum Φ (K(G, V )) is trivial.  The following ’completion theorem’ is a reformulation of the combined work of Greenlees-May [7] and La Vecchia [9]. It is somewhat more precise in that an unspecified ‘sufficiently large’ finitely generated ideal ∗ of MUG is replaced by the ideal generated by the Chern classes of a faithful G-representation. The proof is immediate from the properties of the Koszul complex K(G, V ) listed in Proposition 3.4. We emphasize, however, that our proof is just a different way of arranging the arguments of [7] and [9], taking advantage of the Chern class formalism. Since the morphism G : K(G, V ) −→ MUG is a non-equivariant equivalence of underlying spectra, the morphism EG+ ∧ MUG −→ MUG that collapses the universal space EG to a point admits a unique lift to a morphism of G-equivariant MUG-modules ψ : EG ∧ MUG −→ K(G, V ) across V . Theorem 3.5 (Greenlees-May, La Vecchia). We let V be a faithful complex representation of a compact Lie group G. Then the morphism

ψ : EG ∧ MUG −→ K(G, V ) is an equivalence of G-equivariant MUG-module spectra Proof. Because the underlying space of EG is contractible, the composite

ψ V EG+ ∧ MUG −−→ K(G, V ) −−−→ MUG is an equivariant equivalence of underlying non-equivariant spectra. Since  is an equivariant equivalence of underlying non-equivariant spectra by Proposition 3.4, so is Ψ. For all non-trivial closed subgroups H of G, source and target of ψ have trivial H-geometric fixed points spectra, again by Proposition 3.4. So the morphism ψ induces an equivalence on geometric fixed point spectra for all closed subgroup of G, and it is thus an equivariant equivalence.  Remark 3.6. Applying HomG (−, MU ) to the equivalence ψ : EG ∧ MU −→ K(G, V ) of Theorem MUG G G 3.5 yields an equivalence of non-equivariant spectra HomG (K(G, V ), MU ) ' HomG (EG ∧ MU , MU ) ' MUBG . MUG G MUG + G G 10 STEFAN SCHWEDE

By filtering K(G, V ) by the partial smash products K(cj(G, V )) ∧MUG ... ∧MUG K(cm(G, V )) for j = 0, . . . , m, and then applying HomG (−, MU ), Greenlees and May [7, Corollary 1.6] obtain a local MUG G homology spectral sequence p,q I ∗ p+q E2 = H−p,−p(MUG) =⇒ MU (BG) .

The regularity results about Chern classes from Theorem 2.3 imply that whenever G = U(m1, . . . , ml) is p,q a product of unitary groups, the E2 -term vanishes for all p 6= 0, and the spectral degenerates into the isomorphism Ep,0 ∼ (MU∗ )∧ ∼ MUp+q(BU(m , . . . , m )) 2 = U(m1,...,ml) I = 1 l of Theorem 2.3 (iii), see the following Proposition 3.7. More generally, we could consider any global MU- module M, and the regularity properties hinted at in Remark 2.5 imply the analogous degeneration of the spectral sequence. The following vanishing result for local homology at regular sequences must be well-known, but I do not know of place where it is stated explicitly. I learned the following proof from John Greenlees.

Proposition 3.7. Let R be a graded-commutative ring and let M be a graded R-module. Let (d1, . . . , dm) be a sequence of homogeneous elements of R that is regular on M. Let I be the ideal generated by d1, . . . , dm. I I Then the local homology groups Hp (M) vanish for p > 0, and the natural homomorphism M −→ H0 (M) is completion at the ideal I. Proof. For a homogeneous element x of R, we write R/x = (x : R −→ R) for the complex of graded I R-modules concentrated in complex degrees 0 and 1. The local homology groups H∗ (M) can be calculated as the homology of the homotopy inverse limit of the tower of complexes of graded R-modules a a M ⊗R (R/d1) ⊗R ... ⊗R (R/dm) , see [6, page 447]; the limit is formed along the chain maps a+1 a+1 a a M ⊗R (R/d1 ) ⊗R ... ⊗R (R/dm ) −→ M ⊗R (R/d1) ⊗R ... ⊗R (R/dm) a+1 a obtained by tensoring M with the m chain maps R/di −→ R/di given di on the top cell, and by the I identity on the bottom cell. The associated Milnor short exact sequence shows that Hp (M) surjects onto the inverse limit a a lima≥0 Hp(M ⊗ (R/d1) ⊗R ... ⊗R (R/dm)) 1 with kernel a suitable lim -term. Since the sequence (d1, . . . , dm) is regular on M, so is the sequence a a (d1, . . . , dm) for every a ≥ 0. Hence ( a a a a ∼ M/(d1, . . . , dm) for p = 0, and Hp(M ⊗ (R/d ) ⊗R ... ⊗R (R/d )) = 1 m 0 for p 6= 0. Moreover, the transition maps in the inverse system are the quotient maps. Hence the inverse limit of the sequence is the completion of M at the ideal I = (d1, . . . , dm), and the derived inverse limit vanishes.  References [1] T Br¨ocker, E C Hook, Stable equivariant bordism. Math. Z. 129 (1972), 269–277. [2] G Comeza˜na,J P May, A completion theorem in complex bordism. Chapter XXVII in: J P May, Equivariant homotopy and cohomology theory. With contributions by M Cole, G Comeza˜na,S Costenoble, A D Elmendorf, J P C Greenlees, L G Lewis, Jr., R J Piacenza, G Triantafillou, and S. Waner. CBMS Regional Conference Series in Mathematics, 91. Amer. Math. Soc., 1996. xiv+366 pp. [3] G Comeza˜na, Calculations in complex equivariant bordism. Chapter XXVIII in: J P May, Equivariant homotopy and cohomology theory. With contributions by M Cole, G Comeza˜na,S Costenoble, A D Elmendorf, J P C Greenlees, L G Lewis, Jr., R J Piacenza, G Triantafillou, and S. Waner. CBMS Regional Conference Series in Mathematics, 91. Amer. Math. Soc., 1996. xiv+366 pp. [4] P E Conner, E E Floyd, The relation of cobordism to K-theories. Lecture Notes in Mathematics, No. 28 Springer-Verlag, Berlin-New York 1966 v+112 pp. [5] M Feshbach, The transfer and compact Lie groups. Trans. Amer. Math. Soc. 251 (1979), 139–169. [6] J P C Greenlees, J P May, Derived functors of I-adic completion and local homology. J. Algebra 149 (1992), no. 2, 438–453. [7] J P C Greenlees, J P May, Localization and completion theorems for MU-module spectra. Ann. of Math. (2) 146 (1997), no. 3, 509–544. CHERN CLASSES IN EQUIVARIANT BORDISM 11

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