Proceedings of Symposia in Pure Mathematics

Problem Session, Seattle 1996

Compiled by Dave Benson

Introduction

On one of the last days of the Seattle conference on Cohomology, Representations and Actions of Finite Groups, I hosted a problem session in which speakers were invited to present problems related to the area of the conference. I got a good response, and so I decided to collect these problems and invite further contributions from participants who were not able, for one reason or another, to contribute in person to that problem session.

The problems are presented here in alphabetical order of contributor. They have been lightly edited for uniformity of style.

Group actions (A. Adem)

Let X denote a finite complex. Determining which groups can act freely on X even for familiar examples can be quite difficult. The following is a basic question first raised by Conner:

1. If (ℤ/p)^{r} acts freely on S^{n_{1}} × ⋯ × S^{n_{k}}, does this imply that r ≤ k?

This has been settled in the equidimensional case in work by Carlsson [2] and Adem– Browder [1], except when p = 2 and the dimension of the spheres is 1, 3 or 7.

The following is a general homological conjecture due to G. Carlsson which would imply the above:

2. Let k be a field of characteristic p and C_{∗} a connected finite chain complex of finitely generated free k(ℤ/p)^{r}-modules. Is it true that

∑_{i} dim H_{i}(C) ≥ 2^{r}?

The one dimensional case has recently been established by Adem and Swan. Conversely we have the following question first raised by Benson and Carlson:

3. Let G is a finite group of rank r (i.e., r is the maximum of the p-ranks as p runs over

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the primes dividing |G|). Does G act freely on a finite complex

X ≃ S^{n_{1}} × ⋯ × S^{n_{r}}?

This looks like a difficult problem, even in the case where G = P is a p-group. However, one can show that if every element of order p in P is central, then it in fact acts freely on a product of r equidimensional spheres. The version of this problem using actual spheres arising from representations has been studied by U. Ray [3].

4. Does there exist a fixed integer N such that if G is a finite group with

⨁_{i=1}^{N} H_{i}(G, ℤ) = 0

then G = {1}?

This was conjectured by Loday for N = 3 but Milgram recently showed that the sporadic group M_{23} is a counterexample.

5. Describe the finite groups that can act freely and homologically trivially on a closed 4-manifold.

6. If S_{p}(G) is the Brown complex (see [5] for example) of nontrival p-subgroups of G, calculate

K_{G}^{∗}(|S_{p}(G)|)

in terms of algebraic invariants.

This is a generalized topological version of Alperin’s conjecture (which can be attributed to Thévenaz [4]).

[1] A. Adem and W. Browder. The free rank of symmtry of (S^{n})^{k}. Invent. Math. 92 (1988), 431–440.

[2] G. Carlsson. On the rank of abelian groups acting freely on (S^{n})^{k}. Invent. Math. 69 (1982), 393–408.

[3] U. Ray. Free linear actions of finite groups on products of spheres. J. Algebra 147 (1992), 456–490.

[4] J. Thévenaz. Equivariant K-theory and Alperin’s conjecture. J. Pure & Applied Algebra 85 (1993), 185–202.

[5] P. J. Webb. Subgroup complexes. Arcata Conf. on Representations of Finite Groups, Proc. Symp. Pure Maths 47 (Part 1), A.M.S. 1987, 349–365.

The B conjecture (J. L. Alperin)

The study of representations of finite groups and the study of group actions and cohomology of finite groups both deal with p-local subgroups. Many of the great insights and fundamental discoveries about p-local subgroups discovered in the course of the classification of finite simple groups have yet to play a role in these areas. The B conjecture is such a result; it is actually a theorem, a deep one, and its proof, for odd primes, depends on the entire classification.

7. Give a direct proof of the B conjecture.

The motivation here is to publicize the result and create curiosity about the classification.

If H is any finite group then E(H) is the largest normal subgroup with the following two properties: E(H) is perfect (so E(H) = E(H)′); E(H)/Z(E(H)) is the direct product of (non-abelian) simple groups. Groups with these two properties should be thought of as analogs of reductive groups so E(H) is the largest normal “reductive” subgroup of H, the “reductive part” of H.

Assume now that G is a finite group which has no non-identity normal subgroup of order prime to p and let L be a p-local subgroup of G (so L = N(Q) for a non-identity p-subgroup Q of G). Let L^{∗} be the quotient of L by its largest normal subgroup of order prime to p. In general, our hypothesis on G implies that L^{∗} is very close to L, even equal to L, and that the largest normal subgroup of L of order prime to p is insignificant. The B conjecture makes explicit one aspect of this.

Theorem (B-conjecture) E(L^{∗}) = E(L)^{∗}

That is, the “reductive part” of L^{∗} is nothing more than the image E(L)^{∗} of E(L) in L^{∗}, the “reductive part” of L! As easy as this is to state, it seems so hard to attack. A direct proof would also completely revolutionize the classification proof.

A Yoneda description of the Steenrod operations (R. R. Bruner)

Let k = 𝔽_{p}, the prime field of characteristic p > 0, and let A be a cocommutative Hopf algebra over k. Products can be defined in Ext_{A}(k, k) in two fundamentally different ways.

First, if ⋯ ⟶ P_{1} ⟶ P_{0} ⟶ k ⟶ 0 is a projective resolution of k, then, given cocycles x : P_{n} ⟶ k and y : P_{m} ⟶ k, we can lift x to a chain map x̃ : P ⟶ P and define xy to be represented by the Yoneda composite yx̃_{m} : P_{n+m} ⟶ P_{m} ⟶ k. This method applies to any augmented algebra A over k, and, in general, yields a noncommutative product.

Second, the Hopf algebra structure on A allows us to consider P ⊗ P as an A-module by pullback along the diagonal Δ : A ⟶ A ⊗ A. The isomorphism k ⟶ k ⊗ k lifts to a chain map Δ : P ⟶ P ⊗ P and, given cocycles x and y as above, we may define xy to be represented by the cocycle

(x ⊗ y)π_{n,m}Δ : P_{n+m} ⟶ (P ⊗ P)_{n+m} ⟶ P_{n} ⊗ P_{m} ⟶ k.

It is easily shown that this produces the same product as the first definition.

When A is cocommutative the chain map Δ : P ⟶ P ⊗ P is chain homotopy cocommutative and the chain homotopies give rise to Steenrod operations denoted

Sq^{i} : Ext_{A}^{s,t}(k, k) ⟶ Ext_{A}^{s+i,2t}(k, k)

when p = 2, and similarly for odd p.

8. Find a description of the Steenrod operations in terms of Yoneda composites, avoiding any use of Δ : P ⟶ P ⊗ P.

The motivation for this is computational efficiency. In mechanical calculations of Ext [6, 7], when P is large enough to strain available memory, P ⊗ P is hopelessly out of reach. Worse, even when P is not overly large, the map Δ is so redundant that it is useless. For example, in my calculations of the cohomology of the mod 2 Steenrod algebra, the calculation of Ext out to internal degree t = 60 can be done in a few hours on a small workstation. However, the calculation of Δ on the class h_{3}^{3}h_{5} ∈ Ext^{4,44} required 14 days and contained 25,000,000 terms. This happens because Δ must capture all possible decompositions of an element.

In contrast, the calculation of the product structure by computing chain maps which lift cocycles is quite efficient because we need only lift those cocycles which are indecomposable, and we need only keep track of their values on an A-basis for the resolution. Using this approach, I have calculated the entire product structure of the cohomology of the Steenrod algebra through internal degree t = 140. The largest of the vector spaces P_{s,t} involved are roughly 50,000 dimensional over 𝔽_{2}, so that any attempt to do this calculation using Δ would be pointless.

[6] R. R. Bruner. Calculation of large Ext modules. Computers in Geometry and Topology, ed. Martin C. Tangora, Marcel Dekker (1989), 79-104.

[7] R. R. Bruner. Ext in the nineties. Proc. 1992 Oaxtepec Conf on Alg. Top., ed. Martin C. Tangora, Contemp. Math. 146 (1993), 71-90.

Generating thick subcategories (J. F. Carlson)

Let G be a finite group and k a field of characteristic p > 0. Let B be a block of kG and let stmod-B be the stable category of modules in B.

Suppose that C is a thick (or épaisse) subcategory of stmod-B (see Rickard [9] for background material on the stable category and thick subcategories). Let G(C) be the Grothendieck group of C and

θ : G(C) → G(stmod-B)

be the natural map induced by the inclusion of C in stmod-B.

9. Under what circumstances would the surjectivity of θ imply that C = stmod-B?

The case that C is the thick subcategory generated by the trivial module k was considered in [8]. In this case the surjectivity of θ would mean that only one principal divisor of the Cartan matrix for the principal block is not a unit. Jeremy Rickard has pointed out that it is possible to construct subcategories C ≠ stmod-B such that θ is surjective. But the construction seems somewhat contrived. A more natural question would be whether the subcategory C generated by the Scott modules is the whole of the principal block. This question came up in discussions with Raphaël Rouquier.

[8] D. J. Benson, J. F. Carlson and G. R. Robinson. On the vanishing of group cohomology. J. Algebra 131 (1990), 40–73.

[9] J. Rickard. Idempotent modules in the stable category. J. London Math. Soc., to appear.

Parameters in cohomology (J. F. Carlson)

Let G be a finite group and k a field of characteristic p > 0. A sequence ζ_{1}, …, ζ_{r} ∈ H^{∗}(G, k) is said to be quasi-regular (see [10]) if

(1) ζ_{1} is a regular element,

(2) for each i > 1 the map

ζ_{i} : H^{a}(G, k)/(ζ_{1}, …, ζ_{i−1}) → H^{a−n_{i}}(G, k)/(ζ_{1}, …, ζ_{i−1})

given by multiplication by ζ_{i}, is an injection provided a ≥ ∑_{j=1}^{i−1} n_{j}, and

(3) H^{∗}(G, k) is finitely generated as a module over k[ζ_{1}, …, ζ_{r}].

10. (A) Does H^{∗}(G.k) always have a quasi-regular sequence?

(B) Given a homogeneous system of parameters (satisfying (3) with r = p-rank(G)) ζ_{1}, …, ζ_{r}, is some ordering of these elements a quasi-regular sequence?

(C) Given a homogeneous system of parameters ζ_{1}, …, ζ_{r} with ζ_{1}, …, ζ_{d} a regular sequence for d = Depth(H^{∗}(G, k)), is this a quasi-regular sequence?

These questions are of interest in computations of cohomology since, among other things, an affirmative answer would imply that all of the generators of H^{∗}(G, k) lie in degrees at most ∑_{i} n_{i}.

[10] D. J. Benson and J. F. Carlson. Projective resolutions and Poincaré duality complexes. Trans. Amer. Math. Soc. 342 (1994), 447–488.

Abelian subgroups of discrete groups (J. Cornick)

Recall that G is of type FP_{∞} if the trivial ℤG-module ℤ has a projective resolution of finite type (i.e. each projective in the resolution is finitely generated). If, in addition, the group is finitely presented this condition is equivalent to the existence of a K(G, 1) of finite type (i.e. finitely many cells in each dimension).

11. What are the possible abelian subgroups A of a group G of type FP_{∞}?

12. Same question where G has the additional property of being in the (large) class H𝔉 introduced by Kropholler [11]?

In the second case we know that there is a bound on the cohomological dimensions of the torsion free subgroups, and on the orders of the finite subgroups. It follows that A ≅ F ⊕ V where F is finite and V is a subgroup of a finite dimensional rational vector space.

13. Can every possibility occur?

It would be enough to show that the additive rationals ℚ can be embedded in such a G (this is the case where infinitely many primes are inverted). It is well known that ℤ[\frac{1}{n}] can be embedded in ⟨x, y|x^{y} = x^{n}⟩ which is a group of type FP_{∞} (this is the case where finitely many primes are inverted).

The H𝔉 condition certainly makes a difference. For example, the Thompson group ⟨x_{1}, x_{2}, … |x_{n}^{x_{i}} = x_{n+1}, i < n⟩ is of type FP_{∞}, but not in H𝔉, and contains a free abelian subgroup of infinite rank.

14. Can one embed ℚ/ℤ in an FP_{∞}-group? Such a group can not be in H𝔉 because there is no bound on the orders of the finite subgroups.

[11] P. H. Kropholler. On groups of type (F P)_{∞}. Journal of Pure & Applied Algebra 90 (1993), 55–67.

Constructing classifying spaces (W. G. Dwyer)

Let G be a connected compact Lie group and T ⊆ G a maximal torus. It is known that G is determined up to isomorphism by the normalizer NT of T.

15. Find a direct homotopy theoretic way to construct BG (or a p-completion of BG) from data visible in NT.

Known methods of building BG from NT are not homotopy theoretic: they involve first building the group G, either by generators and relations or by some Lie algebra technique.

Let G be a finite group and p a prime. Let F_{G} be the functor from the category of finite p-groups to the category of finite sets which assigns to each finite p-group P the set of conjugacy classes of monomorphisms P → G. This functor captures the p-fusion in G. The problem is to find an explicit way to construct the p-completion of BG from the functor F_{G}.

Centralizers and cohomology (H.-W. Henn)

The following problem is concerned with the question to what extent centralizers C_{G}(E) of elementary abelian p-subgroups E of suitable discrete groups G can be used to study cohomology with coefficients in nontrivial 𝔽_{p}[G]-modules.

For example assume G admits a cocompact action on a finite dimensional mod-p acyclic space G–CW-complex X such that all isotropy groups are finite, e.g. G itself is finite. Let A_{p}(G) be the Quillen category of elementary abelian p-subgroups of G and let O be a full subcategory of A_{p}(G) with the following properties:

• If E is an object of O and E′ is conjugate to E in G then E′ is also an object of O.

• If E is an object of O and E ⊂ E′ then E′ is also an object of O.

Theorem [12] Under these assumptions the restriction maps induce a natural map

H^{∗}(G, 𝔽_{p}) ⟶ lim_{O}H^{∗}(C_{G}(E); 𝔽_{p})

whose kernel and cokernel are torsion with respect to the ideal

𝔞 := ⋂_{E∉O} Ker(H^{∗}(G; 𝔽_{p}) ⟶ H^{∗}(E; 𝔽_{p})).

Furthermore the higher derived functors lim_{O}^{i}H^{∗}(C_{G}(E); 𝔽_{p}) are also torsion with respect to 𝔞.

Example: O = A_{p}^{∗}(G) := {E ∈ A_{p}(G)|E ≠ {1}}. In this case we have 𝔞 = H̃^{∗}(G; 𝔽_{p}) and the kernel, cokernel and the higher derived functors are actually finite. In the case G is finite they are even trivial by a result of Jackowski and McClure [13], but this might be a red herring for the following question.

16. Find an extension of the Theorem which holds for H^{∗}(G; M) if M runs through a suitable class of k[G]-modules. If G is finite, one would like to be able to take any finitely generated k[G]-module M.

[12] H.-W. Henn. Commutative algebra of unstable K-modules, Lannes’ T -functor and equivariant mod-p-cohomology. Heft Nr. 126, Forschergruppe Topologie und nichtkommutative Geometrie Univ. Heidelberg, September 1995, to appear in Journal für die reine und angewandte Mathematik.

[13] S. Jackowski and J. McClure. Homotopy decomposition of classifying spaces via elementary abelian subgroups. Topology 31 (1992), 113-132.

Conjugacy and Hall subgroups (S. Jackowski)

17. Let H be a Hall π-subgroup of a finite group G for some set of primes π. Assume that for all h_{1} and h_{2} of prime power order in H, if h_{1} and h_{2} are conjugate in G then they are conjugate in H. Is the same then true for all h_{1} and h_{2} of composite order?

Homomorphisms preserving conjugacy (S. Jackowski)

18. Let G be a finite p-group and R = ℤ or ℤ/p. Let ϕ : G → G be an automorphism inducing the identity map on H^{∗}(G; R) . Does ϕ preserve conjugacy classes of elements of G? Is ϕ inner?

Varieties and exponents of cohomology (I. J. Leary)

19. Let I_{n}(G) be the annihilator of p^{n} in the p-local cohomology, H^{∗}(G; ℤ_{(p)}), of a finite group G. Let J_{n} = J_{n}(G) be the radical of the image of I_{n} in H^{∗}(G; 𝔽_{p}). Is J_{n} closed under the Steenrod power operations?

Let V_{G}(k) be the variety of homomorphisms from H^{∗}(G; 𝔽_{p}) to an algebraically closed field k and let V_{n}(G, k) be the subvariety of V_{G}(k) corresponding to J_{n}. The question is equivalent to: Is V_{n}(G, k) equal to a union of the images of V_{E}(k), where E ranges over some family of elementary abelian subgroups of G?

The V_{n}(G, k) were introduced by J. F. Carlson in [14]. In [15] it is shown that J_{n}(G≀C_{p}) is closed under the Steenrod powers provided that J_{n}(G) and J_{n−1}(G) are.

[14] J. F. Carlson. Exponents of modules and maps. Invent. Math. 95 (1989) 13–24.

[15] I. J. Leary. On the integral cohomology of wreath products. Preprint, 1996.

Quantum Galois Theory (G. Mason)

Let V be a simple vertex operator algebra (VOA) and let G be a finite group of automorphisms of V.

20. Is there a Galois correspondence between subgroups of G and subVOAs of V containing V^{G} given by sending H ≤ G to V^{H}.

The answer is known to be yes if G is dihedral or nilpotent. There is a similar conjecture for all groups, which should be endowed with a suitable (Krull) topology.

Classifying spaces of wreath products (R. J. Milgram)

The wreath product

ℤ/2 ≀ ℤ/2 ≀ ⋯ ≀ ℤ/2 = Wr_{n}(ℤ/2)

has the property that Out(Wr_{n}(ℤ/2)) is a finite 2-group for each n. Consequently it has a single dominant summand (see for example [16] or [17] for background material on stable splittings and dominant summands)

X_{Wr_{n}(ℤ/2),𝔽_{2}} ⊆ Σ^{∞}BWr_{n}(ℤ/2).

21. Find the structure of this summand.

Remark: One can see that Out(Wr_{n}(ℤ/2)) is a 2-group by induction. There exists a pair of subgroups (ℤ/2)^{2^{n−1}} ⊆ Wr_{n}(ℤ/2) which are permuted by any outer automorphism. Hence an index two subgroup fixes one and is determined up to an obvious 2-group by its action on the quotient Wr_{n}(ℤ/2)/(ℤ/2)^{2^{n}} ≅ Wr_{n−1}(ℤ/2).

Example: For D_{8} = ℤ/2 ≀ ℤ/2 the dominant summand is X_{D_{8},𝔽_{2}} = B̄L_{3}(2).

22. Is it true that X_{Wr_{3}(ℤ/2),𝔽_{2}} = BL_{3}(2) ∧ BL_{3}(2) ⋉_{ℤ/2} Eℤ/2 ∩ BS_{8}/ℝℙ^{∞}? Here, what is meant by this is that the intersection is taken in BW_{3}(ℤ/2), and the obvious splitting summand ℝℙ^{∞} is removed from BS_{8} before the intersection is taken.

23. Find the complete splitting at the prime two of BÃ_{8}.

Remark: Syl_{2}(Ã_{8}) = Syl_{2}(M_{22}) = Syl_{2}(M_{23}) = Syl_{2}(McL).

24. Is it true that the dominant summand X_{Syl_{2}(Ã_{8}),𝔽_{2}} is B̄McL (at the prime 2), that is to say, that B̄McL is indecomposable at 2?

[16] D. J. Benson. Stably splitting BG. Bull. of the American Math. Society 33 (1996), 189–198.

[17] J. Martino. Classifying spaces and their maps. Contemp. Math. 188 (1995), 161–198.

New Doomsday Conjecture (N. Minami)

25. For each s, does there exist some integer n(s) such that no nontrivial element in the image of

(P^{0})^{n(s)}(Ext_{A_{∗}}^{s,∗}(ℤ/p, ℤ/p)) ⊆ (Ext_{A_{∗}}^{s,p^{n(s)}∗}(ℤ/p, ℤ/p))

is a nontrivial permanent cycle?

Remarks: (1) This is true for s = 1 by the Adams Hopf invariant one theorem and its odd primary analogue by Liulevicius and Shimada–Yamanoshita.

(2) This is true for s = 2 when p ≥ 5, by Miller–Ravenel–Wilson and Ravenel.

(3) An iterated transfer analogue of this conjecture holds (Minami).

(4) No counterexample is known!

(5) The first unsolved case is when s = 2 and p = 2, 3. From the viewpoint of homotopy theory, the case p = 3 appears to be more tractable. On the other hand, the case p = 2 appears to be more difficult and it essentially claims that there are just finitely many Kervaire invariant one elements. Recall that Milgram has also predicted the same claim, from his philosophy: maximal subgroups of symmetric groups should play essential roles in the stable homotopy groups of the sphere (via the Barratt–Priddy–Quillen theorem BΣ_{∞}^{+} ≃ Q_{0}S^{0}). He then speculated that there are just finitely many Kervaire invariant one elements from the finiteness of the finite sporadic simple groups. Conversely, we may ask the following:

26. Can the new doomsday conjecture offer any interesting speculations about finite group theory?

27. Study the odd primary Mahowald η_{j}-elements, using the Hopkins–Miller spectrum EO_{p−1}.

Remarks: (1) Unlike the original 2 primary case due to Mahowald, the existence of the odd primary η_{j}, which is detected by h_{0}h_{j} in the Adams E_{2} term, is still unknown.

(2) When p = 2, Mahowald used bo to study his η_{j} family. On the other hand, EO_{p−1} has been constructed by Hopkins–Miller so as to be a p primary periodic analogue of bo.

(3) We hope such a contribution would provide us with a better idea how to prove (or disprove) the existence of odd primary η_{j}’s. In fact, there is some relationship between η_{j} and the Kervaire invariant one element (and its odd primary analogue). Furthermore, Hopkins–Miller EO_{p−1} at least contains the information about Ravenel’s odd primary Kervaire invariant one theorem for p ≥ 5 and Toda’s differential (which was a starting point in Ravenel’s proof) for p ≥ 3. In this way, this problem might provide us with some clue how to generalize Ravenel’s result to the case p = 3.

Cohomology for restricted simple Lie algebras (D. K. Nakano)

Let 𝔤 be a restricted Lie algebra and u(𝔤) be the restricted enveloping algebra corresponding to 𝔤. Block and Wilson [21] have proven that any restricted simple Lie algebra over an algebraically closed field k of characteristic larger than 7 is either classical or of Cartan type. The classical Lie algebras arise as the Lie algebra of a connected reductive algebraic group scheme. For a reductive group scheme G with Lie(G) = 𝔤, let G_{1} be the kernel of the Frobenius morphism. It is well-known that representations for G_{1} are the same as looking at representations for u(𝔤). The cohomology ring H^{•}(u(𝔤), k) has been computed by Friedlander and Parshall [22], and Andersen and Jantzen [18]. They prove that if p is larger than the Coxeter number of the Weyl group of G, then H^{odd}(u(𝔤), k) = 0 and H^{2•}(G_{1}, k) ≅ H^{2•}(u(𝔤), k) ≅ k[N], where N is the well-studied variety of nilpotent elements in 𝔤. The variety N is often referred to as the nullcone.

Almost nothing is known about the cohomology ring H^{•}(u(𝔤), k) when 𝔤 is a Lie algebra of Cartan type. The Lie algebras of Cartan type can be divided into four infinite families of algebras denoted by W, S, H and K. Let R = k[x]/(x^{p} = 0). The derivations of R form a restricted Lie algebra of dimension p called W (1, 1). A basis for W (1, 1) is given by ⟨x^{i} \frac{d}{dx} : i = 0, 1, …, p − 1⟩ with Lie multiplication

[x^{i} \frac{d}{dx}, x^{j} \frac{d}{dx}] = (i − j)x^{i+j−1} \frac{d}{dx}.

In general the restricted Lie algebras of type W are denoted by W (m, 1), and constructed by taking derivations of k[x_{1}, x_{2}, …, x_{m}]/(x_{j}^{p} = 0 : j = 1, 2, …, m). The other families of Cartan type Lie algebras arise as subalgebras of W (m, 1) which stabilize a specific differential form.

Friedlander and Parshall [23] have shown there exists a finite map Φ^{•} : S^{•}(𝔤^{♯}) → H^{2•}(u(𝔤), k) where 𝔤^{♯} is the dual of the Lie algebra 𝔤. Let |𝔤|_{k} be the maximal ideal spectrum of H^{2•}(u(𝔤), k). The map Φ^{•} induces a map on varieties: Φ : |𝔤|_{k} → 𝔤 ≅ 𝔸^{dim_{k} 𝔤}. We propose the following fundamental problems. After the statement of the problem remarks are given. From this point on, 𝔤 refers to a Lie algebra of Cartan type.

28. Calculate the cohomology ring H^{•}(u(𝔤), k). for 𝔤.

For 𝔤 = W (1, 1), the dimensions of the graded components of H^{•}(u(𝔤), k) were computed for p = 5, 7 [26]. In particular the Poincaré series for H^{•}(u(𝔤), k) for p = 5 is given by the formula

P (t) = ∑_{n=0}^{∞} \frac{1}{6}(2n + 2)(4n^{2} + 2n + 3) t^{2n}.

The obstruction in generalizing this result for larger primes is in the calculation of H^{•}(u(𝔟^{+}), k) where 𝔟^{+} = ⟨x^{i} \frac{d}{dx} : i = 1, …, p − 1⟩. Evidence via calculations indicates that this computation will become increasingly difficult as p gets large. For classical Lie algebras, the computation of the cohomology ring depends heavily on using results about the ambient algebraic group. A formal setting using algebraic groups is developed for Lie algebras of Cartan type in [24], [25]. This might prove to be useful in attacking this problem.

29. Calculate the Krull dimension of the ring H^{•}(u(𝔤), k).

30. Determine when cohomology ring H^{•}(u(𝔤), k) is Cohen–Macaulay.

In case when 𝔤 = W (1, 1), it was shown that the Krull dimension of H^{•}(u(𝔤), k) is p−1. Recall that dim_{k} W (1, 1) = p and the dimension of a maximal torus is 1. In the classical case the dimension of N is the dimension of the Lie algebra minus the dimension of a maximal torus. One is tempted to conjecture that the Krull dimension of the cohomology ring for Cartan type Lie algebras (perhaps with a lower bound on the prime) will also be equal to the dimension of the Lie algebra minus the dimension of a maximal torus. With information about the Poincaré series of the cohomology ring, one might want to look at the Cohen–Macaulay question using ideas given in [19], [20]. One should note that the cohomology ring for classical Lie algebras is Cohen–Macaulay by using the aforementioned results.

31. Give a nice geometric description of Φ(|𝔤|_{k}) ⊂ 𝔤. When is this variety irreducible?

Once again we will return to the example when 𝔤 = W (1, 1). Let G = Aut(𝔤). One can show that G is the semidirect product of a one-dimensional torus T and a unipotent algebraic group U. In [24], it was shown that Φ(|𝔤|_{k}) = \overline{G · e_{−1}} where e_{−1} = \frac{d}{dx}. Moreover, this variety is irreducible. One might wonder if the variety Φ(|𝔤|_{k}) can be expressed as the G-closure of some nilpotent element for other Lie algebras of Cartan type.

[18] H. H. Andersen, J. C. Jantzen. Cohomology of induced representations for algebraic groups. Math. Ann. 269, (1984), 487-525.

[19] D. J. Benson, J. F. Carlson. Projective resolutions and Poincaré duality complexes. Trans. AMS 342, no. 2, (1994), 447-488.

[20] D. J. Benson, J. F. Carlson. Functional equations for Poincaré series in group cohomology. Bull. London Math. Soc. 26 (1994), 438-448.

[21] R. E. Block, R. L. Wilson. Classification of restricted simple Lie algebras. J. Algebra 114 (1988), 115-259.

[22] E. M. Friedlander, B. J. Parshall. Cohomology of Lie algebras and algebraic groups. Amer. J. Math. 108 (1986), 225-253.

[23] E. M. Friedlander, B. J. Parshall. Support varieties for restricted Lie algebras. Invent. Math. 86 (1986), 553-562.

[24] Z. Lin, D. K. Nakano. Algebraic group actions in the cohomology theory of Lie algebras of Cartan type. J. Algebra 179 (1996), 852-888.

[25] Z. Lin, D. K. Nakano. Representations of Hopf algebras arising from Lie algebras of Cartan type. J. Algebra (to appear).

[26] D. K. Nakano. On the cohomology of the Witt algebra: W (1, 1). Proc. of the III International Conference on Non-associative algebra and its applications (ed. S. Gonzales), Kluwer, 1994.

Counting characters (G. R. Robinson)

For a fixed prime p, a finite group G and a non-negative integer d, let k_{d}(G) denote the number of irreducible ordinary characters χ of G such that p^{d}χ(1)_{p} = |G|_{p}.

32. Is there a “natural” complex C on which G acts such that dim_{F} (H_{n}(C)) = k_{n}(G) for each non-negative integer n, where F is a suitable field?

Possible Evidence: Let G, H be finite groups. Note that

k_{d}(G × H) = ∑_{i=0}^{d} k_{i}(G)k_{d−i}(H).

This suggests that if C(G) and C(H) exist, perhaps C(G × H) is the product C(G) × C(H) (up to suitable homotopy).

Transfers in Morava K-theory (N. Strickland)

Let 𝔾 be the formal group associated to K(n). Formally, 𝔾 is the functor from algebras over K(n)^{∗} to abelian groups defined by

𝔾(R) = lim_{\overrightarrow{k}} Hom_{K(n)^{∗}}(K(n)^{∗}ℂP^{k}, R).

For many interesting spaces X (including BU(k), MU(k), K(ℤ, k), ΩS^{2k+1}) there is a simple conceptual interpretation of the functor represented by K(n)^{∗}X in terms of 𝔾. In particular, with suitable definitions, K(n)^{∗}BA represents Hom(A^{∗}, 𝔾) (where A is a finite Abelian group with Pontrjagin dual A^{∗} = Hom(A, S^{1})) and K(n)^{∗}BΣ_{p^{k}}/(transfers from partition subgroups) represents the scheme Sub_{k}(𝔾) of subgroups of 𝔾 of order p^{k}.

To extend this to K(n)^{∗}BH for more general finite groups H, it seems necessary to have a conceptual understanding of transfers.

33. Give a purely algebraic construction of transfers between the rings representing Hom(A^{∗}, 𝔾) and Hom(B^{∗}, 𝔾) when B ≤ A. This should be visibly independent of any choice of coordinate on 𝔾 or decomposition of A, B or B/A as a product of cyclic groups.

In the Greenlees–May theory of Tate spectra, it turns out that (t_{H}K(n))^{H} = 0. It follows that there is a natural isomorphism K(n)_{∗}BH ≅ K(n)^{∗}BH of modules over K(n)^{∗}BH, making K(n)^{∗}BH into a Poincaré duality algebra over K(n)^{∗}. Greenlees and May never write down explicitly what the map is, but I suspect that it is given by a slant product with tr_{Δ}^{H^{2}}(1) ∈ K(n)^{∗}BH ⊗ K(n)^{∗}BH (where Δ ≤ H^{2} is the diagonal subgroup). A good understanding of transfers should explain why the resulting map K(n)_{∗}BH → K(n)^{∗}BH is an isomorphism.

The Kriz group (N. Strickland)

Igor Kriz has proved that K(2)^{∗}B(U_{4}𝔽_{3}) is not concentrated in even degrees (where U_{4}𝔽_{3} is the group of 4 × 4 upper triangular matrices over 𝔽_{3} with ones on the diagonal, and K(2) is defined at p = 3). However, Kriz does not compute the full answer, he merely exhibits an element in odd degree.

34. Compute K(2)^{∗}B(U_{4}𝔽_{3}) completely. Can the odd degree elements be described as Massey products?

Finite rings (N. Strickland)

Let R be a finite p-local ring. The general structure theory of such rings says that R/Rad(R) ≅ ∏_{k,l} M_{k}(𝔽_{p^{l}})^{n_{kl}} where n_{kl} = 0 for almost all (k, l) (Here we have used Wedderburn’s theorem, that a finite division ring is a field). For want of a better word, I’ll say that R is unextended if n_{kl} = 0 whenever l > 1, so R/Rad(R) is a product of matrix algebras over 𝔽_{p}.

35. Let X be a finite p-torsion spectrum, so that [X, X] is a finite ring. Is this always unextended? Are there elementary ways to test whether a ring is unextended? Note that a commutative ring R is unextended if and only if the sequence {a^{p^{k}}} is eventually constant for all a ∈ R, but the fact that 𝔽_{p^{n}} ⊆ M_{n}(𝔽_{p}) prevents any simple-minded extension of this.

Spectra and Q-algebras (N. Strickland)

Let T be the category of pointed, compactly generated, weakly Hausdorff spaces. Consider the functor Q : T → T defined by QX = lim_{\overrightarrow{n}} Ω^{n}Σ^{n}X. It is important here that

we use Q itself (not one of the combinatorial approximations to Q), and that we regard it as a functor on T rather than the homotopy category. The functor Q is a monad, so we can consider the category T^{Q} of Q-algebras. We can also consider the category S of spectra (over the universe ℝ^{∞}) as defined by Lewis and May. It is easy to see that Ω^{∞} gives a functor S → T^{Q}. I have explained to a number of people a proof that Ω^{∞} : S → T^{Q} is full and faithful.

36. Is this functor an equivalence of categories?

Tensor induction for Mackey functors (T. Yoshida)

37. Define tensor induction for Mackey functors. Let G be a finite group and H a subgroup of G. Then for a kH-module L, we have a tensor induced kG-module Jnd_{H}^{G}(L). Tensor induction preserves tensor product, but not direct sum. It is natural to expect that Mackey functors have also tensor induction. The tensor induction has to preserve tensor product of Mackey functors and furthermore to satisfy Tambara’s axiom for multiplicative transfer.

[27] C. W. Curtis and I. Reiner. Methods of Representation Theory (I). Wiley, 1981.

[28] D. Tambara. On multiplicative transfer. Commun. in Algebra 21 (1993), 1393–1420.

G-sets and functors (T. Yoshida)

Let Sp(G) and Binom be the 2-categories of spans and bimodules, respectively [33]. Here, an object of the 2-category Sp(G) is a finite G-set, a 1-morphism from Y to X is a pair of G-maps (X ⟵ A ⟶ Y), and a 2-morphism from (X ⟵ A ⟶ Y) to (X ⟵ A′ ⟶ Y) is a G-map from A to A′ which makes two triangles commutative. We are familiar to an example similar to the above problem, that is, a 2-functor from Sp(G) to CATS, the 2-category of all categories. In fact, the assignment

G/H ⟼ Mod_{kH},
(G/H ⟵ G/A ⟶ G/K) ⟼ (Mod_{kH} ⟶ Mod_{kA} ⟶ Mod_{kK})

gives a 2-functor.

38. Build the theory of (non-monoidal) 2-functors from Sp(G) to Binom. More generally, build (2-)representation theory of a monoidal (2-)category.

The Mackey algebra μ_{R}(G) is the path algebra of Sp(G). A representation of Sp(G) is nothing but a Mackey functor. Furthermore, there is a tensor product of two Mackey functors, so it is natural to expect that the representation category Sp(G) has the following property.

39. Is the category Sp(G) of spans of a finite group G (and the Mackey ring μ_{R}(G)) Morita equivalent to a Hopf algebra? (Mackey Hopf algebra!)

40. Let G be a finite group and S a finite G-monoid. Under what conditions is the monoidal category of crossed G-sets over S symmetric or braided?

A crossed G-set X is a finite G-set equipped with a G-map called a weight function X ⟶ S; x ⟼ ∥x∥. The tensor product X ⊗ Y of crossed G-sets X, Y is the cartesian product X × Y with weight ∥(x, y)∥ := ∥x∥ · ∥y∥. The monoidal category of crossed G-sets is braided if S is commutative or S = G^{c}, the G-monoid G with G-action defined by G-conjugation.

If the monoidal category is braided, then (C[C_{S}(H)])^{N_{G}(H)}, the ring of N_{G}(H)-fixed points in the semigroup algebra C[C_{S}(H)], is commutative for any subgroup H of G.

[29] J. Thévenaz and P. Webb. The structure of Mackey functors. Trans. Amer. Math. Soc. 347 (1995), 1865–1961.

[30] A. Joyal and R. Street. Braided tensor categories. Advances in Math. 102 (1993), 20–78.

[31] F. Oda and T. Yoshida. On crossed G-sets and the crossed Burnside ring of a finite group. preprint.

[32] D. N. Yetter. Topological quantum field theory associated to finite groups and crossed G-sets. J. Knot Theory 1 (1992), 1–20.

[33] T. Yoshida. Idempotents and transfer theorems of Burnside rings, character rings and span rings. Algebraic and Topological Theories - to the memory of Dr. Takehiko Miyata (1985), 589-615.

The Dijkgraaf–Witten invariant (T. Yoshida)

Let M be a closed oriented d-manifold, G a finite group, and [α] ∈ H^{3}(G, U(1)). The Dijkgraaf–Witten invariant is defined as follows:

Z_{G,α}(M) := \frac{1}{|G|} ∑_{γ:π_{1}(M)→G}⟨γ^{∗}(α), [M]⟩,

where γ^{∗} : H^{3}(G, U(1)) ⟶ H^{3}(M, U(1)) is the map induced by γ. In particular, when α = 1,

Z_{G,1}(M) = \frac{|Hom(π_{1}(M), G)|}{|G|},

and so this is defined for any d-manifold M.

We have the integrality conjecture:

|G| Z_{G,1}(M) ≡ 0 mod gcd(|H_{1}(M)/|G|H_{1}(M)|, |G|).

Perhaps, this conjecture will be solved if M is a 3-manifold because its fundamental group has some special properties.

41. Let M be a closed oriented 3-manifold. Find an integrer valued function f(m, n) for which

|G| Z_{G,α}(M) ≡ 0 mod f(|H_{1}(M)|, |G|).

[34] T. Asai and T. Yoshida. |Hom(A, G)| (II). J. Algebra 169 (1993), 273–285.

[35] M. Wakui. On Dijkgraaf–Witten invariant for 3-manifolds. Osaka J. Math. 29 (1992), 675–696.

Department of Mathematics, University of Georgia, Athens GA 30606, USA

    E-mail address: djb@byrd.math.uga.edu