_______________________________________________________________________________* *_|| | * * | | * * | | ____________________________________________________________________________* *||| | | classifyingcspaceslofacompactsliesgroupsiandffiniteyloopispacesngcspace* *slofacompactsliesgroupsiandffiniteyloopispacesng|spaces|of compact lie gr@ | | * * || | | * * || | | augusta26-30,u1991gusta26-30,u1991gust 26-30, 1* *991|| | | * * || | | g"ottingeng"ottingeng"ottingen * * || | | * * || | | * * || | | problem-sessionproblem-sessionproblem-session* * || | |___________________________________________________________________________* * || | * * | |______________________________________________________________________________* *_ | Classifying Spaces The following problems were proposed by Guido Mislin. 1. Let G and H be compact connected Lie groups with BG homotopy equivalent to B* *H. Are G and H isomorphic as Lie groups? The following problem was proposed by Clarence Wilkerson. 1. Given a p-adic reflection group W GL(n; Z^p) which extensions 1 --! (Zp1 )n --! NT -- ! W -- ! 1 ; with W acting via the given inclusion can occur as the normalizer of a maximal * *torus of a finite loop space at p? That is, for which such extensions is there a p-complet* *e finite loop space X and a map BNT -- ! BX with homotopy fibre F the homotopy type of the p-completion of a finite complex* * and O(F ) = 1? The following problem was proposed by Jaume Aguade. In number 2 of the list of Shephard and Todd of reflection groups there is an i* *nfinite family of groups G(m; n; r) defined as follows: G(m; n; r) is the group of all linear * *maps of the form xi 7! i xoe(i); where oe is a permutation of the basis x1; : :;:xn, is an m-root of unity and * *1; : :;:n are such that 1 + . .+.n 0 (r). If r|m|p - 1, G(m; n; r) can be realized over * *the p-adic integers and the invariants mod p form a polynomial algebra R = Fp[y1; : :;:yn-1; z]; where y1; : :;:yn are the elementary symmetric functions on xm1; : :;:xmnand z * *= (x1 . .x.n)m=r. If we choose a prime p which does not divide |G(m; n; r)| then one can easily c* *onstruct a space X such that H*(X; Fp) ~=R. In the modular case, Quillen and Zabrodsky * *(by different methods) constructed spaces realizing the invariants of the groups G(* *m; n; 1). It is an open problem to realize the invariants of the groups with r > 1. There seems to be two approaches to the problem: One could try to construct X i* *nduc- tively as a homotopy limit of a large diagram. Alternatively, there may be some* * discrete group G such that BG has the desired mod p cohomology. (This is Quillen's metho* *d of realizing the invariants of G(m; n; 1) as the cohomology of some general linear* * group over some infinite field of positive characteristic.) The following problems were proposed by Stefan Jackowski and Bob Oliver. 1. Construct unstable Adams operations for a compact connected Lie group G, us* *ing the methods in [JMO]. If T is a maximal torus in G, and Np(T ) is a maximal p -* *toral subgroup, then the main problem is to extend the k -th power map on the torus (* *for any k prime to the order of the Weyl group) to an "Rp -invariant representation" fk : Np(T )- ! G: In other words, for any p-toral subgroup P Np(T ) and any g such that gP g-1 * *Np(T ), the two restrictions of fk to homomorphisms P -! G must be conjugate in G. 2 2. Find more explicit definitions of the exotic maps which so far are construc* *ted using completions, homotopy colimits, and/or computations of higher limits. Closely r* *elated to this are the following more explicit problems: 2a. Construct a pair of homomorphisms r; s : G- ! G0 (e.g., with G finite and * *G0 con- nected), such that r and s are equal on a maximal p -toral subgroup (or Sylow p* *-subgroup) of G, but such that Br and Bs are not homotopic as maps BG- ! BG0. Note that th* *is situation requires a nonvanishing higher limit (and probably would require the * *explicit computation of an obstruction in that group). 2b. Find a nonvanishing existence obstruction. More precisely, find an exampl* *e where one of the obstruction groups to the existence of a map defined on a homotopy c* *olimit is nonzero, and where the only way to determine the existence or nonexistence of t* *he map is by explicitly determining the obstruction. 3. Let G; G' be compact simple Lie groups with maximal tori T and T 0respectiv* *ely. Assume dim (T 0) dim (T ) + 1. Let [BG; BG0]i denote the set of homotopy class* *es of maps which are "rationally injective" in some appropriate sense: e.g. maps f : * *BG- ! BG0 such that H*(BG ; Q|) is finitely generated as a module over H*(BG0 ; Q|). _ Describe [BG; BG0]i. _ Is homotopy detected by rational cohomology in this case? Comments. The idea of "rationally injective" should be that f lifts to a map be* *tween the maximal tori which has finite kernel. Apply methods of [JMO]. 4. Determine (or study) the space of maps map (BG; XG ), when G is a compact co* *nnected simple Lie group and X is a G -complex (and XG is the Borel construction). This* * is closely related to the problem of describing the homotopy fixed point set XhG in this s* *ituation. 5. For any pair of compact [connected?] Lie groups P; G, and any f : P -! G, do* *es the 3 set Rep(P; G)f = {f0 : P -! G : f0(g) G-conjugate f(g) 8g 2 P }=Inn(G) have an abelian group structure in some natural way? Comment: If this could be done in general, then the problem of constructing* * Rp - invariant representations would be in part "reduced" to studying an obstruction* * in H1(Rp(G); --), where the coefficients are these Rep(P; G)f for appropriate f. (* *Reduced in quotes because there would still also be the problem of showing that appropr* *iate pairs of homomorphims are elementwise conjugate.) 6. Show the rigidity of the [JM2] diagram: i.e., the functor Ap(G)- ! T op whi* *ch sends a nontrivial elementary abelian subgroup A G to BCG (A). Here, rigidity means * *that any diagram (functor) Ap(G)- ! T op, which is equivalent to the first one in th* *e homotopy category, is weakly equivalent to it as functors to T op. This means showing th* *e vanishing of appropriate higher limits of the functor A --! ss*(Z(CG (A))) defined on elementary abelian p -subgroups of G. 7. Find useful decompositions of spaces as homotopy inverse limits. If a spac* *e has a hocolim decomposition, is there a "dual" holim decomposition? 8. [Rothenberg] Let G0; G be a pair of compact Lie groups, where G is simple. D* *escribe [B(G0; G); B(G0; G)]. Here, B(G0; G) is the classifying space for G -fiber bun* *dles with structure group G0. An answer should generalize corresponding results in [JMO].* * Of main interest is the case where G0= (S)O(n); (S)U(n); etc. (for studying equivarian* *t vector bundles). 4 References: [JMO] S.Jackowski, J.McClure, and R.Oliver, Homotopy classification of self-map* *s of BG via G-actions, Annals of Math (to appear) [JM2] S.Jackowski and J.McClure, Homotopy decomposition of classifying spaces v* *ia ele- mentary abelian subgroups, Topology (to appear) [McG] C.McGibbon, Self maps of projective spaces, TAMS vol.271 (1982), 325-346 [FG] Feder and Gitler, Mappings of quaternionic projective space, Bol. Soc. Mat* *. Mexi- cana, vol.18 (1973), 33-37 Spaces Related to Classifying Spaces [McGibbon] Describe self-maps of IHIP(n), using some homotopy decomposition. (O* *r, at least, determine which degrees can be realized for self-maps.) See [McG] and [* *FG] for background. Problems proposed by Larry Smith. Let G be a compact connected Lie group with T ,!G a maximal torus. Denote by HE(G=T ) the monoid of homotopy equivalences of the homogenous space G=T . Let SHE(G=T ) denote the identity component of HE(G=T ), and HEid*(G=T ) the submon* *oid of maps inducing the identity map in cohomology (Z or |Q). 1. How many components does HE(G=T ) have? 2. How many components does HEid*(G=T ) have? (Expermental evidence shows that there can be several.) Problem proposed by Clarence Wilkerson. 1. For which simply connected finite complextes is dimHk(SHE(X); Q|) < 1. The s* *pirit 5 of the problem is to decide for which spaces X SHE(X) has properties analagous * *to Lie groups or arithmetic groups. Finite Loop Spaces Problems proposed by Larry Smith. 1. If a finite loop space X has a maximal torus, is BX homotopy equivalent to B* *G for some Lie group G ? 2. If a finite loop space X has a maximal torus, must the Weyl group be crystal* *ographic? 3. If a finite loop space has a maximal torus, what should be the normalizer of* * that torus be? References 1. Smith, L. , Finite Loop Spaces with maximal Tori have Finite Weyl Groups, M* *ath. G"ottingensis 1991. The following problems were proposed by Richard Kane. Let p be a prime and X a finite loop space at the prime p. 1. Does H*(BX; Z) have only elementary torsion: i.e. is all the p-torsion of at* * most order p? Comment: Kono has raised this question for Lie groups. He has shown that H*(BSp* *in(n); Z) has only elementary 2-torsion. He also observes that transfer arguments severel* *y restrict torsion in H*(BG; Z). 2. Does k(n)*(BX) have only vn torsion: i.e. does vn annhilate the torsion subm* *odule. Here K(N)*(--) denotes Morave k-theory, and k(n)*(*) = IFp[vn], deg(vn) = 2pn -* * 2. 3. Let BT -- ! BX be the Dwyer-Wilkerson maximal torus and let X=T be the fibre* * of this map (everything completed at p). Determine H*(X=T ; IFp) and more generall* *y the 6 Serre spectral sequence of the fibration: X --! X=T -- ! BT This was done by Kac and Peterson for the Lie group case. In particular * *well behaved Chevalley operators play a major role in their argument. In the following tw* *o problems p denotes an it odd prime. 4. In particular Kac and Peterson show define the generalized invariants in H* **(BT ; `IFp) and show that these generalized invariants determine H*(G; IFp) as an algebra* *. Show that the coalgebra structure is also determined. 5. The generalized invariants are generated by a regular sequence {x1; : :;:x* *n}, so there is a non-canonical inclusion IFp[x1; : :x:n] H*(BT ; Z). Is there a choice that* * makes this a ring of invariants? The example of E6 shows that if the choice exists then * *the group in question need not be a subgroup of the Weyl group. Invariant Theory The following problems were proposed by David Benson. Let k = Fq, and G be a subgroup of the finite general linear group GLn(Fq) = * *GL(V ). Let c0 be the highest degree Dickson invariant. 1. When is k[V ]G [c-10] a localized polynomial ring? Is it always Cohen-Maca* *ulay*? 2. If G is generated by pseudoreflections (elements fixing a hyperplane point* *wise), what properties does k[V ]G have? Is it always a complete intersection? (Example: * *In the case of the finite symplectic groups in their natural representation, Carlisle and* * Kropholler [1] have shown that k[V ]G is a complete intersection) 3. Is k[V ]G generated by elements of degree |G|? (A theorem of Noether asse* *rts that this is true in the case where |G| is invertible in k.) Find better bounds (i* *n the case where |G| is invertible in k, see the 1989 Comptes Rendues paper of Barbara Schmid). _________________________ * I.e. free and finitely generated as a module over some polynomial subalgebra 7 4. Carlisle and Kropholler [1] have made the following conjecture. If we form t* *he Poincare P series p(t) = i0 tidim k(k[V ]G )i, then the Laurent expansion begins p(t) = _1_|G|(1 - t)-n + _r__2|G|(1 - t)-n+1 + . . . Here, the first term is correct by general degree considerations, and the conte* *nt of the conjecture is the value of r. This is given as follows. For each maximal subs* *pace W of V , let the stabiliser have order pah where h is coprime to p. Then the contrib* *ution from W is a(p - 1) + (h - 1), and the value of r is the sum of the contributions fro* *m all the maximal subspaces. Is there a similar interpretation of the further terms in the Laurent expansion* * in terms of stabilisers of smaller subspaces? Is there a closed expression for the Poincare* * series? References 1. D. Carlisle and P. H. Kropholler, Modular invariants of finite symplectic g* *roups, Preprint The following problems were proposed by Jaume Aguade and Carlos Broto. These problems are connected with our lack of understanding the modular invaria* *nts of the Weyl groups. Let R be a root system and W its Weyl group. Let M be the root lattice generate* *d by R and let S denote the symmetric algebra functor. Then W acts on S(M Fp) for a* *ny prime p and one is interested in the algebra of invariants S(M Fp)W . There ar* *e three types of primes p: a) If p does not divide |W | then everything is as in the characteristic zero c* *ase and S(M Fp)W is a polynomial algebra (Chevalley-Shephard-Todd theorem). b) If p does divide |W | but p is not a torsion prime for R then S(M Fp)W is * *also 8 polynomial (Demazure). Here p is a torsion prime for R if the compact Lie group* * associated to R has torsion at the prime p. Equivalently, there is a purely algebraic desc* *ription of P the torsion primes in the following way: Let J : S(M) ! S(M), J = (det!) ! an* *d let d 2 S be the product of all positive roots of R. Then if t is the smallest inte* *ger such that td 2 Im(J) then the torsion primes are those dividing t (Demazure). c) If p is a torsion prime for R, there are examples with S(M Fp)W polynomial* * and other examples with S(M Fp)W not polynomial. For instance, if R is the root l* *attice associated to P SU(n) then the torsion primes are those dividing n. For n = p * *= 3 the invariants are polynomial. For n = p = 5 they are not. 1. The question of which torsion primes produce polynomial algebras of invarian* *ts is only settled in a few cases. 2. Extend Demazure's theory of torsion primes to complex reflection groups. Since this statement is too imprecise, let us state a very concrete question. * *Let R be a root system and let R0 be a sub-root-system of R. Let W and W 0be the respectiv* *e Weyl groups and assume that p is prime to the index of W 0in W . Then Demazure prove* *s that if p is a torsion prime for R then it is also a torsion prime for R0. Is this a* *lso true for more general groups generated by pseudo-reflections? Demazure's proof uses at a cruc* *ial point the existence of a unique element of maximal length in any Coxeter group and th* *is fails in more general groups. The following problems were proposed by Larry Smith. Let G GL(n; IF) be a (finite) subgroup acting on V = IF . There is then the ri* *ng of coinvariants P (V )G := IFP(V )GP (V ) = P (V )=IG (V ) where P (V )G is the ring of invariants and IG (V ) P (V ) the ideal generate* *d by the invariants of positive degree. The group G acts on P (V )G , and so has a ring * *of invariants 9 References 1. Demazure, M. Invariants symetriques des groupes de Weyl et torsion, Inv. M* *ath 21 (1973), pp. 287 - 301. 2. Speerlich, T., Automorphisms of Rings of Coinvariants, Math G"ottingensis 19* *91. P (V )GG , which may or may not be trivial, and hence a ring of coinvariants P (V )G;2 := IFP(V )GGP (V )G Continuing in this way we define inductively P (V )G;i:= IF(P(V )G;i-1)GP (V )G;i-1: If we introduce the notations P (V )G;0 := P (V ) P (V )G;1 := P (V )G we then have a sequence of epimorphisms: P (V )- ! P (V )G;1-! P (V )G;2-! . .-.!P (V )G;i-! . . . Let IGi(V ) := ker{P (V )- ! P (V )G;i} : By Noether's theorem this stops. Define s to be the first integer at which it s* *tops: 1. When is P (V )G;s a Poincare duality algebra? 2. When is IGs(V ) generated by a regular sequence? 3. For Weyl Groups in characteristic zero Papadima and Speerlich have shown tha* *t the group of algebra automorphisms of P (V )G is the normalizer of the Weyl group i* *n GL(V ). What happens for the other pseudo reflection groups? What happens in finite cha* *racter- istic? 10 3. Kac, Victor G., Dale N. Peterson, Generalized Invariants of Groups generate* *d by Reflections , Journees de Geometria, Birkhauser Verlag 1985. Cohomology of Groups The following problems were proposed by David Benson. 1. Suppose that n is a positive integer and p is a prime. Can one construct a f* *inite group G and an element x 2 H*(G; Fp) with xn = 0 but xn-1 6= 0? If p = 2, Avrunin and Carlson have found a family of 2-groups of nilpotence class 2 which answer this* * problem. If p is odd, no example is known of a nilpotent element of H*(G; Fp) with xp 6=* * 0. 2. Suppose that G is a finite group and k is a field of characteristic p. Suppo* *se further that k[i1; : :;:ir] is a polynomial subring over which H*(G; k) is finitely gen* *erated as a module, with deg(ii) = ni, so that in particular by a theorem of Quillen, r is * *equal to the p-rank of G. Then a theorem of [BC] states that there is a first quadrant spect* *ral sequence whose E2 term is H*(G; k) *("i1; : :;:"ir): Here, H*(G; k) appears on the base, and the "iiare exterior generators on the f* *ibre of degree ni - 1. The spectral sequence converges to the cohomology of a finite P* *oincare P r duality complex, where the dualising class is in degree i=1(ni - 1). In part* *icular, if P H*(G; k) is Cohen-Macaulay, then the Poincare series p(t) = i0 tidim kHi(G; k* *) (as a rational function of t) satisfies the functional equation p(1=t) = (-t)rp(t): (i) Is the converse true? In other words, if the above functional equation hol* *ds, does it follow that H*(G; k) is Cohen-Macaulay? (ii) Is there a generalisation of this to compact Lie groups? In this case, on* *e would P r expect the dualising class to have degree - dimG + i=1(ni- 1), where dim G de* *notes the dimension of G as a manifold. In particular, in the Cohen-Macaulay case, on* *e would 11 obtain the functional equation p(1=t) = (-t)rtdimG p(t): This has been checked for Spin(10) at the conference. 3. Let Z be a central subgroup of (say, finite) groups G1 and G2, and let G be * *the central product of G1 and G2 along Z. Then there is a pullback square of fibrations BG ! BG1=Z # # BG2=Z ! K(Z; 2): and hence an Eilenberg-Moore spectral sequence Tor**H*(K(Z;2))(H*(G1=Z); H*(G2=Z)) ) H*(G): Is there an algebraic construction of this spectral sequence? It is not hard t* *o construct an algebraic model for K(Z; 2). Namely, one regards a (strictly coassociative) * *projective resolution for Z as a differential graded augmented algebra, and constructs a p* *rojective resolution over this. But it seems harder to make a sensible construction of a * *resolution for Gi=Z as a differential graded module over this. 4. Is there a sensible definition of Steenrod operations in Tate cohomology of * *finite groups, H^*(G; Fp)? Preferably, the Steenrod operations should commute with Tate dualit* *y. Is there a sensible definition of an unstable algebra over the Steenrod algebra* * which takes into account the fact that Tate cohomology is graded by both the positive and n* *egative integers? References [BC]. D. J. Benson and J. F. Carlson. Projective resolutions and Poincare * *duality complexes. Submitted to Trans. A.M.S. Problem proposed by Larry Smith. 1. If H*(BG; IFp)) is Cohen-MacCauley, is it Cohen-MacCauley over the Steenrod * *algebra? I.E. Does there exist a polynomial subalgebra closed under the action of the St* *eenrod algebra over which it is finitely generate. 12 Steenrod Operations Problems proposed by Clarence Wilkerson. 1. In H*(IRIP(1)n; Z2) find ideals that are A(2) invariant and generated by a * *regular sequence, with generators of the same degree. 2. In H*(IRIP(1)N ; Z2) let I be an ideal generated by elements {fi | i = 1; : * *:n:} and {gk | k = 1; : :m:} of dimension 2. Suppose Sq1(fi) I 8 i. Let J be the sma* *llest A(2) ideal containing I. Show that for 2m + n < N there are non-nilpotent eleme* *nts in H*(IRIP(1)N ; Z2)=J . 13