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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