PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 12, December 1997, Pages 3735–3739 S 0002-9939(97)04190-7

HOMOTOPY FIXED POINTS FOR CYCLIC p-GROUP ACTIONS

W. G. DWYER AND J. M. MØLLER

(Communicated by Thomas Goodwillie)

Abstract. The homotopy fixed point p-compact groups for cyclic p-group actions on nonabelian connected p-compact groups are not homotopically discrete.

1. Introduction

It is a classical result that cyclic groups acting on nonabelian compact connected Lie groups have no isolated fixpoints [2, Lemme 1, p. 46]:

Theorem 1.1. Let X be a nonabelian connected compact Lie group equipped with an action of a cyclic group G. Then the identity component of the fixed point group X^{G} is nontrivial.

In this note we prove an analog for p-compact groups of this statement. First, we need a few concepts.

Suppose that X is a p-compact group [4] with classifying space BX and that G is a finite group.

Definition 1.2. A G-action on X is a sectioned fibration

[figure: BX; (BX)_{hG}; BG; Ba]

over BG with fibre BX.

If G is a finite p-group, it is known [4, 5.8] that each component of the section space (BX)^{hG} is the classifying space of a p-compact group. We define the homotopy fixed point p-compact group for the G-action to be the p-compact group

X^{hG} = Ω((BX)^{hG}, Ba)

whose classifying space is the component containing the section Ba.

Having introduced these concepts, we can now formulate the main result of this note. (A connected p-compact group is nontrivial if its classyfying space is noncontractible.)

Theorem 1.3. Let X be a nonabelian connected p-compact group equipped with an action of a cyclic p-group G. Then the identity component of the homotopy fixed point p-compact group X^{hG} is nontrivial.

\begin{matrix}c \\ ◯ \end{matrix}1997 American Mathematical Society

Received by the editors July 10, 1996.

1991 Mathematics Subject Classification. Primary 55P35.

Key words and phrases. Fixed point, p-compact group, Lefschetz number, Weyl group.

The following consequence of this theorem, whose proof relies on a Lefschetz number calculation, is immediate.

Corollary 1.4. Let ν: G → Y be a monomorphism of a cyclic p-group into a (not necessarily connected) p-compact group Y which is not a p-compact toral group. Then the identity component of the centralizer C_{Y} (G) of ν is nontrivial.

The self-centralizing diagonal subgroup (ℤ/2ℤ)^{n} of O(n) shows that noncyclic subgroups may have discrete centralizers.

Corollary 1.4 plays an important role in the proof of the main result of [9].

2. A Lefschetz number calculation

Let X be a connected p-compact group, G = ℤ/p^{r}, r ≥ 0, a cyclic p-group and

[figure: BX; (BX)_{hG}; BG; Ba]

an action of G on X. The homotopy fixed point p-compact group X^{hG} is the section space of the fibrewise looping

\xymatrix{X \ar[r] & X_{hG} \ar[r] & BG \ar[l]}

of the G-action. Consider the associated monodromy homomorphisms

G → Aut_{∗}(BX), (1)

G → Aut_{∗}(X) (2)

of G into the groups of based homotopy classes of based self-homotopy equivalences of the fibres and the induced representations

G → Aut H_{ℚ_{p}}^{∗}(BX), (3)

G → Aut H_{ℚ_{p}}^{∗}(X) (4)

of G in the p-adic rational cohomology algebras. Of course, representation (4) induces yet another representation

G → QH_{ℚ_{p}}^{∗}(BX) (5)

of G in the graded vector space of indecomposables.

Let Bg: BX → BX and g: X → X be the self-homotopy equivalences induced by a generator g ∈ G. We shall compute the Lefschetz number

Λ(X; G) = ∑(−1)^{i} trace H_{ℚ_{p}}^{i}(g)

for the action of G on X in terms of the irreducible summands of the representation (5).

Recall [4, §4] that the cyclic group G admits r + 1 essentially distinct irreducible representations ρ_{0}, ρ_{1}, …, ρ_{r} over the p-adic numbers. Here, ρ_{0} is the trivial representation and ρ_{i}, 1 ≤ i ≤ r, is the composition of the reduction map G = ℤ/p^{r} → ℤ/p^{i} with the action of ℤ/p^{i}, regarded as the group of p^{i}th roots of unity, on the extension field ℚ_{p}(ω_{i}) of ℚ_{p} by a primitive p^{i}th root of unity ω_{i}. The dimension of ρ_{i}, 1 ≤ i ≤ r, is [ℚ_{p}(ω_{i}) : ℚ_{p}] = p^{i} − p^{i−1}.

Proposition 2.1. Suppose that the G-representation QH_{ℚ_{p}}^{∗}(BX) contains the irreducible representation ρ_{i} with multiplicity n_{i}, 0 ≤ i ≤ r. Then

Λ(X; G) = \begin{cases}p^{n_{1}+···+n_{r}} & if n_{0} = 0, \\ 0 & if n_{0} ≠ 0 \end{cases}

is the Lefschetz number for the action of G on X. In particular, Λ(X; G) = 0 if and only if G fixes a nonzero vector of QH_{ℚ_{p}}^{∗}(BX).

Proof. Note that the monodromy action (2) of G on X = ΩBX is the looping of the monodromy action (1) on BX and that the Eilenberg–Moore spectral sequence provides a functorial isomorphism between the graded object Gr(H_{ℚ_{p}}^{∗}(X)) associated to a filtration of H_{ℚ_{p}}^{∗}(X) and the exterior algebra E(Σ^{−1}QH_{ℚ_{p}}^{∗}(BX)) on the desuspension of QH_{ℚ_{p}}^{∗}(BX). Combining this with the isomorphism

QH_{ℚ_{p}}^{∗}(BX) ≅ n_{0}ρ_{0} ⊕ n_{1}ρ_{1} ⊕ ⋯ ⊕ n_{r}ρ_{r}

of G-representations induces yet another isomorphism

Gr(H_{ℚ_{p}}^{∗}(X)) ≅ E(Σ^{−1}ρ_{0})^{⊗n_{0}} ⊗ E(Σ^{−1}ρ_{1})^{⊗n_{1}} ⊗ ⋯ ⊗ E(Σ^{−1}ρ_{r})^{⊗n_{r}}

of G-representations. By the additivity [4, 4.12] of traces in exact sequences, then, the Lefschetz number

Λ(X; G) = ∏_{i=0}^{r} Λ_{i}^{n_{i}}

where Λ_{i} is the trace for the action of G on E(Σ^{−1}ρ_{i}).

Since E(Σ^{−1}ρ_{0}) is the trivial representation, Λ_{0} = 0.

When i > 0, we pass to an algebraic closure of ℚ_{p}. Then ρ_{i} splits into 1dimensional representations and we see that Λ_{i} = Φ_{i}(1) where Φ_{i} is the characteristic polynomial for g acting on ρ_{i} or, equivalently, for ω_{i} acting on ℚ_{p}(ω_{i}). Hence Φ_{i} is the p^{i}th cyclotomic polynomial so Φ_{i}(1) = p and the proposition follows.

The consequence below is evident if we recall [4, 4.5, 5.7, 5.10] that the Lefschetz number Λ(X; G) computes the Euler characteristic of X^{hG} and that a p-compact group is homotopically discrete if it looks so in p-adic rational cohomology.

Corollary 2.2. The following conditions are equivalent:

(1) X^{hG} has a nontrivial identity component.

(2) χ(X^{hG}) > 0.

(3) Λ(X; G) > 0.

(4) G fixes a nonzero vector of QH_{ℚ_{p}}^{∗}(BX).

The proof of Theorem 1.3 has now been reduced to the following

Lemma 2.3. Suppose that X is nonabelian (i.e. not a p-compact torus). Then G fixes a nonzero vector of QH_{ℚ_{p}}^{∗}(BX).

Proof. Let T → X be a maximal torus with Weyl group W. The dual weight lattice L = π_{2}(BT) is then a ℤ_{p}[W]-module whose rationalization L ⊗ ℚ exhibits W as a reflection group over ℚ_{p}. The action of G on the symmetric invariants Sym((L ⊗ ℚ)^{∗})^{W} ≅ H_{ℚ_{p}}^{∗}(BX) factors [4, 8.11, 9.5], [8, §3] through N(W)/W where N(W) is the normalizer of W < Aut(L ⊗ ℚ).

Suppose first that X is almost simple, i.e. [5, 1.6] that the center of X is finite and that L⊗ℚ is a simple ℚ_{p}[W]-module. Then the reflection group W is one of the irreducible reflection groups on the Shephard–Todd–Clark–Ewing list as presented e.g. in [6, p. 165]. The list provides information about the indecomposables of the invariant ring in that the degrees of each reflection group are given.

If p > 2, the list shows that dim_{ℚ_{p}} QH_{ℚ_{p}}^{i}(BX) < p − 1 for all i. (In fact QH_{ℚ_{p}}^{i}(BX) has dimension ≤ 2 with dimension 2 occurring only in case 2a (where the degrees given in [6] are incorrect) and in case 19, neither of which are realizable for p = 3.) Since a nontrivial p-adic representation of a cyclic p-group requires at least p − 1 dimensions, G must act trivially on all of QH_{ℚ_{p}}^{∗}(BX) (which is nonzero if X is nontrivial [4, 5.10]).

The case p = 2 requires separate treatment. The only irreducible 2-adic reflection groups are the classical Coxeter groups together with group number 24 of rank 3, W = ℤ/2ℤ × GL_{3}(𝔽_{2}), realized by DI(4) [3]. If W is one of the classical Coxeter groups, the effect of an element of the normalizer N(W) on the degree 4 invariants is multiplication by u^{2}, 2u^{2}, or 3u^{2}, where u ∈ ℚ_{2}^{∗} is a 2-adic unit [7, 1.7]. Since −1 doesn’t have this form, the 1-dimensional G-representation H_{ℚ_{p}}^{4}(BX) = QH_{ℚ_{p}}^{4}(BX) is the trivial one. Generators for the ring of invariant polynomials of the unique nonclassical 2-adic reflection group are [1, p. 101]

y_{8} = x_{1}x_{2}^{3} + x_{2}x_{3}^{3} + x_{3}x_{1}^{3},
y_{12} = det (\frac{∂^{2}y_{8}}{∂x_{i}∂x_{j}} ) ,

y_{28} = det ( \frac{∂y_{12}}{∂x_{j}}  0^{\frac{∂^{2}y_{8}}{∂x_{i}∂x_{j}}  \frac{∂y_{12}}{∂x_{i}}} ) ,

where the subscript on the variable y denotes the dimension of the corresponding indecomposable cohomology class. Note that if an element of N(W) takes y_{8} to its opposite, then also y_{12} is taken to its opposite but y_{28} remains fixed. Thus any element of 2-power order in N(W)/W must fix either y_{8} or y_{28} (considered as elements of H_{ℚ_{p}}^{∗}(BX)).

This proves the lemma for all almost simple p-compact groups.

Next suppose that X is simply connected and nontrivial. Then there exist, by the splitting theorem [5], almost simple p-compact groups X_{1}, …, X_{n} with dual weight lattices L_{1}, …, L_{n} and Weyl groups W_{1}, …, W_{n} such that X ≅ X_{1} × ⋯ × X_{n} and L ≅ L_{1} × ⋯ × L_{n} as W ≅ W_{1} × ⋯ × W_{n}-modules. The efffect of Bg on H_{ℚ_{p}}^{∗}(BX) = ⨂ H_{ℚ_{p}}^{∗}(BX_{i}) has, cf. [8, 3.5], the form

H_{ℚ_{p}}^{∗}(Bg) = (A_{1} ⊗ ⋯ ⊗ A_{n}) ∘ σ

where A_{i} is an automorphism of H_{ℚ_{p}}^{∗}(BX_{i}), 1 ≤ i ≤ n, and σ is a permutation within the isomorphism classes of these algebras. Hence

QH_{ℚ_{p}}^{∗}(Bg) = (QA_{1} ⊕ ⋯ ⊕ QA_{n}) ∘ σ

on QH_{ℚ_{p}}^{∗}(BX) = ⨁ QH_{ℚ_{p}}^{∗}(BX_{i}). There are now essentially two distinct cases to consider. Namely, the case where σ is trivial and the case where σ is a cyclic permutation of p -power order > 1. The first case was treated above and in the second case, QH_{ℚ_{p}}^{∗}(Bg) fixes the diagonal. Hence the fixed point vector space QH_{ℚ_{p}}^{∗}(BX)^{G} is nontrivial for any nontrivial simply connected p-compact group X.

Finally, up to isogeny any connected p-compact group has the form X × S [11, 5.4] where X is simply connected and S is a p-compact torus and any automorphism is a product of an automorphism of X with an automorphism of S [10, 4.3]. Hence

QH_{ℚ_{p}}^{∗}(BX × BS)^{G} ≅ QH_{ℚ_{p}}^{∗}(BX)^{G} ⊕ QH_{ℚ_{p}}^{∗}(BS)^{G}

is nontrivial if X is nontrivial.

We conclude this note with the easy proof of Corollary 1.4.

Proof of Corollary 1.4. Let π be the component group and X the identity component of Y. The p-compact group extension

X^{hG} → C_{Y} (G) → C_{π}(G)

shows that X^{hG} and C_{Y} (G) have isomorphic identity components.

References

1. D.J. Benson, Polynomial invariants of finite groups, London Mathematical Society Lecture Note Series, vol. 190, Cambridge University Press, Cambridge, 1994. MR 94j:13003

2. N. Bourbaki, Groupes et algèbres de Lie, Chp. 9, Masson, Paris, 1982. MR 84j:22001

3. W.G. Dwyer and C.W. Wilkerson, A new finite loop space at the prime 2, J. Amer. Math. Soc. 6 (1993), 37–64. MR 93d:55011

4. , Homotopy fixed point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), 395–442. MR 95e:55019

5. , Product splittings for p-compact groups, Fund. Math. 147 (1995), 279–300. MR 96h:55005

6. R. Kane, The homology of Hopf spaces, North-Holland Mathematical Library, vol. 40, Elsevier Science Publishers B.V., Amsterdam–New York–Oxford–Tokyo, 1988. MR 90f:55018

7. J.M. Møller, The normalizer of the Weyl group, Math. Ann. 294 (1992), 59–80. MR 94b:55010

8. , Completely reducible p-compact groups, The Čech Centennial. A conference on homotopy theory. Contemporary Mathematics, vol. 181 (Providence, Rhode Island) (M. Cenkl and H. Miller, eds.), American Mathematical Society, 1995, pp. 369–383. MR 97b:55020

9. , Normalizers of maximal tori, Preprint, March 1995.

10. , Rational isomorphisms of p-compact groups, Topology 35 (1996), 201–225. MR 97b:55019

11. J.M. Møller and D. Notbohm, Centers and finite coverings of finite loop spaces, J. Reine Angew. Math. 456 (1994), 99–133. MR 95j:55029

Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556

E-mail address: William.G.Dwyer.1@nd.edu

Matematisk Institut, Universitetsparken 5, DK–2100 København Ø, Denmark

E-mail address: moller@math.ku.dk