GROUP ACTIONS AND THE SINGULAR SET by DANIEL H. GOTTLIEB and MURAD OZAYDIN

1. Introduction Suppose a compact Lie group G is acting on a G–CW complex X. The singular set

X^{S} consists of all points in X with non-trivial isotropy subgroups. If the action were free, then the singular set would be empty, otherwise X^{S} is non-empty. The main purpose of this paper is to show that the trace of the action, denoted tr(G, X), is precisely equal to the trace of the singular set tr(G, X^{S}) when the action is not free.

The trace of an action was introduced in [G] . It is closely related to the exponent of Browder and Adem, studied in [B1], [B2], and [A]. It is an integer invariant of the action which characterizes a free action of a finite group on a finite complex, and also equals the number of points are in the smallest orbit of an action when G is an elementary abelian p-group.

We first recall briefly the definition of the trace of an action as defined in [G]. Then we note that the main results of [G] may be expressed in the category of G–CW complexes. The key lemma is established. It gives an obstruction to adjoining an equivariant cell without changing the trace. This implies the main theorem that tr(G, X) equals tr(G, X^{S}).

We apply the key lemma to investigate the relationship between the trace of an action, and the orbit size of the action which we define to be the greatest common divisor of the cardinalities of the orbits. While these two numbers are always equal when an elementary abelian p-group acts on a finite complex (indeed we will show here that these numbers are equal if the group has elementary abelian Sylow p-subgroups for all p), we will show that for cyclic subgroups examples can be found in which the trace and orbit size assume any arbitrary integers subject only to the restriction that the first integer divides the second.

Finally in the last section, we compare the trace and the exponent. We observe they are equal when G is a finite group and X is a finite complex. It is an open question if they agree when X is only finite dimensional. While the trace is defined for every action, the exponent as defined in [A] is only defined for finite group actions on finite dimensional complexes. We propose an extension of the definition of the exponent to compact Lie groups.

Many of these results comprise chapter III of [O], the Ph.D thesis of Murad Özaydın , Purdue University.

2. The trace of an action Given a map f : Y → X between two topological spaces, a transfer τ of trace k (k ∈ ℤ)

for f_{∗} is a graded homomorphism

τ : H_{∗}(X; ℤ) → H_{∗}(Y; ℤ)

so that f_{∗} ∘ τ(x) = kx for all x in H_{∗}(X; ℤ). The set of all traces associated to transfers for f_{∗} form a subgroup of ℤ, and we define deg(f) (degree of f) to be the non-negative generator of this subgroup.

Let F → E → B be a fiber bundle. The trace of this bundle is the least common multiple of the degrees of projection maps q of the pullback bundles F → f^{∗}(E) ⟶^{q} Y taken over all maps f : Y → B. The trace of the action of a group G on a space X is the trace of the Borel construction X → X_{G} → B_{G}, denoted tr(G, X).

Now a key fact about traces is Proposition 6.6 of [G]:

Proposition 1. Let (G, M) be an action on a compact manifold and suppose B_{G} has finite type. Then tr(G, M) = tr(p) where p : E → B is a pullback of the universal fibration p_{G} : M_{G} → B_{G} by any map f : B → B_{G} which is a (dim M + 2)-homotopy equivalence. If B is a closed oriented manifold, then tr(G, M) = deg(p).

Now this proposition applies when G is a compact Lie group, since B_{G} has finite type in this case. We want to extend this result for manifolds M replaced by a finite G–CW complex X. Recall that a G–CW complex is built from equivariant cells of the form G/K × e (where K, a closed subgroup of G, is the isotropy subgroup of this cell e ) with equivariant attaching maps ([I1]). A smooth compact manifold with a smooth G action has a G–CW structure [I2].

Lemma 2. Let G be a compact Lie group and let X be a finite G–CW complex. Then X is a equivariant retract of a compact oriented G-manifold.

\underline{Proof:} We may equivariantly imbed X as a subspace of a representation space of G. Then Schultz, [S], states that X is an equivariant retract of some open invariant neighborhood V. Using the Haar measure, we can find a smooth invariant map from V to ℝ which is zero on X and 1 near the frontier of V. The inverse image of a suitably chosen interval gives the compact oriented invariant manifold which we are seeking. □

Hence Proposition 1 is true when M is a finite G–CW complex, since traces are preserved by equivariant retractions.

If f : B → B_{G} pulls back a fibre bundle p : E → B so that tr(p) = tr(G, X), then we say that p realizes tr(G, X). When B is a closed oriented m-manifold, then tr(G, X) = deg(p). That is the same as saying that the image of p_{∗} : H_{m}(E) → H_{m}(B) ≅ ℤ is generated by tr(G, X)[B] ∈ H_{m}(B; ℤ).

We can easily find a closed compact oriented manifold B so that f : B → B_{G} is highly connected for G a compact Lie group. It suffices to find a highly connected, closed, oriented manifold E with a free, orientation preserving G-action. Then B = E/G and f is a classifying map for the principal bundle G → E → B. Any compact Lie group G is isomorphic to a subgroup of a special orthogonal group SO(n). But SO(n) acts freely (and preserves the orientation because SO(n) is connected) on the highly connected Stiefel manifold SO(n + N)/SO(N) = E.

3. The obstruction to preserving the trace We investigate the effect of adjoining an equivariant cell to a G–CW complex on the

trace of the action. Let X be G–CW complex. The case to understand is the adjunction of an equivariant cell, because any finite G–CW complex is built in finitely many such steps. Recall that an equivariant n-cell G/K × e^{n}, (where K is a closed subgroup, G acts on the first factor) is adjoined by extending a map f : S^{n−1} → X^{K} equivariantly to f̃ : G/K × S^{n−1} → X, the boundary of our n-cell. The next result establishes an obstruction to removing an equivariant cell and preserving the trace.

Key Lemma. Let G be a compact Lie group, X a finite G–CW complex and Y = (G/K × e^{n}) ∪ ; f̃ X as above. Then there is an α ∈ H^{n+d}(B_{K}; ℤ) such that

tr(G, Y)| tr(G, X)| tr(G, Y) |α|

where d = dim G/K and |α| is the (additive) order of α.

\underline{Proof:} We know that tr(G, Y)| tr(G, X) because there is a G-map from X to Y ([G], 6.2a). To show that tr(G, X)| tr(G, Y) |α|, we first choose a closed oriented manifold E of connectivity N > n + d, where G acts freely on E. Then, as above, we have manifolds, E/G = B and E/K. Now E/K is N-homotopy equivalent to B_{K}, so H^{n+d}(B_{K}) ≅ H^{n+d}(E/K). So the α we are seeking in H^{n+d}(B_{K}), we may regard as in H^{n+d}(E/K). Now E/K is a closed oriented manifold of dimension dim E/K = dim E−dim K = dim B+ dim G − dim K = dim B + dim G/K = m + d, where m is the dimension of B. Now by Poincare duality H^{n+d}(E/K) ≅ H_{m−n}(E/K). Hence we can imagine the α in the theorem as an element in H_{m−n}(E/K).

We construct α as follows. There is an element ω ∈ H_{m}((Y × E)/G) so that p_{∗}(ω) = tr(G, Y)[B]. Consider the Meyer-Vietoris exact sequence of the union (Y × E)/G = ((X × E)/G) ∪ ((G/K × e^{n} × E)/G), noting that (G/K × e^{n} × E)/G = e^{n} × E/K and also the intersection of the two subspaces is equal to (G/K × S^{n−1} × E)/G:

\begin{matrix}→ H_{m}((X × E)/G) ⊕ H_{m}(e^{n} × (E/K)) ⟶^{i_{∗}+j_{∗}} H_{m}((Y × E)/G) ⟶^{δ} \\ H_{m−1}(S^{n−1} × (E/K))| ⟶^{(k,ℓ)_{∗}} H_{m−1}((X × E)/G) ⊕ H_{m−1}(e^{n} × (E/K)) \end{matrix}

Then δ(ω) = [S^{n−1}] × α + 1 × β ∈ H_{m−1}(S^{m−1} × E/K) where α is some element in H_{m−n}(E/K) and β is some element in H_{m−1}(E/K). But note that (k, ℓ)_{∗}(1 × β) = 1 × β ⊕ g where g is some other term contained in the H_{m−1}((X × E)/G)-summand. Hence, by the exactness of the Meyer-Vietoris exact sequence, β must be zero . Hence δ(ω) = [S^{n−1}] × α.

Now |α|ω is in the kernel of δ, hence it is in the image of i_{∗} + j_{∗} where i and j are the inclusions of (X × E)/G and (e^{n} × (E/K)) into (Y × E)/G respectively. So there are u and v so that i_{∗}(u) + j_{∗}(v) = |α|ω. So p_{∗}(i_{∗}(u) + j_{∗}(v)) = p_{∗}(|α|ω). We have |α| tr(G, Y)[B] = p_{∗}(|α|ω) = p_{∗}(i_{∗}(u) + j_{∗}(v)) = a[B] + b[B] = (a + b)[B] for some integers a and b. Now tr(G, X) divides a and tr(G, G/K) divides b, so (a + b) is a multiple of the greatest common divisor of tr(G, X) and tr(G, G/K). But G/K maps equivariantly into X, so tr(G, X) divides tr(G, G/K). Hence tr(G, X) divides a + b and hence divides |α| tr(G, Y) as was to be shown. □

In order to apply the key lemma we need some control over the cohomology of the isotropy subgroup K. The easiest case is when K is trivial, so all cohomology in positive dimensions vanish. We call two G spaces X and Y G–related if there are G-maps f : X → Y and f′ : Y → X. The singular set of a G-space X, denoted X^{S}, consists of all points in X with nontrivial isotropy subgroups.

Theorem 3. Let G be a compact Lie group and let X be a G- space which is G-related to a finite G–CW complex. Then tr(G, X) = tr(G, X^{S}), when X^{S} is not empty.

\underline{Proof:} If X is G-related to Y then the singular sets X^{S} and Y^{S} are also G-related. Hence tr(G, X) = tr(G, Y) and tr(G, X^{S}) = tr(G, Y^{S}), so we only need to prove tr(G, Y) = tr(G, Y^{S}). The finite G–CW complex Y is built from the subcomplex Y^{S} by adding equivariant cells with trivial isotropy subgroup. But the trace does not change when we add free cells, by the key lemma. □

4. Trace and orbit size When G is a finite group, a naive invariant measuring the nontriviality of the G-action

on X is the orbit size, denoted os(G, X). This is defined as the greatest common divisor of the cardinalities of all the orbits. That is

os(G, X) = gcd{[G : G_{x}] : x ∈ X}.

In this section we compare the properties of the trace and the orbit size.

(i) tr(G, X)| os(G, X) if X is an arbitrary space ([G], 6.2a and 6.7b). (ii) Suppose that G_{p} is a Sylow p–subgroup of G. Then tr(G, X) = Π tr(G_{p}, X) where the product is over all primes p dividing the order of G. ([G–O], theorem 14; [O], theorem 2.5) (iii) os(G, X) = Π os(G_{p}, X) where the product is taken over all primes p dividing the order of G. (iv) tr(G, X) = os(G, X). If all Sylow p–subgroups of G are elementary abelian and X is a finite complex. This follows from (ii) and (iii) and ([G], theorem 7.4) or ([B2], theorem 1.1).

(v) os(G, X)|Λ_{f}. where X is a finite complex and f : X → X is a G–map and the Lefschetz number of f is denoted by Λ_{f}. ([O],appendix)

Now we show that the trace equals the orbit size when X is low dimensional. On the other hand for cyclic groups acting on closed oriented manifolds, we can find examples where the trace and orbit size are any pair of positive integers subject to condition (i). These results will follow from the key lemma and theorem 3.

Corollary 4. Let G be a finite group and let X be G-related to a finite, one dimensional G–CW complex (a G-graph). Then tr(G, X) = os(G, X).

\underline{Proof:} Assume X^{S} is not empty. Since tr(G, X) and os(G, X) are invariants of Grelatedness, it suffices to prove this for a finite, one dimensional G–CW complex X. For any finite (isotropy) group K, we know H^{1}(K; ℤ) is trivial, hence by the key lemma the trace of the zero-skeleton is equal to tr(G, X). But for a finite discrete set X^{∘}, tr(G, X^{∘}) = os(G, X^{∘}). When the singular set is empty, i.e., when the action is free then tr(G, X) = |G| = os(G, X) (G a finite group and X a finite G–CW complex). The first equality follows from the key lemma by peeling off free equivariant cells till we get to the zero skeleton. □

Corollary 5. Let G be a finite group acting smoothly on a connected manifold M. Assume that there is a free orbit and either: a) M is a compact surface; or b) M is an oriented 3-manifold with an orientation preserving action. Then tr(G, M) = os(G, M).

\underline{Proof:} M is a finite G–CW complex [I2]. Since there is a free orbit the hypothesis implies that the singular set M^{S} is at most one dimensional, (because the action is orientation preserving in the three manifold case the codimension of the singular set must be greater than 1). Therefore Theorem 3 and Corollary 4 imply tr(G, M) = os(G, M). □

Lemma 6. Let ϕ : ℤ_{p^{n+1}} → ℤ_{p^{n}} be the cononical epimorphism. Thinking of ℤ_{p^{n}} as a subgroup of S^{1}, we have the standard action on S^{2k+1} given by complex multiplication on the unit sphere in complex (k + 1)–space. Let ℤ_{p^{n+1}} act on S^{2k+1} via ϕ. Then os(ℤ_{p^{n+1}}, S^{2k+1}) = p^{n} and tr(ℤ_{p^{n+1}}, S^{2k+1}) = max(p^{n−k}, 1).

\underline{Proof:} Since the action is orientation preserving, the trace equals the fiber number ([G], 6.4) The fiber number is given by the exponent of the transgression of the fundamental class [\overline{S}^{2k+1}] in the cohomology Serre spectral sequence of the Borel construction:

H^{2k+1}(S^{2k+1}) ≅ E_{2}^{0,2k+1} → E_{2}^{2k+2,0} ≅ H^{2k+2}(ℤ_{p^{n+1}})

(because there are only 2 nonzero rows at the E^{2} level). The action being induced from the action of ℤ_{p^{n}}, the transgression map factors through H^{2k+2}(ℤ_{p^{n}}) → H^{2k+2}(ℤ_{p^{n+1}}), by the naturality of the Serre spectral sequence. H^{∗}(ℤ_{p^{n}}) is generated by the powers of α in H^{2}(ℤ_{p^{n}}) ≅ ℤ_{p^{n}}, and ϕ^{∗}(α) = pβ where β is an analogous generator for H^{2}(ℤ_{p^{n+1}}). So ϕ^{∗}(α^{k+1}) = p^{k+1}β^{k+1}. The image of [\overline{S}^{2k+1}] under the transgression in the Borel construction of the action of ℤ_{p^{n}} is a generator of H^{2k+2}(ℤ_{p^{n}}) because ℤ_{p^{n}} is acting freely. Thus the exponent of the image of [\overline{S}^{2k+1}] in H^{2k+2}(ℤ_{p^{n+1}}) ≅ ℤ_{p^{n+1}} =< β^{k+1} > is p^{n+1}/p^{k+1} = p^{n−k} if k ≤ n and 1 otherwise. Also os(ℤ_{p^{n+1}}, S^{2k+1}) = p^{n} because every orbit has p^{n+1}/p = p^{n} elements. □

Let G and H act on spaces X and Y respectively. We say that G × H acts on X × Y with the product action if g × h(x × y) = g(x) × h(x) for all g ∈ G, h ∈ H, x ∈ X, y ∈ Y.

Lemma 7.. If (G × H) is a product action, and if the order of G is relatively prime to the order of H, then

a) tr(G × H, X × Y) = tr(G, X) · tr(H, Y)

b) os(G × H, X × Y) = os(G, X) · os(H, Y)

\underline{Proof:} We identify G with the subgroup G×1 ⊂ G×H and H with the subgroup 1×H. Then the composition X → X × Y → X given by x ↦ (x × y_{0}) ↦ x is a composition of G–maps. Thus X and X × Y are G–related. Both trace and orbit size are preserved under G–relatedness, hence tr(G, X) = tr(G, X × Y) and os(G, X) = os(G, X × Y). Similarly tr(H, Y) = tr(H, X×Y) and os(H, Y) = os(H, X×Y). Now it follows from (ii) and the fact that |G| and |H| are relatively prime that tr(G × H, X × Y) = tr(G, X × Y) · tr(H, X × Y) which equals tr(G, X)·tr(H, Y) as required. Similarly, using (iii) and the relative primeness of |G| and |H|, we get os(G × H, X × Y) = os(G, X) · os(H, Y).

Theorem 8. Let m and n be positive integers with m dividing n. There exists an action (G, M) where G is cyclic, M is a closed oriented manifold and the action is orientation preserving and effective, such that tr(G, M) = m and os(G, M) = n.

\underline{Proof:} We factor m and n into prime factors. Then m = p_{1}^{e_{1}}p_{2}^{e_{2}} … p_{k}^{e_{k}} and n = p_{1}^{e_{1}+f_{1}}p_{2}^{e_{2}+f_{2}} … p_{k}^{e_{k}+f_{k}}, where 0 ≤ e_{i}, f_{i} for all i = 1, …, k. For each i consider the action (ℤ_{p_{i}}^{e_{i}+f_{i}+1}, S^{2f_{i}+1}) as described in Lemma 6. So the trace of the action is p_{i}^{e_{i}} and the orbit size is p_{i}^{e_{i}+f_{i}}. Now let G be the product of the ℤ_{p_{i}}^{e_{i}+f_{i}+1}, so G is a cyclic group. Let X be the product of the S^{2f_{i}+1}. Then by lemma 7, the product action (G, X) has trace equal to m and orbit size equal to n . This proves the theorem except for the conclusion that the action should be effective. We attach a fee G–cell to X to get the action (G, Y) which is effective. By the key lemma, tr(G, Y) equals m; and os(G, X) obviously equals n. But now Y is not a manifold. Using lemma 2 we can find a compact oriented manifold on which G acts effectively, preserving orientation, and which retracts equivariantly onto Y . Then doubling the manifold and the action we get a compact oriented manifold M on which G acts effectively, preserving orientation, and which retracts equivariantly onto Y. Thus the trace and orbit size of (G, M) still equal m and n . □

Finally we show that corollary 4 is true only for dimension of X equal to one, and corollary 5 needs the hypothesis that there is a free orbit, by considering the following example which also gives a non-trivial use of the key lemma.

Example 9. Let ℤ_{4} act on S^{2} through the epimorphism ℤ_{4} → ℤ_{2} and then via the antipodal action. Then tr(ℤ_{4}, S^{2}) = 1, and os(ℤ_{4}, S^{2}) = 2 .

\underline{Proof:} By lemma 6, tr(ℤ_{4}, S^{3}) = 1. But S^{3} is obtained from S^{2} by adding an equivariant 3 cell of isotropy type ℤ_{2}. Since H^{3}(ℤ_{2}) = 0 we get tr(ℤ_{4}, S^{2}) = tr(ℤ_{4}, S^{3}) = 1 by the key lemma. □

5. The trace and the exponent We have already mentioned that the trace is closely related to the exponent of ([A],

definition 1.4 and 3.1) . The trace is defined for any action while the exponent is defined only for actions of finite groups acting on connected finite dimensional CW–complexes. We will show that the trace equals the exponent, but only for finite complexes. Whether they are equal for actions on non-compact, finite dimensional complexes is open. We will suggest an extension of the definition of the exponent to the case of compact Lie groups acting on possibly disconnected G–CW complexes.

Proposition 10. If G is a finite group and X is a connected compact G–CW complex, then tr(G, X) = e_{G}(X), i.e. the trace equals the exponent.

\underline{Proof:} If X is a connected orientable manifold and the finite group G acts preserving orientation, then the trace and the exponent both equal the fibre number ([G], theorem 6.4; [A], corollary 3.13). Now lemma 2 states that a compact G–CW complex is an equivariant retract of of a closed oriented G–manifold. Since both the trace and the exponent are invariant under equivariant retracts, they are equal. □

It would be useful to know if the trace equaled the exponent for non-compact finite dimensional G–CW complexes since the exponent works just as well in that case as in the compact case. For example e_{G}(X) = e_{G}(X^{S}) . So if trace equals exponent, then our main theorem, theorem 3, could be extended to non-compact situations for finite G.

We propose to define e_{G}(X) in the case where X is a possibly disconnected finite dimensional G–CW complex and G is a connected Lie group. First if G is finite and acts transitively on the set of connected components, where e_{G}(X) is the subgroup of G which fixes a component G_{0}, we define

e_{G}(X) = [G : G_{0}] · e_{G_{0}}(X)

Then if X is generally disconnected, we note that X is the union of spaces X_{i} on which G acts transitively and we define

e_{G}(X) = gcd{e_{G}(X_{i})}

Finally if G is a compact Lie group we propose the definition

e_{G}(X) = lcm{e_{H}(X)} for all finite subgroups H of G.

The new exponent e_{G}(X) satisfies all the relevant properties listed in the introduction of [A] when G is finite. Also, for G finite and X compact, the trace equals the exponent. This follows because the trace satisfies the first two equations above ([O], proposition 3.3; [G],p. 397). In the case of G compact and X compact we no longer know if they are equal, but e_{G}(X) divides tr(G, X) since tr(H, X) divides tr(G, X) by ([G], 6.2b). (It is unusual for a useful invariant to divide the trace, it is usually the other way.) The trace would be equal to the exponent if the answer to the following question is yes. Does

tr(G, X) = lcm {tr(H, X)} for all finite subgroups H of G ?

A similar question, whose affirmative answer would imply that trace equals exponents when G is finite and X is finite dimensional, is the following. Does

tr(G, X) = gcd {tr(G, Y)} for all finite G-subcomplexes Y of X ?

BIBLIOGRAPHY

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[G] D.H. Gottlieb, The trace of an action and the degree of a map, Transactions Amer. Math. Soc., 293 (1986), 381-410.

[G-O] D. H. Gottlieb and M. Özaydın, Intersection numbers, transfers, and group actions, (preprint)

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[I1] S. Illman, Equivariant algebraic topology, Ph. D. Thesis, Princeton University (1972).

[I2] S. Illman, The Equivariant triangulation theorem for actions of compact Lie groups, Math. Ann., 262 (1984), 487-501.

[O] M. Özaydın, An integer invariant of a group action, Ph. D. Thesis, Purdue University (1984).

[S] R. Schultz, Homotopy decompositions of equivariant function spaces II, Math. Zeitschrift, 132 (1973), 69-90.

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