INTERSECTION NUMBERS, TRANSFERS, AND GROUP ACTIONS

by

Daniel H. Gottlieb and Murad Ozaydin

Purdue University University of Oklahoma

Abstract

We introduce a new definition of intersection number by means of umkehr maps. This definition agrees with an obvious definition up to sign. It allows us to extend the concept parametrically to fibre bundles. For fibre bundles we can then define intersection number transfers. Fibre bundles arise naturally in equivariant situations, so the trace of the action divides intersection numbers. We apply this to equivariant projective varieties and other examples. Also we study actions of non-connected groups and their traces.

AMS subject classification 1991: Primary: 55R12, 57S99, 55M25 Secondary: 14L30 Keywords: Poincare Duality, trace of action, Sylow subgroups, Euler class, projective variety.

1. Introduction

When two manifolds with complementary dimensions inside a third manifold intersect transversally, their intersection number is defined. This classical topological invariant has played an important role in topology and its applications. In this paper we introduce a new definition of intersection number.

The definition naturally extends to the case of parametrized manifolds over a base B; that is to fibre bundles over B . Then we can define transfers for the fibre bundles related to the intersection number. This is our main objective , and it is done in Theorem 6. The method of producing transfers depends upon the Key Lemma which we prove here. Special cases of this lemma played similar roles in the establishment of the Euler–Poincare transfer, the Lefschetz number transfer, and the Nakaoka transfer.

Two other features of our definition are: 1) We replace the idea of submanifolds with the idea of maps from manifolds, and in effect we are defining the intersection numbers of the maps; 2) We are not restricted to only two maps, we can define the intersection number for several maps, or submanifolds.

We can apply Theorem 6 to equivariant topology. When the maps involved are equivariant, or even if they are only homotopy equivariant, the intersection number is an equivariant invariant. This is seen by using the Borel construction to get the fibre bundle situation for which theorem 6 provides transfers. Transfers tell us much about equivariant topology. Much of this information can be concentrated in a single integer called the trace of the action.

The trace of an action is an integer which is defined by looking at all the possible transfers arising from the Borel construction. It was defined in [G2] and is very closely related to the exponent of an action as defined by W. Browder in [B]. In fact the trace equals the exponent for orientation preserving actions on manifolds. However the trace, in contrast to the exponent, is always defined in any equivariant setting. When the equivariant situation allows intersection numbers, the trace must divide the intersection number. This result is our main application of transfers to group actions and is found in Theorem 7 and is a consequence of Theorem 6.

In addition to showing that the trace divides intersection numbers, we examine how the trace behaves when the acting group is disconnected. In particular, we show that the trace of an action of a finite group G on a space X must be the product of the traces of the Sylow p-subgroups acting on X. This latter part is contained in the thesis of Murad Özaydin at Purdue University [O].

Finally, we give some examples of trace in action. In the case of nonsingular projective varieties, the degree of the variety is an intersection number. Thus in suitable equivariant situations the trace divides the degree of the variety. We obtain generalizations of a theorem of W. Browder and N. Katz [B-K] . Also we look at examples of elementary p–groups acting on products of spheres. In these situations we find that appropriate G–submanifolds imply that the trace divides associated determinants, permanents, and Pfaffians.

We would like to thank L. Avramov for helpful discussions.

2. Multiple functional intersection number DEFINITION 1. Let f_{i} : A_{i} → M be a finite set of maps between closed oriented manifolds A_{i} and M. We suppose the sum of the co-dimensions of A_{i} equals the dimension of M. That means, assuming there are k maps f_{i}, that ∑^{k} ; i=1(dim M − dim A_{i}) = dim M, or

∑^{k} ; i=1 dim A_{i} = (k − 1) dim M.

Next define F : A_{1} × ⋯ × A_{k} → M × ⋯ × M by F = f_{1} × ⋯ × f_{k}. Define the \underline{multiple intersection number}, denoted f_{1} • f_{2} • ⋯ • f_{k}, as the integer

[Δ^{∗}F^{!}(1)] ∩ [M] ∈ H_{0}(M; ℤ) ≅ ℤ.

Here Δ : M → M × ⋯ × M is the diagonal map, F^{!} denotes the Umkehr map on cohomology and 1 ∈ H^{0}(A_{1} × ⋯ × A_{k}). We denote by [M] the orientation class of M. We require M to be connected, so that H_{0}(M; ℤ) ≅ ℤ, but the A_{i} need not be connected. So the 1 ∈ H^{0}(A_{1} × ⋯ × A_{k}) is the sum of the generators 1 ∈ H^{0} (connected components of A_{1} ×⋯×A_{n}) ≅ ℤ, and each of those generators is equal to a product 1_{A} × 1_{B} × …, where the A, B, … are connected components of A_{1}, A_{2}, … respectively. The Umkehr map F^{!} is defined as the composition D_{M^{×}}^{−1}F D_{A^{×}} where D_{M} denotes the Poincare duality map H^{∗}(M; ℤ) ⟶^{≅} H_{∗}(M; ℤ) given by D_{M}(x) = x ∩ [M]. Here M^{×} = M × ⋯ × M where the product is taken k-times and A^{×} = A_{1} × ⋯ × A_{k}.

Obviously the multiple intersection number depends on the choices of orientation on A^{×} and on M. For the case of two connected closed oriented submanifolds A and B of complimentary dimensions in M, the usual intersection number A • B agrees with f • g (up to sign) where f and g are the inclusion maps. That is because the usual intersection number ends up being given by the cup product of the Poincare duals of the fundamental classes [A] and [B] in H_{∗}(M; ℤ). Using the concept of the Umkehr map, this is just f^{!}(1_{A}) ∪ g^{!}(1_{B}) which the following lemma implies is equal to ±(f • g)[M]. REMARK: The reader may ask why the multiple functional intersection number f_{1} • f_{2} • ⋯ • f_{k} is defined using Umkehr maps instead of the obvious definition as the cup product of the Poincare duals of the homology classes f_{i∗}[A_{i}] capped with [M]? As one can see from Lemma 3, the obvious definition differs from our definition by a complicated sign. The reason we need Umkehr maps is to construct the transfer in Theorem 6. Note that the Key Lemma cannot be stated without Umkehr maps. In general, if one can express an invariant in terms of Umkehr maps, then there is a good chance transfers can be defined. This is especially true since the Key Lemma holds. The sign difference in Lemma 3 is not important; it is just there.

Lemma 2. Δ^{∗}(f × g)^{!}(1_{A} × 1_{B}) = (−1)^{nq}f^{!}(1_{A}) ∪ g^{!}(1_{B}) where q = n − dim B and n = dim M.

Proof:

Δ^{∗}(f × g)^{!}(1_{A} × 1_{B})
= Δ^{∗}D_{M×M}^{−1}(f_{∗}([A]) × g_{∗}([B]))
= Δ^{∗}D_{M×M}^{−1}((f^{!}(1_{A}) ∩ [M]) × (g^{!}(1_{B}) ∩ [M])))
= (−1)^{nq}Δ^{∗}D_{M×M}^{−1}((f^{!}(1_{A}) × g^{!}(1_{B})) ∩ ([M] × [M]))
= (−1)^{nq}Δ^{∗}(f^{!}(1) × g^{!}(1)) = (−1)^{nq}f^{!}(1) ∪ g^{!}(1).

In fact, we have in general

Lemma 3. Δ^{∗}F^{!}(1) = (−1)^{a}f_{i}^{!}(1_{A_{1}}) ∪ ⋯ ∪ f_{k}^{!}(1_{A_{k}}) where

a = n[(k − 1)q_{k} + (k − 2)q_{k−1} + ⋯ + q_{2}] and q_{i} = n − dim A_{i}, the codimension of A_{i}.

Proof: The same as Lemma 2, only the notation is much more involved. Counting the sign change is the only complication.

Next we extend the definition of multiple intersection number to the case where M is a connected oriented manifold with boundary ∂M and the f_{i} : A_{i} → M are maps of pairs (A_{i}, ∂A_{i}) ⟶^{f_{i}} (M, ∂M). We assume that at least one of the A_{i} has an empty boundary. Also the A_{i} are all oriented and the sum of the codimension (n − dim A_{i}) is equal to n as before. We let DM denote the double of M. It is a closed oriented manifold. We let Df_{i} : DA_{i} → DM be the double of the map f_{i} : (A_{i}, ∂A_{i}) → (M, ∂M) in the case where A_{i} has a boundary, and if ∂A_{i} = ϕ we let Df_{i} : A_{i} ⟶^{f_{i}} M ⟶^{j} DM be the composition of f_{i} with the standard inclusion j of M into its double. DEFINITION 4. We define the multiple intersection number f_{1} • f_{2} • ⋯ • f_{k} to be the intersection number Df_{1} • Df_{2} • ⋯ • Df_{k} . REMARKS: a) We required at least one of the A_{i} to have an empty boundary. We can define the intersection number as above in the case when all the A_{i} have non-empty boundaries, but in this case it will always be zero.

b) The intersection number is obviously preserved under homotopy of the f_{i}.

c) If f_{1} • ⋯ • f_{k} ≠ 0, then the intersection of the images of the f_{i} is not empty, that

is ∩^{k}_{i=1}f_{i}(A_{i}) ≠ ϕ.

d) A geometric interpretation of multiple intersection number for the case when the A_{i} are submanifolds of M should go as follows: choose orientations of the normal bundles ν_{i} consistent with the choices of orientation of M and A_{i}. Assuming the A_{i} are mutually transversal at the points x_{j} ∈ ∩^{k}_{i=1}A_{i}, assign to the point x_{j} the number +1 if the orientations at x_{j} combine to give the orientation of M at x_{j}, and assign x_{j} the number −1 otherwise. The sum of these local multiple intersection numbers is the global intersection number.

e) Let ν^{k} be a k bundle over the closed oriented manifold M^{n}. If n = kr, the selfintersection number of the zero section s is defined to be s • ⋯ • s . Let χ(ν) denote the Euler class of a bundle ν. Then

χ(ν ⊕ ⋯ ⊕ ν) = [χ(ν)]^{r} = ±(s • ⋯ • s)[\overline{M}] ∈ H^{n}(M; ℤ)

An application of the idea of intersection number can be used in a dual situation to calculate the degree of a map. Suppose that M is a closed smooth oriented connected manifold and suppose that we have a family of k smooth maps f_{i} : M → A_{i} where the A_{i} are connected smooth closed oriented manifolds the sum of whose dimensions equals the dimension of M. The relevant invariant, a “co-intersection number”, is the degree of the composition

G : M ⟶^{Δ} M × ⋯ × M ⟶^{f_{1}×···×f_{k}} A_{1} × ⋯ × A_{k}.

Corollary 5. With the notation of the preceding paragraph, ± deg G = i_{1} • ⋯ • i_{k} where i_{j} : f_{j}^{−1}(a_{j}) → M and a_{j} is a regular value of f_{j} in A_{j}. Thus the degree of G is the intersection number of the fibres of the f_{i}.

Proof: For any smooth map f : M → N between closed oriented manifolds and fibre F = f^{−1}(b) where b is a regular point, we have i_{∗}[F] = f^{∗}([\overline{N}]) ∩ [M] where i : F → M is the inclusion, [G1]. So, in other words, i^{!}(1_{F}) = f^{∗}([\overline{N}]). Now (deg G)[\overline{M}] = G^{∗}([\overline{A}_{1}] × ⋯ × [\overline{A}_{k}]) = f_{1}^{∗}([\overline{A}_{1}]) ∪ ⋯ ∪ f_{k}^{∗}([\overline{A}_{k}]) = i_{1}^{!}(1_{F_{1}}) ∪ ⋯ ∪ i_{k}^{!}(1_{F_{k}}) = ±(i_{1} • ⋯ • i_{k})[\overline{M}].

3. Transfers and intersection numbers Consider the following commutative diagram

F ⟶^{f} F_{1}
i↓  ^{i_{1}}↓
E ⟶^{\overline{f}} E_{1}
p↓  ^{p_{1}}↓
B ⟶_{1_{B}} B

Here p and p_{1} are fibre bundle projections and i and i_{1} are inclusion of a fibre. The map \overline{f} is a fibre preserving map which restricts to f on the fibre. In addition every space is a closed oriented manifold

Key Lemma. f^{!}i^{∗} = (−1)^{B(F_{1}−F)}i_{1}^{∗}\overline{f}^{!} where (−1)^{X} means (−1)^{dim X}.

Proof: Let U be a small ball in B centered on the base point ∗ where p^{−1}(∗) = F. Then V = p^{−1}(U) and V_{1} = p_{1}^{−1}(U) are open sets of E and E_{1} respectively.

The following diagram commutes except for the top rectangle. The undecorated arrows are induced by inclusions.

H^{∗}(F)  ^{f^{!}}  H^{∗}(F)
↑  ↑
H^{∗}(\overline{U} × F) ⟶_{≅}^{∩[\overline{U},U̇]×[F]} H_{∗}((\overline{U}, U̇) × F) ⟶^{(1×f)_{∗}} H_{∗}((\overline{U}, U̇) × F_{1}) ⟵_{≅}^{∩[\overline{U},U̇]×[F_{1}]} H^{∗}(\overline{U} × F_{1})
↑_{i^{∗}}  ≅↓^{ex}  ≅↓^{ex}  ↑^{i_{1}^{∗}}
H^{∗}(E) ⟶^{∩[E,E−V]} H_{∗}(E, E − V) ⟶^{\overline{f}_{∗}} H_{∗}(E_{1}, E_{1} − V_{1}) ⟵^{∩[E_{1},E−V_{1}]} H^{∗}(E_{1})
≅↑^{1_{E}}  ↑  ↑  ≅↑^{1_{E_{1}}}
\overline{f}^{!} : H^{∗}(E)  ⟶_{≅}^{∩[E]}  H_{∗}(E)  ⟶^{\overline{f}_{∗}}  H_{∗}(E_{1})  ⟵_{≅}^{∩[E_{1}]}  H^{∗}(E_{1})

Now the top rectangle commutes up to (−1)^{B(F_{1}−F)}, since the second row gives the following

1 × x ↦ (1 × x) ∩ [\overline{U}, U̇] × [F] = (−1)^{xB}[\overline{U}, U̇] × (x ∩ [F])
↦ (−1)^{xB}[\overline{U}, U̇] × f_{∗}(x ∩ [F]) = (−1)^{xB}[\overline{U}, U̇] × (f^{!}(x) ∩ [F_{1}])
= (−1)^{xB+(x+B(F_{1}−F))B}(1 × f^{!}(x)) ∩ ([\overline{U}, U̇] × [F_{1}])
↦ (−1)^{B(F_{1}−F)}1 × f^{!}(x)

This proves the lemma.

Suppose we have the following k fibre bundle squares.

\begin{matrix}A_{i} & ⟶^{f_{i}} & M \\ ↓^{i_{i}} & & ↓^{i} \\ E_{i} & \underrightarrow{f̃_{i}} & E \\ ↓^{p_{i}} & & ↓^{p} \\ B & ⟶_{1_{B}} & B \end{matrix} i = 1, …, k

where each space is a closed oriented manifold, M and B are connected and the codimension of the A_{i} sum to the dimension of M. More generally we may assume that M and B and hence E have non-empty boundaries and that some of the E_{i} have non-empty boundaries and in those cases f̃_{i} : (E_{i}, ∂E_{i}) → (E, ∂E). Of course all spaces are compact.

Theorem 6. There is a transfer associated to the fibre bundle p with trace (f_{1}•⋯•f_{k}). That is, there is a homomorphism τ_{∗} : H_{∗}(B, G) → H_{∗}(E; G) so that p_{∗}∘τ_{∗} is multiplication by (f_{1} • ⋯ • f_{k}). Similarly there is a homomorphism τ^{∗} : H^{∗}(E, G) → H^{∗}(B, G) so that τ^{∗} ∘ p^{∗} is multiplication by (f_{1} • ⋯ • f_{k}).

Proof: Let E_{1} ∘ ⋯ ∘ E_{k} denote the multiple fibre product of the fibre bundles p_{i} : E_{i} → B. That is,

E_{1} ∘ ⋯ ∘ E_{k} = {(e_{1}, …, e_{k}) ∈ E_{1} × ⋯ × E_{k}|p_{1}(e_{1}) = ⋯ = p_{k}(e_{k})}

Now the map \overline{p} : E_{1} ∘ ⋯ ∘ E_{k} → B defined by p(e_{1}, …, e_{k}) = p(e_{1}) ∈ B is a fibre bundle projection with fibre A_{1} × ⋯ × A_{k}. Also we have the multiple fibre product E ∘ ⋯ ∘ E → B with fibre M × ⋯ × M. Define F̃ : E_{1} ∘ ⋯ ∘ E_{k} → E ∘ ⋯ ∘ E by F̃(e_{1}, …, e_{k}) = (f̃_{1}(e_{1}), …, f̃_{k}(e_{k})). The diagonal Δ̃ : E → E ∘ ⋯ ∘ E given by Δ̃(e) = (e, …, e). All this fits into a commutative diagram

A_{1} × ⋯ × A_{k} ⟶^{F} M × ⋯ × M ⟵^{Δ} M
↓^{k} ↓^{j} ↓^{i}
E_{1} ∘ ⋯ ∘ E_{k} \underrightarrow{F̃} E ∘ ⋯ ∘ E \underleftarrow{Δ̃} E
↓  ↓  ↓^{p}
B  ⟶^{1_{B}}  B  ⟵^{1_{B}} B

Now we can define the class Δ̃^{∗}F̃^{!}(1) ∈ H^{n}(E; G) in the case of closed E. Then i^{∗}(Δ̃^{∗}F̃^{!}(1)) = Δ^{∗}j^{∗}F̃^{!}(1) = ±Δ^{∗}F^{!}k^{∗}(1) = ±Δ^{∗}F^{!}(1) = ±(f_{1} • ⋯ • f_{k})[\overline{M}]. The equalities are true by commutativity and the key lemma and the last equality is the definition. The transfer maps are defined as usual: τ^{∗}(x) = p^{!}(x ∪ Δ̃^{∗}F̃^{!}(1)) and τ_{∗}(x) = p_{!}(Δ̃^{∗}F̃^{!}(1) ∩ x).

In the case that M has a non-empty boundary, and B is closed, taking the doubles Df̃_{i} : DE_{i} → DE and Df̃_{j} : E_{j_{i}} → E → DE gives a transfer for DM → DE → B. Since M → E → B is a fibrewise retraction DM → DE → B, the transfer defined for the latter restricts to the former with the same trace, f_{1} • ⋯ • f_{k} . In the remaining case where B has non-empty boundary, we consider the pullback of M → E ⟶^{p} B by the retraction map r : DB → B. Suppose this pullback is denoted M → DE ⟶^{Dp} DB. Then Dp admits a transfer with associated number f_{1} • ⋯ • f_{k} , and so p, which is a fibrewise retract of Dp, admits a transfer with associated number f_{1} • ⋯ • f_{k}.

4. The trace of an action Suppose G is a group acting on spaces A_{i} and M and suppose f_{i} : A_{i} → M are

homotopy G-maps. Let us say that i : W → V is a homotopy G-map if there is a fibre bundle square

W ⟶_{i} V
↓ ↓
E ⟶_{f} V_{G}
↓  ↓
B_{G} ⟶^{1} B_{G}

which extends i to a map f over 1_{B_{G}}. Here V_{G} is the Borel construction E_{G}×_{G}V

In [G2] and [O] we introduced the integer tr(p), read the trace the fibration p. Let deg(p) be the smallest positive integer N so that there is a homomorphism τ so that p_{∗} ∘ τ = multiplication by N. If no such integer exists then set deg(p) equal to zero. In fact, deg(p) is the greatest common divisor of these numbers. Next the trace of p, denoted tr(p), is the least upper bound of all the positive deg(p_{f}) where the p_{f} are the pullbacks of p by arbitrary maps f : X → B. If the least upper bound does not exist set tr(p) equal to zero. In fact the tr(p) is the least common multiple of these numbers. Then we define the trace of an action, denoted tr(G, M), by tr(G, M) = tr(p_{G}) where p_{G} : E_{G} ×_{G} M → B_{G} is the Borel construction of the universal G-principal bundle G → E_{G} → B_{G} with M.

Theorem 7. Suppose G is a group so that the classifying space B_{G} has finite type and suppose that M is an oriented compact G-manifold and f_{i} : A_{i} → M are homotopy Gmaps from oriented compact manifolds A_{i} which preserve boundaries. Then tr(G, M)|(f_{1} • ⋯ • f_{n}).

Proof: Let B be a regular neighborhood of the (Dim M + 2)- skeleton embedded in some Euclidean space. So B is an oriented manifold with boundary. Let M → E ⟶^{p} B be the pullback of M → M_{G} → B_{G} onto B where the map B → B_{G}, which induces the pullback, is homotopic to the inclusion of the (Dim M + 2)-skeleton. By Proposition (6.6) of [G2] we have tr(G, M) = deg(p). By theorem 6 and the definition of deg p , we see that f_{1} • ⋯ • f_{k}| deg(p) = tr(p) = tr(G, M).

Theorem 7 allows us to add multiple intersection numbers to the list of topological and group theoretical properties divided by the trace of an action. Most of this list so far may be found in Theorem (1.1) of [G2]. REMARK: By Theorem (1.5) of [G2], if G acts orientably on a closed manifold M^{n} the trace tr(G, M) is the order of quotient group of the top dimensional integer cohomology group divided by the image of fibre inclusion of the cohomology of H^{n}(EG×_{G}M), if this is finite, and zero otherwise. This is just what Browder’s exponent is in [B]. In this special case the trace appears more tractible than in the general definition. But the general definition works for any action and therefore appears in theorems which are not restricted to closed manifolds and orientation preserving actions. The list of properties of the general trace is quite substantial and they work together well, so that despite its strange looking definition it is actually quite easy to use the concept. This is exemplified by the arguments in the next section which do not use the exponent definition above and so result in facts which are true in more generality and which yield even more additions to the list of properties of the trace.

5. Traces and subgroups Let G be a group, X a G-space and let H be a subgroup of G. By ([G2], Theorem

6.2a), tr(H, X) divides tr(G, X). We will now show that the quotient divides tr(G, G/H) where G/H is the G space consisting of a single orbit of isotropy type H.

Proposition 8. Let H be a subgroup of G, and let X be a G-space. Then

tr(H, X)| tr(G, X)| tr(G, G/H) tr(H, X).

Proof: We already know the left division by [G2], Theorem 6.2a. For the second, given any map f : Y → B_{G} , let Ỹ be the pullback fitting into the diagram:

G/H  G/H
↓  ↓
Ỹ \underrightarrow{f̃} (G/H)_{G} = B_{H}
↓^{q} ↓
Y ⟶^{f}  B_{G}.

From the definition of tr(G, G/H) as the l.c.m. of the degrees of the projection maps over all pullbacks of the Borel construction, it follows that deg(Ỹ → Y) divides tr(G, G/H). Hence there is a transfer τ : H_{∗}(Y; ℤ) → H_{∗}(Ỹ; ℤ) so that q_{∗} ∘ τ = tr(G, G/H).

From the diagram

X  X
↓ ↓
G/H ⟶ X_{H} ⟶ X_{G}
∥ ↓ ↓
G/H ⟶ B_{H} ⟶ B_{G}

we get the pullback diagram (via f and f̃)

X  X
↓ ↓
G/H ⟶ f̃^{∗}(X_{H}) \underrightarrow{q̃} f^{∗}(X_{G})
∥ ↓p̃ ↓^{p}
G/H ⟶ Ỹ ⟶^{q} Y

If σ is a transfer realizing deg(p̃) then ρ = q̃_{∗} ∘ σ ∘ τ satisfies p_{∗}ρ = p_{∗} ∘ q̃_{∗} ∘ σ ∘ τ = q_{∗} ∘ p̃_{∗} ∘ σ ∘ τ = tr(G, G/H) deg(p̃). Hence deg(p)| tr(G, G/H) deg(p̃). Now taking the lowest common multiple over all f : Y → B_{G}, and noting that tr(H, X) is defined as the l.c.m. of degrees over a set containing all f̃ : Y → B_{H}, we get tr(G, X) divides tr(G, G/H) tr(H, X). □

The G space G/H consisting of a single orbit is perhaps the simplest case we should consider, also important in view of the proposition above. In general, with some finiteness assumptions, we have tr(G, G/H) dividing the Euler characteristic χ(G/H) ([G2], Proposition 6.7b). When G/H is finite we have tr(G, G/H) equalling χ(G/H) = [G : H], but tr(G, G/H) does not equal χ(G/H) in general.

Lemma 9. Let H be a subgroup of G such that G/H is finite and discrete. Then tr(G, G/H) = [G : H].

Proof: We have tr(G, G/H)|χ(G/H) = [G : H]. Hence it will suffice to find a pullback of the Borel construction G/H → B_{H} = (G/H)_{G} → B_{G} whose projection has degree [G : H]. Let σ_{1}, …, σ_{n} be loops representing classes in π_{1}(B_{G}) mapping onto π_{0}(G/H) = G/H in the homotopy exact sequence

1 → π_{1}(B_{H}) → π_{1}(BG) → G/H → ∗.

Let r : M → ∨^{i}S^{1} be a retraction of a closed, connected, oriented manifold on a bouquet of n circles. Let f be given by

f : M ⟶^{r} ∨^{i}S^{1} ⟶^{∨_{σ}} B_{G}.

Then f^{∗}(B_{H}) is a closed, connected, oriented manifold, covering M with fiber G/H. Thus the projection ρ : f^{∗}(B_{H}) → M has degree exactly [G : H]. □

Combining Propositions 8 and 9 we obtain the following.

Corollary 10. If G/H is finite then

tr(H, X)| tr(G, X)|[G : H] tr(H, X)

Theorem 11. Let G be a finite group and let X be a G space. Then

tr(G, X) = Π tr(G_{p}, X)

where G_{p} is a Sylow p-subgroup of G and the product is over all primes dividing the order of G.

Proof: Since tr(G_{p}, X)| |G_{p}| and |G_{p}| is relatively prime to [G : G_{p}] we see that the divisions

tr(G_{p}, X)| tr(G, X)|[G : G_{p}] tr(G_{p}, X)

implies that the p-primary factor of tr(G, X) is given by tr(G_{p}, X).

In the proof above, the relevant fact about a collection G_{p} was that their orders are relatively prime to each other and the order of G is the products of the orders of the G_{p}. Hence if {H_{i}} is a collection of subgroups satisfying this condition, we have

tr(G, X) = ∏_{i} tr(H_{i}, X)

Now we consider the consequences of Proposition 8 for a compact Lie group G. We will denote the connected component of the identity by G_{0}, a maximal torus of G_{0} by T, the normalizer of T in G_{0} by N(T), and the Weyl group N(T)/T by W.

Lemma 12. Let X be a G-space (G compact Lie). Then tr(G, X) divides [G : G_{0}] tr(N(T), X). In particular if G is connected then tr(G, X) equals tr(N(T), X).

Proof: Note that tr(G_{0}, G_{0}/N(T))|χ(G_{0}/N(T)) by ([G2], 6.7b). But it is known that χ(G_{0}/N(T)) = 1, hence tr(G_{0}/N(T)) = 1 thus Proposition 8 tells us that

tr(N(T), X)| tr(G_{0}, X)| tr(N(T), X)

Hence tr(N(T), X) = tr(G_{0}, X). Now Corollary 10 gives us tr(G, X)|[G : G_{0}] tr(G_{0}, X) = [G : G_{0}] tr(N(T), X).

Now Lemma 12 gives us tr(N(T), X)||W | tr(T, X). When X is a compact manifold M, we know from ([G2], 6.11) that tr(T, M) = 1, if the action has a stationary point; or tr(T, M) = 0, if the action does not have a stationary point. Thus we obtain

Corollary 13. Let G be a compact Lie group acting on a compact manifold M. Then tr(G, M) is either zero, or it divides [G, G_{0}]|W | where W is the Weyl group of G_{0}.

Theorem 14. Let G be a compact Lie Group acting on a compact manifold M. Let W_{p} be the Sylow p-subgroups of the Weyl group W, and let Ŵ_{p} is the pre-image in N(T) of W_{p} in W = N(T)/T. Then

tr(G, M) = ∏_{p} tr(Ŵ_{p}, M)

where the product is taken over all the p dividing |W |.

Proof: We may assume that tr(Ŵ_{p}, M) ≠ 0 because otherwise tr(T, M) = 0, which implies tr(G, M) = 0; then both sides of the equation would be zero, proving the theorem. So since tr(Ŵ_{p}, M) ≠ 0, we have tr(T, M) = 1, so from Proposition 8, tr(Ŵ_{p}, M)| |Ŵ_{p}|. Now tr(Ŵ_{p}, M)| tr(N(T), M)|[W : W_{p}] tr(Ŵ_{p}, M) by ([G2], 6.2.b) for the first division and by Corollary 10 for the second division. Hence tr(Ŵ_{p}, M) is a common divisor of |W_{p}| and tr(G, M) = tr(N(T), M), and in fact is the greatest common divisor since tr(G, M)/ tr(Ŵ_{p}, M) is not a multiple of p. Since tr(G, M) divides |W | = ∏_{p}W_{p}, the theorem follows. REMARKS: a) Theorem 14 can be generalized for a family {H_{i}} of subgroups of W such that Π|H_{i}| = |W | and the |H_{i}| are pairwise relatively prime.

b) The following example shows that tr(G, G/H) does not always equal χ(G/H), in fact it shows that tr(G, G/T) ≠ χ(G/T) = |W |.

(c) In fact tr(S^{3}, M) is always zero or 1 as M is a compact manifold. This follows from the following example where tr(S^{3}, S^{3}/S^{1}) = 1. Hence tr(S^{3}, M) = tr(S^{1}, M) EXAMPLE. Let G = S^{3} and H = S^{1}. Then tr(S^{3}, S^{3}/S^{1}) = 1 but χ(S^{3}/S^{1}) = 2.

Proof: This can be seen by considering the fibration S^{2} ⟶^{i} B_{S^{1}} ⟶^{p} B_{S^{3}}. This is a fibration which arises from the action of S^{3} on S^{2} = S^{3}/S^{1} by the Borel construction. Now H_{i}(B_{S^{3}}) ≅ 0 for 0 < i < 4, so i^{∗} is an isomorphism i^{∗} : H^{2}(B_{S^{2}}) → H^{2}(S^{2}). Thus the generator of H^{2}(S^{2}) is in the image of i^{∗}. Hence by theorem 1.5 of [G2] we see that tr(S^{3}, S^{3}/S^{1}) = 1.

6. Applications of the trace theorems An application of Theorem 11 gives an extension of a result of Browder and Katz [B-K]

to the nonfree case. Recall that a nonsingular complex projective variety V of dimension k is a closed, complex analytic submanifold of a complex projective space P^{n}, given as the zero set of a set of homogeneous polynomials (in n + 1 variables). The Chern class of the canonical line bundle c(ξ) generates the cohomology of P^{n}. The hyperplane section α in H^{2}(V; ℤ) is the image of c(ξ) under the map induced by inclusion i^{∗} : H^{2}(P^{n}; ℤ) → H^{2}(V; ℤ). We also have α^{k} = deg(V) [\overline{V}] where [\overline{V}] is the fundamental class in cohomology. The degree of the variety V, deg(V) (not to be confused with the degree of a map), depends on the particular embedding of V in the projective space, since the hyperplane section α depends on the embedding.

Proposition 15. Let G be a finite group acting on a nonsingular complex projective variety V. Assume H^{1}(V) = 0 and the hyperplane section α in H^{2}(V) is left invariant by G. Then

tr(G, V)^{2}| |G| (deg V)^{2}.

If all Sylow subgroups of G are cyclic, then tr(G, V) divides deg V.

Proof: The p-primary part of tr(G, V) is exactly tr(G_{p}, V) for the Sylow p-subgroup G_{p}. Hence we may assume that G is a p-group. The action of G is orientation preserving because α and thus α^{k} = (deg V)[\overline{V}] is invariant (k = dimension of V). In the cohomology Serre spectral sequence of the Borel construction V → V_{G} → B_{G} we have E_{2}^{0,2} ≅ H^{2}(V)^{G} and E_{2}^{∗} ≅ H^{∗}(G). These survive to E_{3} since H^{1}(V) is zero. Let dαϵH^{3}(G) be the image of α under

d : E_{3}^{0,2} ≅ H^{2}(V)^{G} → H^{3}(G) ≅ E_{3}^{3,0}.

Now there is a subgroup K of G such that [G : K]^{2} divides |G| and d α maps to 0 under the restriction map H^{3}(G) → H^{3}(K) ([B-K], 2.1). Restricting the action to K, from the functoriality of the Serre spectral sequence we deduce that α survives to E_{∞} (for V → V_{K} → B_{K}). Hence α and α^{k} = deg V [\overline{V}] are in the image of H^{2k}(V_{K}) and thus tr(K, V) divides deg V. Then tr(G, V)|[G : K] tr(K, V)|[G : K] deg V, and squaring the first and last terms, tr(G, V)^{2}|[G : K]^{2}(deg V)^{2}| |G|(deg V)^{2}. When G is cyclic, H^{3}(G) is zero so dα is zero. We don’t have to restrict to K, and we conclude tr(G, V)| deg V. □

When G acts freely, tr(G, V) = |G| and we have |G| divides (deg V)^{2} as in [B-K]. Now the degree of V is just the self intersection number of W with itself where W =

V ∩ P^{k−1}. So if G is acting on V such that W is left invariant, then by Theorem 7 we have tr(G, V)| deg(V). In fact we can relax the condition that the inclusion i : W → V is a G-map. Recall that i : W → V is a \underline{homotopy G-map} if there is a fibre bundle square

W ⟶_{i} V
↓ ↓
E ⟶_{f} V_{G}
↓  ↓
B_{G} ⟶^{1} B_{G}

which extends i to a map f over 1_{B_{G}}. Then we have shown the following.

Proposition 16. Let G be compact Lie group acting on a non-singular projective variety V so that the hyperplane section inclusion i : W → V is a homotopy G-map. Then tr(G, V)| deg(V).

Browder and Katz have an example ([B-K]) where G acts freely on V and satisfies the hypothesis of Proposition 15, so |G| | (deg V)^{2}. But |G| does not divide deg(V). Hence we can conclude that in that example the inclusion of the hyperplane section is not a homotopy G-map. EXAMPLE: Suppose that M = S^{2n} ×⋯×S^{2n} is a product of k spheres each of dimension 2n. Suppose G acts on M leaving a 2n(k − 1)-dimensional submanifold W invariant. The Poincare dual of W is a class α ∈ H^{2n}(M) and so α = a_{1}α_{1} + ⋯ + a_{k}α_{k} where the a_{i} are integers and α_{i} generate H^{2n}(S^{2n}) of the i-th sphere in the product M. Now α_{i}^{2} = 0 for all i and α_{i}α_{j} = α_{j}α_{i} since the α_{i} are even dimensional. Hence α^{k} = k!(a_{1} … a_{k})α_{1} ∪⋯∪α_{n} = k!(a_{1} … a_{k})[\overline{M}]. Thus tr(G, M) divides k!(a_{1} … a_{k}). More generally, if we have k G– invariant submanifolds W_{i} (repetitions allowed), each having co-dimension 2n, then their Poincare duals are of the form Σa_{ij}α_{j}. Their intersection number is the permanent of (a_{ij}), and it is divisible by tr(G, M).

As a particular example, suppose G is an elementary abelian p-group. By ([G2], 7.4) we have tr(G, M) = (the number of points in the smallest orbit). Thus (the number of points in the smallest orbit) divides k!(a_{1} … a_{k}).

In the odd dimensional situation, where M = S^{2n+1} × ⋯ × S^{2n+1} is a product of k odd dimensional spheres each of dimension 2n + 1, suppose we have a set {W_{i}} of k Ginvariant submanifolds of co–dimension 2n + 1. Then the Poincare dual of each manifold is given by β_{i} = Σa_{ij}α_{j} ∈ H^{2n+1}(M) where the a_{ij} are integers and each α_{j} generates H^{2n+1}(S^{2n+1}). Now α_{j} ∪ α_{k} = −α_{k} ∪ α_{j}. Hence W_{1} • ⋯ • W_{k} = det(a_{ij}). So if G is an elementary abelian p-group we see that (the number of points in the smallest orbit) divides det(a_{ij}). In addition, suppose that k = 2l and W is an invariant oriented closed manifold of codimension 2(2n + 1). Suppose the Poincare dual of W is given by β = Σa_{ij}α_{i}α_{j}. Now β^{l} = l! Pf(a_{ij})α_{1} … α_{k}. Thus tr(G, M) divides l! Pf(a_{ij}). Here Pf(a_{ij}) is the Pfaffian.

Theorem 7.4 of [G2] is greatly generalized by theorem 1.1 of [B]. In particular, Browder’s result implies that the trace of an action of an elementary abelian p–group on a G–CW complex is equal to the number of points in the smallest orbit.

BIBLIOGRAPHY

[B] W. Browder, Actions of elementary abelian p–groups, Topology 27,(1988) pp. 459-472. [B-K] W. Browder and N. Katz, Free actions of finite groups on varieties I, in Current Trends in Algebraic

Topology, Vol. 2, part 2 (1982), pp. 23-33. [G1]D.H. Gottlieb, Partial transfers, geometric applications of homotopy theory I (Proc. Evanston Conf.

1977), Lecture Notes in Math., Vol. 657, Springer-Verlag, Berlin and New York (1977), 255-266. [G2]D.H. Gottlieb, The trace of an action and the degree of a map, Transactions of the American Math.

Soc., 293 (1986), 381-410.

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

Purdue University University of Oklahoma