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Isometric Group Actions and the Cohomology of Flat Fiber Bundles

Using methods originating in the theory of intersection spaces, specifically a de Rham type description of the real cohomology of these spaces by a complex of global differential forms, we show that the Leray-Serre spectral sequence with real coefficients of a flat fiber bundle of smooth manifolds collapses if the fiber is Riemannian and the structure group acts isometrically. The proof is largely topological and does not need a metric on the base or total space. We use this result to show further that if the fundamental group of a smooth aspherical manifold acts isometrically on a Riemannian manifold, then the equivariant real cohomology of the Riemannian manifold can be computed as a direct sum over the cohomology of the group with coefficients in the (generally twisted) cohomology modules of the manifold. Our results have consequences for the Euler class of flat sphere bundles. Several examples are discussed in detail, for instance an action of a free abelian group on a flag manifold.

A fiber bundle with given structure group is flat, if the transition functions into the structure group are locally constant. We show that the method of intersection spaces introduced in [Ban10], specifically the de Rham description of intersection space cohomology given in [Ban11], implies the following result, by a concise and topological proof: holds, where the H q (F; R) are local coefficient systems on B induced by π, whose groups are the real cohomology groups of the fiber.
By a result of R. Palais ([Pal61]), condition (1) implies that F can be endowed with a Riemannian metric such that (2) holds. The isometry group of an m-dimensional, compact, Riemannian manifold is a compact Lie group of dimension at most 1 2 m(m + 1). Thus (2) implies (1) by taking H to be the isometry group of F, and the two conditions are essentially equivalent. In Section 6, we provide an example of a flat, smooth fiber bundle whose structure group does not act isometrically for any Riemann metric on the fiber, and whose spectral sequence does not collapse at E 2 . Hence, conditions (1), (2) in the theorem cannot be deleted without substitution.
Let us indicate two immediate consequences of the theorem. Let π : E → B be an oriented, flat sphere bundle with structure group SO(n) over a closed, smooth manifold B. The transgression d n : E 0,n−1 n = H 0 (B; H n−1 (S n−1 ; R)) = H n−1 (S n−1 ; R) −→ E n,0 n = H n (B; H 0 (S n−1 ; R)) = H n (B; R) sends a certain element σ ∈ E 0,n−1 Corollary. For every n > 1, there is a flat manifold M 2n whose linear tangent sphere bundle is not flat.
For the tangent bundles of these manifolds M 2n , the structure group GL + (2n, R) can be reduced to GL + (2n, R) with the discrete topology and can also be reduced to SO(2n) (by using a metric), but there are no further reductions to SO(2n) with the discrete topology.
Our theorem may be applied to the equivariant cohomology H • G (−; R) of certain discrete groups G: Theorem. (See Theorem 7.1.) Let F be an oriented, closed, smooth manifold and G a discrete group, whose Eilenberg-MacLane space K(G, 1) may be taken to be a closed, smooth manifold. If for a smooth action of G on F, (1) the action factors through a proper, smooth Lie group action, or (2) F is Riemannian and G acts isometrically on F, then there is a decomposition where the H q (F; R) are G-modules determined by the action.
The assumption on G can be paraphrased as requiring G to be the fundamental group of a closed, (smooth) aspherical manifold. Examples of such groups include finitely generated free abelian groups, the fundamental groups of closed manifolds with non-positive sectional curvature, the fundamental groups of surfaces other than the real projective plane, infinite fundamental groups of irreducible, closed, orientable 3-manifolds, torsionfree discrete subgroups of almost connected Lie groups, and certain groups arising from Gromov's hyperbolization technique. Only torsionfree G can satisfy the hypothesis of the above theorem. Note that in (1) we do not assume that the image of G is closed in the intermediary Lie group, nor that one can identify G with a subgroup of the intermediary group, and in (2) we do not assume that the image of G is closed in the isometry group of F, nor that one can identify G with a subgroup of the isometries. For instance, the integers G = Z with K(Z, 1) = S 1 , the circle, act isometrically and freely (and ergodically) on the unit circle by powers of a rotation by an angle which is an irrational multiple of 2π. Theorem 7.1 thus emphasizes the discrete dynamical systems viewpoint. This example also satisfies (1), since the powers of the irrational rotation are a subgroup (which is not closed) of the compact Lie group S 1 which acts on itself by (e.g. left) multiplication. The actions to which the theorem applies need not be proper, nor need our G-spaces be G-CW complexes, but in many geometric situations, nonproper actions factor through proper actions in a natural way. (The above Z-action on the circle is not proper, since e.g. the orbit space is not Hausdorff and orbits are not closed in S 1 .) Section 7 contains a number of corollaries to, applications of, and examples illustrating Theorem 7.1. For instance, if an integral Heisenberg group H n acts isometrically on an oriented, closed, connected, Riemannian manifold F, then rk H k H n (F) ≥ 2, k = 1, 2, and H 3 H n (F; R) does not vanish (Corollary 7.8).
In Section 8, we illustrate the use of our results by calculating explicitly an equivariant cohomology group of a certain action of a free abelian group on the flag manifold F 8 = U(8)/T 8 of real dimension 56. The outcome is verified by an alternative, logically independent, computation of the same group, relying on a recursive scheme introduced in the appendix (Section 9). This recursive scheme is only available for finitely generated free abelian groups.
The proof of Theorem 5.1 relies on the complex of multiplicatively structured forms constructed in [Ban11], and the fact that fiberwise truncation in both directions, yielding a subcomplex in both cases (and not a quotient complex in one case), can be performed on the multiplicatively structured forms. Such truncations, carried out homotopy theoretically on the space level (generally a much harder problem), are also required to build the intersection space IpX of a stratified pseudomanifold X. The only analytic tool we need to prove the theorem is the classical Hodge decomposition -hence the assumption that the fiber must be closed, oriented and Riemannian. Otherwise, our argument is purely topological in nature, usingČech complexes. No connections, tensor fields, etc. are used on the base or total space; in particular we need not assume that B and E are Riemannian. The combinatorial nature of our proof may lend itself to an extension of our theorem to nonsmoothable PL manifolds B and E and flat PL bundles π : E → B. In this situation, smooth forms have to be replaced by Sullivan's complex A • (K) of piecewise C ∞ -forms on a simplicial complex K. We may also recall at this point that there are closed, aspherical PL manifolds which are not homotopy equivalent to closed, smooth manifolds, [DH89].
Our Theorem 5.1 is closely related to results of [Dai91] and [Mül11]. Dai and Müller work with Riemannian E, B and a Riemannian submersion π : E → B. Their metric on F is allowed, to a certain extent, to vary with points in the base. Using Dai's spectral sequence degeneration result from [Dai91], Müller proves that if a flat Riemannian submersion π is locally a warped product, or has totally geodesic fibers, then the spectral sequence of π collapses at E 2 . Let us put this into perspective. A Riemannian submersion whose total space is complete is a locally trivial fiber bundle. The geometry of a Riemannian submersion is largely governed by two tensor fields T and A. Let V and H denote the vertical and horizontal distributions, respectively, that is, at each point x ∈ E there is an orthogonal decomposition V x ⊕ H x = T x E of the tangent space and V x is tangent to the fiber over π(x). Let V and H also denote the projection of a vector onto V x and H x , respectively. With ∇ the Levi-Civita connection of the metric on E, one sets for vector fields V,W on E, T V W = H∇ VV VW + V∇ VV HW and A V W = H∇ HV VW + V∇ HV HW.
If V,W are vertical, then T V W is the second fundamental form of each fiber. The identical vanishing T = 0 is equivalent to each fiber being totally geodesic, that is, geodesics in the fibers are also geodesics for E. This implies that all fibers are isometric and the holonomy group (which agrees with the structure group of π, at least when E is complete) is a subgroup of the isometry group of the fiber. The identical vanishing A = 0 is equivalent to the integrability of H. If H is integrable, then E is locally isometric to B × F with a metric g B + g F,b , where g B is a metric on B and {g F,b } b∈B is a smooth family of metrics on F. In this situation, the horizontal foliation yields a flat (Ehresmann) connection for π. For a flat connection, the holonomy along a path depends only on the homotopy class of the path. Indeed, for path-connected B, flat bundles with structure group G acting effectively on a fiber G are in one-to-one correspondence with homomorphisms π 1 (B) → G. In particular, one may take G to be discrete. A warped product is a Riemannian manifold B × F, whose metric has the form g B + f g F , where g F is a fixed metric on F and f : B → R is a positive function. If the Riemannian submersion π is locally a warped product, then H is integrable. When T ≡ 0 and A ≡ 0, the total space E is locally a product g B + g F , where g F does not depend on points in the base. From [Dai91], Müller isolates a technical condition, called condition "(B)" in [Mül11, Section 2.3], which for a flat Riemannian submersion implies collapse of the spectral sequence at E 2 . He shows that this condition is satisfied for locally warped products as well as for totally geodesic fibers, while Dai shows that collapse at E 2 happens for flat Riemannian submersions satisfying (B).

A COMPLEX OF MULTIPLICATIVELY STRUCTURED FORMS ON FLAT BUNDLES
This section reviews the multiplicatively structured form model introduced in [Ban11]. The proofs of the cited results can be found in that paper. For a smooth manifold M, Ω • (M) denotes the de Rham complex of smooth differential forms on M. Let F be a closed, oriented, Riemannian manifold and π : E → B a flat, smooth fiber bundle over the closed, smooth n-dimensional base manifold B with fiber F and structure group the isometries of F. An open cover of an n-manifold is called good, if all nonempty finite intersections of sets in the cover are diffeomorphic to R n . Every smooth manifold has a good cover and if the manifold is compact, then the cover can be chosen to be finite. Let U = {U α } be a finite good open cover of the base B such that π trivializes with respect to U. Let {φ α } be a system of local trivializations, that is, the φ α are diffeomorphisms such that commutes for every α. Flatness implies that the transition functions The maps g β α : F → F are isometries. If X is a topological space, let π 2 : X × F → F denote the second-factor projection. Let V ⊂ B be a U-small open subset and suppose that V ⊂ U α .
Flatness is crucial for the following basic lemma.
The lemma follows from the transformation law The lemma shows that the property of being multiplicatively structured over V is invariantly defined, independent of the choice of α such that V ⊂ U α . We will use the shorthand notation U α 0 ...α p = U α 0 ∩ · · · ∩U α p for multiple intersections. (Repetitions of indices are allowed.) Since U is a good cover, every U α 0 ...α p is diffeomorphic to R n , n = dim B. A linear subspace, the subspace of multiplicatively structured forms, of Ω q (E) is obtained by setting We shall eventually see that this inclusion is a quasi-isomorphism, that is, induces isomorphisms on cohomology. For any α, set If p is positive, then we set | ω is α 0 -multiplicatively structured }. Lemma 2.2 implies that for any 1 ≤ j ≤ p, In particular, if σ is any permutation of 0, 1, . . . , p, then Ω • MS (U α σ(0) ...α σ(p) ) = Ω • MS (U α 0 ...α p ). The components of an element will be written as . We impose the antisymmetry restriction ξ ...α i ...α j ... = −ξ ...α j ...α i ... upon interchange of two indices. In particular, if α 0 , . . . , α k contains a repetition, then ξ α 0 ...α k = 0. The difference operator and satisfying δ 2 = 0, restricts to a difference operator Since the de Rham differential d commutes with restriction to open subsets, we have dδ = δ d. Thus is a double complex with horizontal differential δ and vertical differential d. The associ- in degree j and differential D = δ + (−1) p d on C p (U; Ω q MS ). We shall refer to the double complex (C • (U; Ω • MS ), δ , d) as the multiplicatively structuredČech-de Rham complex. Lemma 2.3. (Generalized Mayer-Vietoris sequence.) The sequence Let us recall a fundamental fact about double complexes. This fact is applied in showing: Proposition 2.5. The restriction map r : can be used to compute the cohomology of the total space E. The restriction map makes C • (π −1 U; Ω • ) into an augmented double complex. By the generalized Mayer-Vietoris sequence, Proposition 8.5 of [BT82], the rows of this augmented double complex are exact. From Proposition 2.4, we thus deduce: Proposition 2.6. The restriction map r : Regarding R n × F as a trivial fiber bundle over R n with projection π 1 , the multiplica- Let s : R n−1 ֒→ R × R n−1 = R n be the standard inclusion s(u) = (0, u), u ∈ R n−1 . Let q : R n = R × R n−1 → R n−1 be the standard projection q(t, u) = u, so that qs = id R n−1 . Set are chain homotopy inverses of each other and thus induce mutually inverse isomorphisms . An induction on n using Proposition 2.7 shows:

are chain homotopy inverses of each other and thus induce mutually inverse isomorphisms
Using the classical Poincaré lemma, this Proposition readily implies: induces an isomorphism on cohomology (with respect to the de Rham differential d).
is a morphism of double complexes. It induces an isomorphism on vertical (i.e. d-) cohomology by Proposition 2.10. Whenever a morphism of double complexes induces an isomorphism on vertical cohomology, then it also induces an isomorphism on the D-cohomology of the respective simple complexes. Consequently, using Propositions 2.5 and 2.6, one gets: on cohomology.

FIBERWISE TRUNCATION
Before we discuss fiberwise (co)truncation, we must first discuss (co)truncation over a point. Again, we refer to [Ban11] for complete proofs of the facts cited in this section. We shall use the Riemannian metric on F to define truncation τ <k and cotruncation τ ≥k of the complex Ω • (F). The bilinear form where * is the Hodge star, is symmetric and positive definite, thus defines an inner product on Ω • (F). The codifferential is the adjoint of the differential d, (dω, η) = (ω, d * η). The classical Hodge decomposition theorem provides orthogonal splittings where Harm r (F) = ker d ∩ ker d * are the closed and coclosed, i.e. harmonic, forms on F. In particular, Let k be a nonnegative integer.
(2) If f is an isometry, then f induces an automorphism f * of τ ≥k Ω • F.
is given by requiring every γ j to lie in τ <t Ω • (F). This is well-defined by the transformation law (1) together with Lemma 3.4 (1) is given by requiring every γ j to lie in τ ≥t Ω • (F). This is well-defined by the transformation law and Lemma 3.4 (2). (At this point, it is used that the transition functions of the bundle are isometries.) Lemma 3.5. The maps

are chain homotopy inverses of each other and thus induce mutually inverse isomorphisms
As in Section 2, let S 0 : An induction on n using Lemma 3.5 shows: Lemma 3.6. (Poincaré Lemma for truncated forms.) The maps

are chain homotopy inverses of each other and thus induce mutually inverse isomorphisms
An analogous argument, replacing τ <t Ω • (F) by τ ≥t Ω • (F), proves a version for fiberwise cotruncation:

are chain homotopy inverses of each other and thus induce mutually inverse isomorphisms
and aČech differential δ : Let us return to the good cover U on our base space B. We shall define threeČech presheaves on U. Define H q (F) by The structural morphisms associated to inclusions are given by restriction of forms. Since all nonempty objects U in C(U) are diffeomorphic to R n , and the bundle π : E → B trivializes over every such U, the classical Poincaré lemma implies that H q (F) is a locally constant presheaf with group H q (F), the de Rham cohomology of the fiber. Define the According to Proposition 2.10, the inclusion Ω

Thus the inclusion of multiplicatively structured forms induces an isomorphism
The isomorphism induces furthermore an isomorphism Since U = ∅ is diffeomorphic to R n and the bundle π : E → B trivializes over U, the Poincaré Lemma 3.7 for cotruncated forms implies that the restriction S * 0 of a form to the fiber over the origin of U ∼ = R n induces an isomorphism (assuming, without loss of generality, that the origin of U lies in V ) induces a commutative diagram Since it commutes with restriction to smaller open sets, we obtain the following result.

THE SPECTRAL SEQUENCE OF A FLAT, ISOMETRICALLY STRUCTURED BUNDLE
defined in Section 2. The spectral sequence of the fiber bundle π : E → B is the spectral , the (singular) cohomology of B with values in the local coefficient system H q (F).
Theorem 5.1. Let F be a closed, oriented, Riemannian manifold and π : E → B a flat, smooth fiber bundle over the closed, smooth base manifold B with fiber F and structure group the isometries of F. Then the spectral sequence with real coefficients of π : E → B collapses at the E 2 -term.
Proof. Using multiplicatively structured forms, we first build a smaller model K MS of K. The spectral sequences of K MS and K will be shown to be isomorphic (from the E 2 -term on). In Section 2, we introduced the multiplicatively structuredČech -de Rham double In bidegree (p, q) it is given by

The vertical cohomology of K MS is
The core of the argument is the construction of a filtration of K MS by cotruncated double complexes K ≥t ⊂ K MS , where t is an integer. The group in bidegree (p, q) is The vertical differential is given by the (restriction of the) de Rham differential d, and the horizontal differential is given by theČech differential δ . The vertical cohomology of K ≥t is A map of double complexes induces a morphism of the associated spectral sequences. Thus the above inclusions induce morphisms Let us show that the differentials d MS,2 vanish. This will then provide the induction basis for an inductive proof that all d MS,r , r ≥ 2, vanish. The term E MS,1 is given by In view of the above identifications of E 2 -terms, this can be rewritten as Our choice of t together with Lemma 4.1 implies that e p,q 2 is an isomorphism. Therefore, we can express d p,q MS,2 as the composition We shall next show that for arbitrary t, d ≥t,2 = 0. Given any bidegree (p, q), there are two cases to consider: q − 1 < t and q − 1 ≥ t.
Again take t = q. Then e p,q 2 is an isomorphism and the factorization d This induction shows that d MS,r = 0 for all r ≥ 2. We conclude that E(K MS ) collapses at the E 2 -term. This will now be used to prove that E(K) collapses at E 2 .
Since f is a morphism of spectral sequences, we have for every (p, q) a commutative square which can be rewritten as By (2), f p,q 2 is an isomorphism for all (p, q). Thus d 2 = 0 on E 2 . The fact that f 2 is an isomorphism implies that f r is an isomorphism for all r ≥ 2. This shows, since d MS,r = 0 for all r ≥ 2, that d p,q r = 0 for all r ≥ 2. Consequently, E(K) collapses at the E 2 -term, as was to be shown.

NONISOMETRICALLY STRUCTURED FLAT BUNDLES
We construct an example of a flat, smooth circle bundle whose Leray-Serre spectral sequence with real coefficients does not collapse at the E 2 -term. The example shows, then, that in Theorem 5.1 one cannot delete without substitution the requirement that the structure group act isometrically. The example is based on constructions of J. Milnor, cf. [Mil58], [MS74].
It follows from Theorem 5.1 that there is no Riemannian metric on S 1 = R ∪ {∞} such that the action of π 1 (B) on R ∪ {∞} is isometric. This statement will now be affirmed directly, without appealing to the theorem.
denotes the differential of X : R ∪ {∞} → R ∪ {∞} at x, then X * (v) = λ v for some nonzero scalar λ ∈ R and where · x is the putative Γ-invariant norm, evaluated at x. Consequently, X * (v) = ±v. Let t be the standard coordinate in R ⊂ R ∪ {∞} and write Taking v to be the standard basis vector v = ∂ t , we have X * (∂ t ) = dX dt (x) · ∂ t and thus dX dt (x) = 1.
In fact, however, the derivative of a hyperbolic Möbius transformation in PSL(2, R) at a finite, real fixed point is never ±1. Indeed, if c = 0, then where ∆ is the discriminant ∆ = tr 2 X − 4 > 0, and if c = 0, then x = −b/(a − d). (Note that c = 0 and X hyperbolic implies that a = d.) The derivative of X is given by dX/dt = (ct + d) −2 . Brief calculations verify that (cx + d) 2 cannot be 1 at the above points x, for hyperbolic X.
This example illustrates that when modifying the structure group of a fiber bundle, there is a tension between flatness and compactness of the structure group: For a flat bundle with noncompact structure group, one can often reduce to a compact group, but in doing so may be forced to give up flatness. Conversely, given a compactly structured bundle which is not flat, one may sometimes gain flatness at the expense of enlarging the structure group to a noncompact group. For example, identifying the flat PSL(2, R)-bundle γ with S(T B), one may give γ the structure group SO(2), but one loses flatness (S(T B) is not a flat SO(2)bundle).

EQUIVARIANT COHOMOLOGY
We turn our attention to isometric actions of discrete, torsionfree groups. The actions considered here are usually not proper, and our G-spaces are generally not G-CW complexes. Concerning condition (1) of Theorem 7.1 below, we remark again that a nonproper action factors in many geometric situations through a proper action; for instance, a discrete, torsionfree group may act nonproperly on a closed manifold, but in such a way that the manifold can be endowed with an invariant Riemannian metric -in that case, the action factors through the (proper) action of the compact isometry group.
Theorem 7.1. Let F be an oriented, closed, smooth manifold and G a discrete group, whose Eilenberg-MacLane space K(G, 1) may be taken to be a closed, smooth manifold.

If for a smooth action of G on F, (1) the action factors through the proper, smooth action of a Lie group, or (2) F is Riemannian and G acts isometrically on F, then the real G-equivariant cohomology of F decomposes as
where the H q (F; R) are G-modules determined by the action.
Proof. If (1) is satisfied, that is, the action of G on F factors as G → H → Diffeo(F) with H a Lie group acting properly, then F can be equipped with an H-invariant Riemannian metric, by [Pal61]. Then H, and thus also G, acts isometrically on F so that it suffices to prove the theorem assuming hypothesis (2). Let B ≃ K(G, 1) be an aspherical, closed, smooth manifold with fundamental group G. The universal cover B → B can serve as a model for the universal principal G-bundle EG → BG because it is a principal G-bundle with contractible total space B, as follows from the fact that B is a simply connected CWcomplex and π i ( B) ∼ = π i (B) = 0 for i ≥ 2. We recall that the Borel construction F × G EG is the orbit space (F × EG)/G, where G acts diagonally on the product F × EG. The G-equivariant cohomology of F is the cohomology of the Borel construction,

Using the model B → B, this may be computed as
whose projection is induced by the second-factor projection F × B → B. Since G acts isometrically on F, Theorem 5.1 applies and we conclude that the Leray-Cartan-Lyndon spectral sequence for real coefficients of the G-space F collapses at the E 2 -term. In particular, is any fibration with B path-connected such that the restriction i * : H • (E; R) → H • (F; R) is surjective (i.e. the fiber is "totally nonhomologous to zero in E"), then the action of π 1 B on H • (F; R) is trivial (see [Hat04, p. 51, Ex. 2]) and the spectral sequence of the fibration collapses at E 2 , yielding the Leray-Hirsch theorem. Our results apply to situations where i * needs not be surjective. In Section 8, we consider an example of a certain Z 3 -action on the flag manifold F 8 = U(8)/T 8 . We compute (in two different ways) that H 2 Z 3 (F 8 ) has rank 5, while H 2 (F 8 ) has rank 7. Thus i * is not surjective in this example. Moreover, in the context of the present paper, the action of π 1 B on H • (F; R) is typically nontrivial.
Let us discuss some immediate consequences of Theorem 7.1. For a G-module H, let H G = {v ∈ H | gv = v for all g ∈ G} denote the subspace of invariant elements.

Corollary 7.3. Let F be an oriented, closed, connected, smooth manifold and G a discrete group as in Theorem 7.1, acting smoothly on F so that hypothesis (1) or (2) is satisfied. Then there is a monomorphism
Proof. For k ≥ 1, the direct sum of Theorem 7.1 contains the term H k (G; H 0 (F; R)) and the term H 0 (G; H k (F; R)). The former is isomorphic to H k (G; R), since G acts trivially on H 0 (F; R) and H 0 (F; R) ∼ = R as F is connected. The latter is isomorphic to H k (F; R) G ; see the appendix.
In particular, we obtain lower bounds for the ranks of the equivariant groups in terms of the ranks of the group cohomology.

Corollary 7.4. In the situation of Corollary 7.3, the inequalities
If a G-space F has a fixed point, then π : F × G EG → BG has a section given by is a monomorphism. Note that our results concern group actions that may be fixed-point-free, even free. (See Example 7.6 below.) Let cd R G denote the R-cohomological dimension of a group G, that is, the smallest n ∈ N ∪ {∞} such that H k (G; R) vanishes for all k > n. For a topological space X, let cd R X be the smallest n ∈ N∪{∞} such that the singular cohomology H k (X; R) vanishes for all k > n.

Corollary 7.5. In the situation of Corollary 7.3, the inequality
Proof. Suppose that n = cd R G is finite. Then H n (G; R) is not zero. By Corollary 7.4, H n G (F; R) is not zero and it follows that cd R (F × G EG) ≥ n. If cd R G = ∞, then for every n ∈ N, there exists an N ≥ n such that H N (F × G EG; R) = 0, whence cd R (F × G EG) = ∞.
Although the underlying spaces of the G-actions considered in this paper are smooth manifolds, and hence can be given a (regular) CW structure, no such structure can usually be found that would make the G-space into a G-complex. A G-complex is a CW complex together with a G-action which permutes the cells. If X is a G-complex on which G acts freely, then H • G (X) ∼ = H • (X/G). Thus, if F were a free G-complex satisfying the hypotheses of Corollary 7.3, then, by Corollary 7.5, The following example of a free G-action shows that this inequality is generally false in the context of Theorem 7.1.
Example 7.6. Let G = Z act freely on F = S 1 by powers of a rotation by an angle which is an irrational multiple of 2π. The quotient topology on the orbit space S 1 /Z is the coarse topology, that is, the only open sets in S 1 /Z are the empty set and S 1 /Z. The coarse topology on a set X has the property that any map Y → X is continuous. In particular, the map H : X × I → X given by H(x,t) = x for t ∈ [0, 1) and H(x, 1) = x 0 for all x ∈ X, where x 0 ∈ X is a base-point, is continuous. Thus X is homotopy equivalent to a point and therefore acyclic. This shows that cd R (S 1 /Z) = 0. Since cd R Z = cd R S 1 = 1, inequality (4) is violated.
This also emphasizes that it is prudent to observe carefully the hypotheses of the Vietoris-Begle mapping theorem in attempting to apply it to the map F × G EG → F/G for a free action.
Example 7.6 illustrates once again that the actions considered in the present paper are generally not proper, since their orbits need not be closed. Furthermore, the isotropy groups for a proper G-CW complex are compact (so finite if G is discrete). The isotropy groups arising for our actions can be infinite. (Consider the trivial Z-action, or a Z-action that factors through a finite cyclic group.) For a proper G-CW complex X, a result of W. Lück [Lüc07,Lemma 6.4], based on [LRV03, Lemma 8.1], asserts that the projection X × G EG → X/G induces an isomorphism For the actions arising in our Theorem 7.1, the groups H n (X × G EG; Q) and H n (X/G; Q) are generally not isomorphic, as Example 7.6 shows.
Let us consider some specific groups.
Corollary 7.7. If Z n acts isometrically on an oriented, closed, connected, Riemannian manifold F, then with equality for k = 0.
Proof. The inequality follows from Corollary 7.4 by observing that we may take K(Z n , 1) = T n , the n-torus, and rk H k (T n ) = n k .

AN EXAMPLE: GROUP ACTIONS ON FLAG MANIFOLDS
We will compute the equivariant second real cohomology of a certain isometric Z 3action on a (complete) flag manifold, using our Theorem 7.1. The result will then be independently confirmed by employing the recursive scheme introduced in the appendix. The flag manifold F n is the space of all decompositions Cv 1 ⊕ · · · ⊕ Cv n of C n into lines, where {v 1 , . . . , v n } is an orthonormal basis of C n . Let U = U(n) be the unitary group and T = T n ⊂ U the maximal torus given by the diagonal matrices. Writing each v i as a column vector and using these as the columns of a matrix, we obtain an element u of U. If Cv ′ 1 ⊕ · · · ⊕ Cv ′ n = Cv 1 ⊕ · · · ⊕ Cv n , then u ′ = ut with t ∈ T . Thus {v 1 , . . . , v n } → u induces a well-defined map F n → U/T, where U/T denotes the space of left cosets uT . This map is a diffeomorphism and exhibits F n as a homogeneous space. The symmetric group S n acts on F n by permuting the lines Cv i . In terms of unitary matrices, this can be described as follows. Since every permutation is a product of transpositions, it suffices to describe the action of a transposition τ. Let p τ be the corresponding permutation matrix. The normalizer N(T ) of T in U consists of all generalized permutation matrices, that is, matrices whose pattern of zero entries is the same as the zero-pattern of a permutation matrix, but whose nonzero entries can be any complex number of modulus one. In particular, p τ ∈ N(T ). Using this, the map τ : U/T → U/T induced by right-multiplication with p τ , τ(uT ) = up τ T, is seen to be well-defined. The diagram commutes. Up to homotopy, τ can also be described by conjugation: Given an element ν ∈ N(T ), conjugation by ν yields a well-defined map c ν : U/T → U/T, c ν (uT ) = νuν −1 T. While right multiplication by an element generally only induces a map on U/T if the element lies in N(T ), left multiplication by any element of U induces an automorphism of the homogeneous space. However, all these maps f g : U/T → U/T, f g (uT ) = guT, are homotopic to the identity, since in U we can choose a path g t , t ∈ [0, 1], from g to the identity (U is connected) and uT → g t uT is a homotopy from f g to the identity. This shows that for ν = p τ , c ν is homotopic to τ, as c ν (uT ) = f p τ (up τ T ). (A similar argument cannot be used to show that right multiplication uT → uνT for fixed ν ∈ N(T ), is homotopic to the identity because a path ν t from ν to the identity would have to be chosen within N(T ), but N(T ) is disconnected.) Permutation matrices define a splitting of the Weyl group N(T )/T ∼ = S n into N(T ). Then the Weyl group acts on U/T by conjugation, and this is up to homotopy the action of S n on F n that permutes the lines.
The effect of this action on the real cohomology H • (F n ) is easily determined. Let x ∈ H 2 (CP n−1 ) be the standard generator. There are n projection maps π i : F n → CP n−1 given by taking the i-th line of an orthogonal decomposition. With (x 1 , . . . , x n ) = 0, i = 1, . . . , n), where e i (x 1 , . . . , x n ) is the i-th elementary symmetric function in x 1 , . . . , x n . The action of the Weyl group S n on F n induces the action on H • (F n ) which permutes the x i .
If a compact Lie group G acts smoothly on a smooth manifold, then that manifold can be equipped with a smooth G-invariant Riemannian metric. Thus, as S n is finite, there exists a Riemannian metric on F n such that S n acts by isometries. We endow F n with such a metric.
Next, let us determine which elements of S n preserve the orientation of F n . Let ξ i ∈ Ω 2 (F n ) be a closed form representing x i , i = 1, . . . , n. The flag manifold F n is a complex manifold of complex dimension d = n(n − 1)/2. By [HH84, Theorem 2.3], there is a nonzero constant C such that for scalars λ 1 , . . . , λ n , Thus if the λ i are all distinct, then (λ 1 ξ 1 + . . . + λ n ξ n ) d is a generator of the real topdimensional cohomology group H 2d (F n ) ∼ = R. When a permutation acts on the scalars λ i by permuting them, then the right hand side of (5), and therefore also the left hand side, changes sign if and only if the permutation is odd. Consequently, the subgroup of orientation preserving automorphisms in S n is the alternating group A n .
The twisted group H 0 (T 3 ; H 2 (F 8 )) thus has rank 2 and basis {x 1 + x 2 + x 3 + x 4 , x 5 + x 6 + x 8 }. Since H 1 (F 8 ) = 0, we have H 1 (T 3 ; H 1 (F 8 )) = 0. Furthermore, as H 0 (F 8 ) is a constant local system of rank one on T 3 , the group H 2 (T 3 ; H 0 (F 8 )) = H 2 (T 3 ; R) has rank 3 generated by the dual basis of the homology basis S 1 × S 1 × pt, S 1 × pt ×S 1 , and pt ×S 1 × S 1 . By Theorem 7.1, We shall confirm this result by applying the recursive scheme of the appendix. Let B 1 = R 1 × Z F 8 be the Borel construction of the Z-action on F 8 by powers of R 1 , B 2 = R 1 × Z B 1 the Borel construction of the Z-action on B 1 by powers of R 2 , and B 3 = R 1 × Z B 2 the Borel construction of the Z-action on B 2 by powers of R 3 (notation as in the appendix). As explained in the appendix, H 2 Z 3 (F 8 ) = H 2 (B 3 ) can then be found by carrying out the following steps: 1. Determine the action of R * 1 , R * 2 and R * 3 on both H 0 (F 8 ) (trivial) and H 2 (F 8 ). (Note Step 3 determines the action of R * 2 and R * 3 on both H 1 (B 1 ) and H 2 (B 1 ). The action of 6. Using the results of steps 4 and 5, determine the action of R * 3 on the four groups computed in step 5. 7. The result of step 6 determines the action of R * 3 on H 1 (B 2 ) and H 2 (B 2 ). 8. Compute (Der Z / Princ Z )H 1 (B 2 ) and H 2 (B 2 ) Z . 9. The direct sum of the two groups obtained in step 8 is H 2 (B 3 ).
We shall now carry out these steps.

APPENDIX: THE CASE OF THE FREE ABELIAN GROUP
We shall describe a recursive scheme for calculating the equivariant cohomology of an isometric action of Z n on a closed, oriented, path-connected, Riemannian manifold F. We begin with some remarks on calculating the action of the homomorphism induced on low-degree local system cohomology by a fiber preserving map. We shall think of a local system H on a base space B as a functor H : Π 1 (B) → R − MOD from the fundamental groupoid Π 1 (B) to the category R − MOD of real vector spaces and linear maps. Thus H assigns to every b ∈ B a real vector space H(b) and to every homotopy class [ω] ∈ π 1 (B; b 1 , b 2 ), where ω is a path ω : yields H p (B; H), the cohomology of B with coefficients in H.
The recursive scheme concerning Z n -actions only requires being able to compute in the degrees p = 0 and p = 1. Assume that B is path-connected and endowed with a base-point b 0 . Let G = π 1 (B, b 0 ) be the fundamental group and let H(b 0 ) G denote the G-invariants of H(b 0 ), that is,  H).
Thus κ G is surjective as well, hence an isomorphism. This isomorphism will be used to compute H p (B; H) for p = 0. The group H 1 (B; H) (p = 1) will be computed as derivations modulo principal derivations. A derivation is a function f : G → V , where V is a real G-vector space, such that f (gh) = f (g) + g · f (h) for all g, h ∈ G. The set of all functions from G to V is a real vector space under pointwise addition and scalar multiplication. The derivations form a linear subspace Der G (V ). A principal derivation is a derivation f of the form f (g) = g · v − v for some v ∈ V and all g ∈ G. The principal derivations form a subspace Princ G (V ) ⊂ Der G (V ). We shall recall the well-known isomorphism θ : H 1 (B; H).
Observe that H 1 (B; H) can be computed by only considering singular 1-simplices that close up. Given a derivation f : is a coboundary. Thus θ is well-defined and it is readily verified that it is an isomorphism.
Let us discuss the naturality of the above constructions. Suppose H and G are local coefficient systems on B. A homomorphism φ : H → G of local systems is a natural transformation of functors. Thus for every point For p = 0, 1 we wish to understand φ * in terms of the above identifications κ G and θ . Let us discuss The calculation shows that H 0 (B; G) commutes. Thus κ G is a natural isomorphism. Let us turn to p = 1 and θ . Moreover, if f is principal, say f (g) = gv − v = H(g)(v) − v for all g ∈ G, then Thus φ * ( f ) is again principal and φ * induces a map φ * : . Let (F, g F ) be a closed, oriented, path-connected, Riemannian manifold. Let R and S be commuting orientation preserving isometries of F, RS = SR, defining a Z 2 -action on F. We endow the real line R 1 with the canonical metric g 1 = dt 2 and give R 1 × F the product metric g 1 + g F . Using S, we define a diffeomorphism S : R 1 × F → R 1 × F by S(t, x) = (t + 1, S(x)), t ∈ R 1 , x ∈ F. This map defines a Z-action on R 1 × F. Let E be the orbit space of this action. Since the action is properly discontinuous, the quotient map R 1 × F → E is a regular covering map and E is a smooth manifold, the mapping torus of S. If M, N are Riemannian manifolds and f : M → M, g : N → N isometries, then f × g : M × M → N × N is an isometry for the product metric. Since the translation t → t + 1 is an isometry and S is an isometry, we conclude that S is an isometry. Thus Z acts by isometries on R 1 × F and there exists a unique metric g E on E such that the quotient map R 1 × F → E is a local isometry. The projection (t, x) → t induces a fiber bundle projection p : E → S 1 such that commutes. Using the isometry R, define a fiber-preserving diffeomorphism R : . This is well-defined as R and S commute: Moreover, R is an isometry of (E, g E ), since (t, x) → (t, R(x)) is an isometry of R 1 × F. Our next goal is to describe the induced map R * : H • (E) → H • (E) in terms of a calculation of H • (E) through the Leray-Serre spectral sequence of p : E → S 1 . Let K ⊂ H q (E) be the subspace K = ker(H q (E) −→ H q (F)).
Since S is orientation preserving, E is orientable and receives an orientation from the canonical orientation of R 1 and the given orientation of F. Thus E is a closed, oriented manifold and Hodge theory applies. On the vector space Ω q (E) of smooth q-forms on E, an inner product is given by ω, η = E ω ∧ * η.
The Hodge theorem asserts that with respect to this inner product, there is an orthogonal decomposition Ω q (E) = ∆Ω q (E) ⊕ Harm q (E), where ∆ is the Hodge Laplacian on E and Harm q (E) are the harmonic q-forms. The inner product restricts to an inner product on Harm q (E), which yields an inner product on H q (E) via the isomorphism Harm q (E) ∼ = −→ H q (E) induced by the inclusion. Let K ⊥ be the orthogonal complement of K in H q (E) with respect to this inner product.
Proof. Let i : F ֒→ E be the inclusion of the fiber over the base-point. By the definition of R, the square
Thus R * (v) ∈ K and K is R * -invariant.
Since R is an orientation preserving isometry, its induced map commutes with the Hodge star, R * • * = * • R * . Thus, for harmonic forms ω, η ∈ Harm q (E) ∼ = H q (E), Consequently, R * is an orthogonal transformation on H q (E), and so is (R * ) −1 . For v ∈ K and w ∈ K ⊥ , Lemma 9.2. The diagram H q (E)/K E p,q 2 = E p,q ∞ , φ 2 = φ ∞ (i.e. R 2 = R ∞ ). Putting the above isomorphisms γ, α, β together, we obtain a commutative diagram H 0 (S 1 ; H q (F)) ⊕ H 1 (S 1 ; H q−1 (F)) to everyone how to proceed if n is larger. Let F be a closed, oriented, path-connected, Riemannian manifold. Let R 1 , R 2 , R 3 be commuting orientation preserving isometries of F, determining a Z 3 -action on F. We wish to calculate the equivariant cohomology H • Z 3 (F). Let B 1 be the Borel construction of the Z-action on F by powers of R 1 , that is, B 1 is the mapping torus B 1 = R 1 × Z F. The closed, oriented manifold B 1 is a fiber bundle over the circle with fiber F. The isometry R 2 defines a fiber preserving isometry R 2 : B 1 → B 1 by R 2 [t, x] = [t, R 2 (x)]. Since the monodromy of the mapping torus B 1 is R 1 , the monodromy of the induced local system H q (F) is R * 1 . Thus, assuming that the action of R * 1 on H • (F), defining a Z-action on H • (F), as well as the action of R * 2 on H • (F), are known, the automorphism R