{'fraction_non_alphanumeric': 0.05744016649323621, 'fraction_numerical': 0.012382934443288241, 'mean_word_length': 3.841435768261965, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 17, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Equivariant Poincar\'e-Alexander-Lefschetz duality and the Cohen-Macaulay property

We prove a Poincare-Alexander-Lefschetz duality theorem for rational torus-equivariant cohomology and rational homology manifolds. We allow non-compact and non-orientable spaces. We use this to deduce certain short exact sequences in equivariant cohomology, originally due to Duflot in the differentiable case, from similar, but more general short exact sequences in equivariant homology. A crucial role is played by the Cohen-Macaulayness of relative equivariant cohomology modules arising from the orbit filtration.


Introduction
Let T = (S 1 ) r be a torus, and let X be a T -space satisfying some fairly mild assumptions (see Section 2.1). Recall that H * T (X) = H * T (X; Q), the equivariant cohomology of X with rational coefficients, can be defined as the cohomology of the Borel construction (or homotopy quotient) X T = (ET × X)/T , and that it is an algebra over the polynomial ring R = H * (BT ).
In [6, p. 23], A. Borel observed that "even if one is interested mainly in a statement involving only cohomology, one has to use in the proof groups which play the role of homology groups, and therefore this presupposes some homology theory". In this spirit we defined in [3] the equivariant homology H T * (X) of X, which is a module over R. In contrast to H * T (X), it is not the homology of any space. Nevertheless, it has many desirable properties: it is related to H * T (X) via universal coefficient spectral sequences, and, in the case of a rational Poincaré duality space X, also through an equivariant Poincaré duality isomorphism for 0 ≤ i ≤ r and zero otherwise. (We set X −1 = ∅.) The differential is the boundary map in the long exact sequence of the triple (X i+1 , X i , X i−1 ). In other words, AB * (X) is the E 1 page of the spectral sequence arising from the orbit filtration and converging to H * T (X), and H * (AB * (X)) is its E 2 page. A principal result of [3] is a natural isomorphism (1.6) H i (AB * (X)) = Ext i R (H T * (X), R) for all i ≥ 0. This is once again a consequence of the Cohen-Macaulay property of H T * (X i , X i−1 ). In [3] we used the isomorphism (1.6) to study syzygies in equivariant cohomology and to relate them to the Atiyah-Bredon complex. Here we again indicate generalizations to (co)homology with the new pair of supports and/ or twisted coefficients. They are used in [16] to prove a "geometric criterion" for syzygies in equivariant cohomology that only depends on the quotient X/T as a stratified space.
The paper is organized as follows. In Section 2 we first review equivariant cohomology with closed supports and equivariant homology with compact supports and then define equivariant (co)homology with the other pair of supports. We also consider homology manifolds and define variants of equivariant (co)homology with twisted coefficients in this case. Theorem 1.1 and its corollaries are proved in Section 3. Applications to the orbit structure are given in Section 4. There we also relate the cohomology of the Atiyah-Bredon complex to the question of uniformity of an action. Given the importance of (1.6), we include a direct proof of it in Section 5. It uses only exact sequences as in Proposition 1.2 and avoids the intricate reasoning with spectral sequences done in [3].
Acknowledgements. We thank an anonymous referee for numerous helpful comments and in particular for suggesting a strengthening of Proposition 2.7.

Equivariant homology and cohomology
2.1. Notation and standing assumptions. We write "⊂" for inclusion of sets and " " for proper inclusion.
Throughout this paper, T = (S 1 ) r denotes a compact torus of rank r ≥ 0, and k a field. From Section 4 on we will assume that the characteristic of k is zero. All (co)homology is taken with coefficients in k unless specified otherwise. C * (−) and C * (−) denote normalized singular chains and cochains with coefficients in the field k, and H * (−) and H * (−) singular (co)homology. We adopt a cohomological grading, so that the homology of a space lies in non-positive degrees; an element c ∈ H i (X) has cohomological degree −i. R = H * (BT ) is the symmetric algebra generated by H 2 (BT ), and m ⊳ R its maximal homogeneous ideal. All R-modules are assumed to be graded. We consider k as an R-module (concentrated in degree 0) via the canonical augmentation. For an R-module M and an l ∈ Z the notation M [l] denotes a degree shift by l, so that the degree i piece of M [l] is the degree i − l piece of M . For the cohomology of some space, we alternatively write H * (X) [l] or H * −l (X). Due to the cohomological grading, we have in homology the identity H * (X)[l] = H * +l (X).
We assume all spaces to be Hausdorff, second-countable, locally compact, locally contractible and of finite covering dimension, hence also completely regular, separable and metrizable. Important examples are topological (in particular, smooth) manifolds, orbifolds, complex algebraic varieties, and countable, locally finite CW complexes. We also assume that only finitely many distinct isotropy groups occur in any T -space X. In addition we will put a finiteness condition on the (co)homology of the spaces and pairs we consider. This will be explained in detail once we have defined equivariant (co)homology.
2.2. The singular Cartan model. Let X be a T -space. We recall from [3,Sec. 3] the definition of equivariant homology and cohomology via the "singular Cartan model". As pointed out in [3], it can be replaced by the usual Cartan model for differentiable actions on manifolds and k = R.
The singular Cartan model of the T -pair (A, B) in X is and R-bilinear product Here t 1 , . . . , t r are a basis of H 2 (BT ) ⊂ R, and a 1 , . . . , a r are representative loops of the dual basis of H 1 (T ); the product a i · γ refers to the action of C * (T ) on C * (X) induced by the T -action on X. The equivariant chain complex C T * (A, B) is the R-dual of (2.1), . Equivariant cohomology and homology are defined as This definition of H * T (A.B) is naturally isomorphic, as an R-algebra, to the usual one based on the Borel construction X T .

Other supports.
Let (A, B) be a closed T -pair in a T -space X. We define the equivariant cohomology of (A, B) with compact supports by (2.8) and the direct limits are taken over all T -stable open neighbourhood pairs (U, V ) of (A, B) such that X \ V is compact. By tautness and excision, H * T,c (A, B) is easily seen to be naturally isomorphic to the Alexander-Spanier cohomology of the closure of (A, B) in the one-point compactification of X relative to the added point. Hence it does not matter whether (A, B) is considered as a closed T -pair in X or in A.
The equivariant homology of (A, B) with closed supports is defined by taking the R-dual of (2.8), 2.4. Properties. We list several important properties of equivariant (co)homology, omitting proofs that were given in [3]. In Section 2.6 we will extend all results of this section to homology manifolds and (co)homology with twisted coefficients, see Remark 2.15.
. For a multiplicative subset S ⊂ R and a T -space X, define the T -stable subset It is closed in X, cf. [4, p. 132]. For example, X S = X T if char k = 0 and S contains all non-zero linear polynomials.

Proposition 2.5 (Localization theorem). Let (A, B) be a T -pair in X, and let S ⊂ R be a multiplicative subset. Then the inclusion X S ֒→ X induces isomorphisms of
. Proof. The localization theorem for equivariant cohomology with closed supports is classical, cf. [4,Ch. 3]. (Recall that only finitely many orbit types occur in X.) By the universal coefficient theorem there is a spectral sequence converging , and similarly for the pair (A S , B S ). The inclusion X S ֒→ X gives rise to a map of spectral sequences, which on the E 2 pages is the canonical map . Since localization is an exact functor, the S-localization of (2.12) is the map In the proof below as well as in that of Proposition 2.7 we will use the following fact, cf. [4, Cor. B.1.13]: Let φ : M → N be a quasi-isomorphism of dg R-modules. If M and N are free as R-modules, then φ is a homotopy equivalence over R.
Proof. Since B is closed in A, we have, by tautness and excision, a quasi-isomorphism of dg R-modules where the direct limit is taken over all T -stable open sets U ⊃ B. Hence we may work with this direct limit, which we denote by M . Now choose a splitting T = K × L. By [3,Prop. 3.4], we may assume that the representatives a i ∈ C 1 (T ) appearing in the "Cartan differential" (2.2) are chosen such that a 1 , . . . , a p ∈ C 1 (K) and a p+1 , . . . , a r ∈ C 1 (L). Since we are using normalized singular (co)chains, C * (K) acts trivially on each C * (A\B, U \B). The differential on M therefore takes the form By the remark made above, the quasi-isomorphism (2.15) is a homotopy equivalence, which is preserved by the functor Hom R (−, R). For equivariant homology we can therefore argue analogously.
The last claim follows by the five-lemma from the previous one, applied to A and B separately, and the long exact sequence of the pair.
At the other extreme, we have the following: ]. Geometrically, the homological isomorphism can be understood as a transfer for the quotient map X → X/K. Since in the singular setting it is delicate to define a transfer map or integration over the fibre on the (co)chain level, we will follow an algebraic approach and postpone the geometrical aspects to our discussion of Poincaré-Alexander-Lefschetz duality (Remark 3.10). The homology isomorphism can also be viewed as a version of the Adams isomorphism in equivariant stable homotopy theory (see [19,§II.7] or [20,§XVI.5]) in our algebraic context.
The proof of Proposition 2.7 requires some preparation. Recall that m = (t 1 , . . . , t r ) is the maximal graded ideal in R. In the proof below we will use the local duality isomorphism [15, Thm. A1.9] (where the generators of the polynomial ring are assigned the degree 1, not 2). The symbol " ∨ " in (2.18) denotes the dual of a graded k-vector space. We will also need the Čech complex computing H * m (M ) by means of some generators of m as in [15, p. 189]. More generally, we consider the Čech complex for a dg R-module M . Thus we obtain a bicomplex C * , * m (M ) with first differential d I coming from M and second differential d II coming from the Čech complex for the canonical generators t 1 , . . . , t r . An element in C i,j m (M ) is a sum of elements of degree j in the i-fold localizations in this Čech complex. While j is unbounded in both directions, we have 0 ≤ i ≤ r, so that both filtrations of the bicomplex are regular [11, p. 452]. Hence both associated spectral sequences converge to H * m (M ), the cohomology of C * , * m (M ) with respect to the total differential.
In the first bicomplex spectral sequence we have since the cohomology of the localization of M is the localization of the cohomology.
Taking the other bicomplex spectral sequence, we get with respect to the differential d II , that is, the local cohomology of the R-module M with trivial differential.
Suppose that M is finitely generated and free as an R-module. By local duality one then has that the E 1 page is concentrated in the column k = r, and therefore , then it is R-homotopy equivalent to a dg R-module M ′ which is finitely generated and free as an R-module. We therefore conclude that Let m L = (t p+1 , . . . , t r ) be the maximal graded ideal of R L . Using the canonical generators, we can similarly define C * , * mL (−) and H * mL (−) for dg R L -modules, hence a fortiori for dg R-modules. Since these generators are among the chosen generators of m, we have a canonical map of bicomplexes ) is an isomorphism. If char k = 0, then it is enough that K acts locally freely and that the isotropy groups in A \ B have the same identity component K ′ .
Proof. Since K acts (locally) freely, the composition K ′ → T → L is injective (or has finite kernel). This implies that the composition i maps to 0. By the localization theorem (Proposition 2.5), this implies that the localization of H * T (A, B) at u i vanishes. We observe that u 1 , . . . , u p , t p+1 , . . . , t r also generate m. Since local cohomology can be computed from any set of generators, cf. [15, Thm. A1.3], we can assume that one has chosen these generators instead of the canonical generators t 1 , . . . , t r . Then the terms in the Čech complex involving at least one of the u i 's drop out, and we are left with the Čech complex computing H * mL (H * T (A, B)). Proof of Proposition 2.7. We choose a splitting T = K × L with compatibly chosen representatives a i ∈ C 1 (T ) as in the proof of Proposition 2.6.
As mentioned already, the isomorphism It follows from the localization theorem that the localization of H * is killed by some power of each t i and therefore that it is finite-dimensional as k-vector space. By taking For the homological statement we start by proving that the canonical map ) is a quasi-isomorphism. We proceed by induction on the number m of (connected) orbit types in A \ B. For m = 0 there is nothing to show as A = B in this case. Otherwise fix an orbit type of maximal dimension in A \ B and let A ′ ⊂ A be the union of B and all other orbit types; A ′ is T -stable and closed in A.
The short exact sequence whose horizontal sequences are again short exact. The right vertical arrow is a quasi-isomorphism by induction. To see that the left one is so as well, we consider the induced map between the E 2 pages of the first bicomplex spectral sequences (2.20). In our case this is the map , and it is an isomorphism by Lemma 2.8 and our choice of A ′ . The map induced in cohomology by the left arrow above therefore is also an isomorphism. Hence the middle arrow is a quasi-isomorphism by the five-lemma, which proves the claim.
The quasi-isomorphism (2.28) is in fact a homotopy equivalence over R L as both sides are free as R L -modules. We therefore get isomorphisms of R L -modules The last claim follows again from the absolute case and the five-lemma.
All these results hold as well for cohomology with compact supports and homology with closed supports and closed T -pairs (A, B), assuming that H * c (A, B) is a finite-dimensional k-vector space, cf. Assumption 2.2. The proofs are identical; the localization theorem for cohomology with compact supports follows from the version for closed supports since direct limits preserve isomorphisms.
Proof. The first identity follows from excision and the fact that a direct limit is an exact functor. The second identity then is a consequence of the universal coefficient theorem.
2.5. Homology manifolds. Let X be a k-homology manifold, say of dimension n. By this we mean a connected space X such that for any x ∈ X one has If in addition H c n (X) ∼ = k, then X is called orientable. Homology manifolds are an appropriate setting for Poincaré duality, see Lemma 3.1 below. Assumption 2.10. We assume that any homology manifold X we consider is orientable or admits an orientable two-fold covering π :X → X.
For non-orientable X, such a covering will be called an orientation cover. Note thatX is necessarily connected. For orientable X we define the trivial two-fold covering to be the orientation cover. We will use orientation covers to define (co)homology with twisted coefficients in Section 2.6. Remark 2.11. Any Z-homology manifold admits an orientation cover, but it seems unclear whether this holds for arbitrary k-homology manifolds, see the discussion in [11, p. 331]. On the other hand, if an orientation cover exists, then it is unique. For orientable X, this is true by definition. For non-orientable X it can be seen as follows: Let γ be a loop at x ∈ X. By transporting local orientations along γ, we get an automorphism of H n (X, X \ {x}), cf. [7, p. 39], which is necessarily multiplication by some non-zero scalar. This induces a morphism φ : π 1 (X) → k × . The connected orientable covers of X are of the formX/G whereX is the universal cover and G a subgroup of the kernel of φ. In particular, there is at most one orientation cover.
The following observation seems to be well-known, but we could not find a suitable reference.  [21] for the definition of an orbifold. By "locally orientable" we mean that locally the orbifold X, say of dimension n, is the quotient of an open ball in R n by a finite subgroup of SO(n).
Proof. Condition (2.36) holds because one locally divides by a finite subgroup of SO(n) and char k = 0.
The existence of an orientation cover can be shown in the same way as for manifolds. Recall that in the smooth case one proceeds as follows, cf. [18,: If X admits an oriented atlas, that is, if the charts of X can be oriented in a way consistent with coordinate changes, then one can integrate differential forms with compact supports, and the integration map provides an isomorphism H c n (X) ∼ = k. Otherwise H c n (X) = 0, and one can construct a connected double cover with oriented atlas by doubling all charts and gluing them according to whether coordinate changes preserve or reverse chart orientations. Hence X is orientable in our sense if and only if it admits an oriented atlas.
For an orbifold X one can also define differential forms with compact supports, and if X is locally orientable and has an atlas of compatibly oriented charts, then one can integrate these forms, cf. [21, §8]. If such an atlas does not exist, then one can again pass to an oriented two-fold cover. Now the proofs for manifolds go through without change. Proof. The case of orientable X is trivial. If X is non-orientable, then by [8,Sec. I.9] the T -action on X lifts to aT -action onX, whereT is a two-fold covering of T and ker(T → T ) ∼ = Z 2 acts by deck transformations. If the non-trivial deck transformation τ were orientation-preserving, then X would have to be orientable because H c n (X) = H c n (X) τ ∼ = k, where n = dim X. So τ does not preserve orientations, which implies thatT cannot be connected. Hence its identity component is T , and the action lifts.
2.6. Twisted coefficients. The aim of this section is to introduce equivariant (co)homology with twisted coefficientsk. To distinguish it from the (co)homology we have considered so far, the latter will be called (co)homology with constant coefficients k from now on. Twisted coefficients are only interesting if the characteristic of the ground field k differs from 2, which we assume in this section. For char k = 2 (co)homology with twisted coefficients is defined to be the same as (co)homology with constant coefficients.
We focus on cohomology with closed supports and homology with compact supports. All results are equally valid for the other pair of supports; we will indicate when proofs for that case need additional arguments.
Let X be a k-homology manifold (which, by our definition, is connected) with orientation cover π :X → X and non-trivial deck transformation τ . For a pair (A, B) in X, we write (Ã,B) = (π −1 (A), π −1 (B)). Moreover, we denote the involution of C * (Ã,B) induced by τ by the same letter. Since 2 ∈ k is invertible, we get a decomposition We now assume that X is equipped with a T -action and that the pair (A, B) is T -stable. Since the decomposition (2.38) is C * (T )-stable, we can define equivariant (co)homology with twisted coefficients in a way analogous to Section 2.2: with the same differential as in (2.2), Assumption 2.2 is extended as follows: Assumption 2.14. For any T -pair (A, B) in a k-homology manifold X and any (co)homology theory we are going to consider, we assume that the non-equivariant cohomology of the cover (Ã,B) is finite-dimensional over k. In light of (2.39), this is equivalent to both the cohomology with constant coefficients and that with twisted coefficients being finite-dimensional. By Proposition 2.3, this in turn implies that the equivariant (co)homology of (A, B) with constant or twisted coefficients is finitely generated over R.
Our definition of H * (A, B;k) does not require A or B to be k-homology manifolds themselves. But if A is connected and open in X, then it is a k-homology manifold as well, and the restriction of π to A is the orientation cover of A. Hence the definition of twisted coefficients is independent of the ambient space in this case.

Remark 2.15.
All results from Section 2.4 (Serre spectral sequences, universal coefficient theorems and localization theorems) carry over to twisted coefficients. To see this, one can either redo the proofs with twisted (co)homology, or one can reduce the new results to the untwisted case by using the splittings (2.44) and (2.45).

Remark 2.16.
An alternative way to define cohomology with twisted coefficients is to use local coefficient systems. This could be done as well in the equivariant setting, and one could even dispense with Assumption 2.10. The drawback of this approach would be that one cannot reduce statements to the case of constant coefficients anymore. In particular, one would need to prove a generalization of Proposition 2.7 (essentially, of the Vietoris-Begle mapping theorem) to local coefficients, which is required to prove Proposition 4.1.

Remark 2.17.
We are mainly interested in applying our results to the fixed point sets X K of subtori K ⊂ T , and because we want to use the Localization Theorem for singular cohomology (Proposition 2.5), we put local contractability into the standing assumptions in Section 2.1 Now it is a small step from (2.8) to using Alexander-Spanier cohomology for all closed invariant pairs (A, B), cf. [3,Rem. 4.7]. Thus it is not, in fact, necessary to assume closed subsets to be locally contractible since the Localization Theorem for Alexander-Spanier cohomology does not need this assumption. We would, however, continue to assume that the ambient space X satisfies the standing assumptions; and we do not know of any torus action on such a space where the fixed point sets are not locally contractible -but nor do we know a proof that they always are.

Poincaré duality.
Let k be a field. We start with the statement of nonequivariant Poincaré duality for orientable homology manifolds in our setting because it is not easy to locate it in the literature in the desired generality. Proof. Recall that X is assumed to be a locally compact and locally contractible second-countable Hausdorff space. Sheaf (co)homology and singular (co)homology (with closed or compact supports) are therefore naturally isomorphic on X, cf. Now let X be a T -space and a not necessarily orientable k-homology manifold of dimension n with orientation coverX. The cup product inX is τ -equivariant, so that we obtain a pairing (3.1) C * T,c (X;k) ⊗ C * T (X) → C * T,c (X;k), hence a cap product Extending the above definition, an orientation of X is a non-zero element o ∈ H c n (X;k) ⊂ H c n (X). An equivariant orientation is an element o T ∈ H T,c n (X;k) that restricts to an orientation under the restriction map H T,c * (X;k) → H c * (X;k). To prove the first isomorphism, let us assume for the moment that X is orientable. Applying the Serre spectral sequence to the map The second isomorphism is a consequence of the first and the universal coefficient theorem (Proposition 2.4 and Remark 2.15).

Remark 3.3.
If X is orientable, then the two eigenspaces of τ in the decomposition (3.5) H * T (X) = H * T (X) ⊕ H * T (X;k) are isomorphic as R-modules and even as modules over H * T (X). Hence (co)homology with twisted coefficients and with constant coefficients are isomorphic in this case, and these isomorphisms are compatible with the two Poincaré duality isomorphisms H * T,c (X) → H T * (X) and H * T,c (X;k) → H T * (X). Of course, in the case char k = 2 there is no difference either.

Poincaré-Alexander-Lefschetz duality.
Classically, Poincaré-Alexander-Lefschetz duality (also called "Poincaré-Lefschetz duality") refers to an isomorphism of vector spaces any closed pair (A, B) in an oriented n-dimensional manifold X, cf. [13,§VIII.7] or [10,§VI.8], for instance. In this section we generalize this to equivariant (co)homology, and we also derive a spectral sequence version of it. Our approach is similar to the one in [10]. We continue to assume that X is an n-dimensional k-homology manifold with a T -action.
Our first result includes Theorem 1.1 from the introduction.
all of whose vertical arrows are isomorphisms. An analogous diagram exists with the roles of homology and cohomology interchanged and all arrows reversed.
Since we also want to prove an extension to spectral sequences, we place ourselves in a slightly more general situation. Consider an increasing filtration of X by closed T -stable subsets. (See Remark 2.17 on how to do without the standing assumption of local contractability.) We set X i = X \ X i , so that the decreasing filtration of X by the open complements of the X i can be written as Let π :X → X be the orientation cover, and let U = (U −1 , U 0 , . . . , U m ) be an increasing sequence of open subsets of X such that X\U −1 is compact and X i ⊂ U i for all i. Any such sequence determines an open cover ofX. We write C * (X | U;k) for the complex of U-small cochains and similarly C * T (X | U;k) = C * (X | U;k) ⊗ R for the corresponding singular Cartan model. We start by establishing a variant of the cup product (3.1).

It is compatible with restrictions in the sense that the diagram
whereβ ∈ C * (X) is a preimage of β.
The commutativity of the diagram is clear by construction.
For i ≤ j, write where the direct limit is taken over all open covers U induced by sequences U as above, and letC T,c * (X j , X i ;k) be the R-dual complex. Note that for i ≤ j ≤ k we have short exact sequences * (X j , X i ;k) are quasi-isomorphisms by tautness. By passing to the direct limit in Lemma 3.5, we get the family of relative cup products . Fix a representative c T ∈C T,c n (X;k) of the equivariant orientation o T ∈ H T,c n (X;k). Composition of (3.13) with c T yields a pairingC * T,c (X, X i ;k)⊗ C * T ( X i ) → R, which we interpret as a map (3.14) f i :C * T,c (X, X i ;k) → C T * ( X i ). Lemma 3.6. The map (3.14) is a quasi-isomorphism. Moreover, for i ≤ j it leads to a commutative diagram fj fi fji whose rows are exact and whose induced map f ji is a quasi-isomorphisms as well.
Proof. The exactness of the top row in the diagram was already observed in (3.12). The compatibility of relative cup products with restrictions stated in Lemma 3.5 implies that the left square in the diagram commutes, which induces the right vertical arrow. Note that Proposition 2.9 remains valid for twisted coefficients, and that the diagram is commutative. Moreover, the restriction of the orientation o T to any component of X i is again an orientation. Hence the map (3.14) corresponds to the map H * T,c ( X i ;k) → H T * ( X i ), which is an isomorphism by Proposition 3.2. Coming back to the commutative ladder, two out of the three maps between the corresponding long exact sequences in (co)homology are isomorphisms, hence so is the third.

Proof of Theorem 3.4.
Consider the filtration ∅ ⊂ B ⊂ A ⊂ X of X and the associated diagram fAB fA fB whose top row is again of the form (3.12). The maps f A and f B are special cases of the map f ji from Lemma 3.6. It follows from their definition that the right square commutes, which induces the map f AB . By passing to (co)homology we get the commutative ladder stated in Theorem 3.4. Since H * (f A ) and H * (f B ) are isomorphisms by Lemma 3.6, so is H * (f AB ). This proves the first part of the theorem. The analogous result with the roles of (co)homology reversed is obtained by applying the functor Hom R (−, R) to the diagram above. Because the short sequences in the diagram split over R, their duals remain exact. Moreover, the natural inclusion of C * T (A, B) into its double dual is a chain homotopy equivalence. This follows from the fact that for the chain-equivalent minimal Hirsch-Brown model, which is free and finitely generated over R, the corresponding map is even an isomorphism.
A spectral sequence version of equivariant Poincaré-Alexander-Lefschetz duality is as follows:

Similarly, the spectral sequences
. We know from Lemma 3.6 that the diagram so that we obtain a map of spectral sequences with It follows as in the proof of Theorem 3.4 that the map (3.18) is an isomorphism. The second part follows analogously by dualizing (3.14) and the filtrations F and F .
Equipped with equivariant Poincaré-Alexander-Lefschetz duality, we can easily deduce the following result, which is asserted in [9, p. 849] without proof. Corollary 3.8. If X is orientable and T acts locally freely, then X/T is an orientable k-homology manifold of dimension n − r.
Proof. As discussed in Remark 2.1, X/T satisfies our assumption on spaces, and it is connected since X is.
To verify condition (2.36), take an x ∈ X with imagex ∈X = X/T . By Proposition 2.7 and Poincaré-Alexander-Lefschetz duality for the T -pair (T x, ∅) in X, we have Again by Proposition 2.7, the equivariant orientation o T ∈ H T,c n (X) descends to a non-zero element in H c n−r (X/T ). Hence X/T is orientable. whenever a subtorus K ⊂ T of rank p and with quotient L = T /K acts freely on A \ B; a locally free action was sufficient in case char k = 0. In the context of orientable homology manifolds, we can now understand this isomorphism in terms of Poincaré-Alexander-Lefschetz duality: Assume that K acts freely (or just locally freely if char k = 0) on the orientable homology manifold X, so that X/K is again an orientable homology manifold by Corollary 3.8. Let n = dim X = dim X/K + p. Using the cohomological part of Proposition 2.7 and Poincaré-Alexander-Lefschetz duality, we get Hence the isomorphism (3.20) can be interpreted as a push-forward map or integration over the fibre in this setting.

Thom isomorphism.
As in the non-equivariant case, the Thom isomorphism is a consequence of Poincaré and Poincaré-Alexander-Lefschetz duality, cf. [13,§VIII.7,§VIII.11]. In fact, one can use our version of equivariant duality to define also Gysin homomorphisms (push forwards), indices, Euler classes etc. in the equivariant setting and to prove their main properties (cf. [4,Sec. 5.3]) for cohomology with different supports. The use of the Cartan model even provides a more functorial approach than the minimal Hirsch-Brown model used in [4]. Here we only develop the theory as far as needed for our applications in Section 4.2.
We continue to assume that X is an n-dimensional k-homology manifold with a T -action.

Proposition 3.11.
(1) Let Y ⊂ X be a closed T -stable k-homology manifold of dimension m. Suppose that the orientation cover of X restricts to the orientation cover of Y . Then there is an isomorphism of R-modules (2) Assume char k = 0, and let K ⊂ T be a subtorus. Then there is an isomorphism of R-modules . This isomorphism has degree m − n if all components of X K are of dimension m; in general it only preserves degrees mod 2.
Proof. We start with the first case. By Poincaré-Alexander-Lefschetz duality for the pair (X, Y ) and Poincaré duality for Y we have isomorphisms of R-modules , whose composition has degree m−n. Note that for the first isomorphism H T,c * (Y,k) is defined via the restriction of the orientation cover of X, and via the orientation cover for Y in the second isomorphism. By assumption, these two covers coincide.
We now consider the fixed point set X K . It has finitely many components, say Y 1 , . . . , Y k , which are k-homology manifolds whose dimensions are congruent to n mod 2 by a result of Conner and Floyd [6,Thm. V.3.2]. By excision we have The claim follows once we know that the restriction of an orientation cover for X to each Y i is an orientation cover for that component. This is the content of the following lemma. Note that each component Y is orientable if and only if its orientation cover is trivial. According to the theorem of Conner and Floyd mentioned previously, each component Y of X T is orientable if so is X. Lemma 3.12 can therefore be seen as a generalization of this part of their result. Also note that for a smooth T -manifold X Lemma 3.12 is a consequence of the fact that the normal bundle of each component Y of X T is orientable, cf. [14, Cor. 2]: By excision one can restrict from X to a T -stable tubular neighbourhood of Y , and, like the normal bundle, this neighbourhood is orientable if and only if Y is.
Proof. LetX → X be an orientation cover for X andZ → Z its restriction to Z = X T . Note thatZ = (X) T . For each component Y of Z, say of dimension m, let Y → Y be the further restriction. We will show H c m (Ỹ ) − = 0, which proves that Y → Y is an orientation cover of Y : If Y is orientable, this condition ensures that Y is disconnected, and if Y is non-orientable, it shows thatỸ is orientable.
Since the cap product (3.2) is natural with respect to proper maps of spaces, we get a commutative diagram where ι :Z ֒→X and b ∈ H T,c * (Z) − . Note that ι * : H c * (Z) → H c * (X) commutes with the involution τ and therefore preserves the ±1 eigenspaces. Let S ⊂ R be the multiplicative subset of homogeneous polynomials of positive degree. We localize the diagram at S and choose b to be a preimage of o T ∈ S −1 H T * (X) − , which is possible by the localization theorem in equivariant homology (Proposition 2.5, here for homology with closed supports). By the same result and equivariant Poincaré duality, this turns the top and vertical arrows into isomorphisms, hence also the bottom arrow. Now b ∈ S −1 H T,c * (Z) − is a sum of elements, one for each component of Z = X T . The summand b Y corresponding to the component Y can be written in the form

Applications to the orbit structure
We assume throughout the rest of this paper that X is a T -space and that the characteristic of the field k is 0. Recall that the orbit filtration (X i ) has been defined in the introduction.

4.1.
General T -spaces. In Sections 4, 5.1 and 5.2 of [3] we established results about the equivariant cohomology with closed supports and equivariant homology with compact supports of the orbit filtration of a T -space X. All these results have analogues for the other pair of supports, i. e., for cohomology with compact supports and homology with closed supports. Moreover, for a k-homology manifold X, one has another set of analogous results for (co)homology with twisted coefficients. The proofs for the new cases are usually identical to the ones given in [3]. In the case of twisted coefficients, one may alternatively derive them from the decompositions (2.44) and (2.45) and the untwisted result for an orientation cover; see Proposition 4.1 below for an example. We therefore content ourselves by stating the most important results in a more general setting. All results in this section are equally valid for the other pair of supports.
We simplify notation in the following way: For a T -pair (A, B) in a homology manifold X we write H * T (A, B; ℓ) to denote either cohomology with constant coefficients (ℓ = k) or with twisted coefficients (ℓ =k). The same applies to homology and (co)chain complexes. If X is not a homology manifold, then ℓ always means constant coefficients.
Proof. The version for constant coefficients and the usual pair of supports is proved in [3,Prop. 4.2], following the ideas of [5,Sec. 7]. The proof for the other pair of supports is identical. The case of twisted coefficients follows from the untwisted version for an orientation cover and the observation that a non-zero direct summand of a Cohen-Macaulay module is again Cohen-Macaulay of the same dimension.   H T * (X, X i−1 ; ℓ) → H T * (X, X i ; ℓ) → H T * (X, X j ; ℓ), which proves the first claim. Taking X = X j in (4.2), we obtain (4.4) 0 −→ H T * (X i , X i−1 ; ℓ) −→ H T * (X j , X i−1 ; ℓ) −→ H T * (X j , X i ; ℓ) −→ 0. The second claim now follows by induction and the way Ext modules (or dimension and depth/projective dimension) behave with respect to short exact sequences.
The spectral sequence for equivariant cohomology induced by the orbit filtration does not degenerate at the E 1 page in general. Since this page of the spectral sequence is of independent interest, we give it a name.
Proof. The proof of [3, Thm. 5.7] carries over. Only the argument for the equivalence (2) ⇔ (3) has to be slightly modified in the case of twisted coefficients: The involution on an orientation coverX induces one on the Borel constructionX T , and H * T (X;k) = H * (X T ) − in the notation of Section 2.6, and analogously for K. (Note that the decomposition of the cohomology into the ±1 eigenspaces of the involution exists even for spaces that do not satisfy our standing assumptions.) Now one considers the map (4.9) H * T (X;k) = H * T /K (X K ) − → H * (X K ) − = H * K (X;k). and applies the Leray-Hirsch argument as used in [3] to the −1 eigenspaces.

Homology manifolds.
In this section we assume that X is a k-homology manifold. Theorem 4.6 and [3, Thm. 4.8] may be combined with Poincaré duality and Poincaré-Alexander-Lefschetz duality in various ways. The following result is an example of this. Recall that X i = X \ X i . Corollary 4.9. The following spectral sequences are isomorphic from the E 2 page on: . Proof. Let n = dim X. By Theorem 3.7, the first spectral sequence is isomorphic, from the E 1 page on, to the spectral sequence  Proof. Since only finitely many isotropy groups occur in X, there is a subtorus K ⊂ T such that X K i−1 = X i \X i−1 . Hence our claim reduces to the Thom isomorphism from Proposition 3.11.
The following result generalizes a theorem of Duflot [14, Thm. 1] concerning smooth actions on differential manifolds. More than the extension to continuous actions on homology manifolds, our main insight is that Duflot's result follows by equivariant Poincaré-Alexander-Lefschetz duality from Proposition 4.3 which is valid for all T -spaces. where all columns come from the long exact sequence for Ext, applied to some row of (5.1). We have used Proposition 4.4 to obtain the zero entries. As a consequence, ker δ i = p −1 i (E i (H T * (X i+1 ))) (5.5) and im δ i−1 = ker p i . H i (AB * (X)) = ker δ i im δ i−1 ∼ = E i (H T * (X i+1 )) ∼ = E i (H T * (X)). The last isomorphism follows from Proposition 4.4 and the short sequence ) → E i+1 (H T * (X, X i+1 )) = 0. This completes the proof.