A fundamental class for intersection spaces of depth one Witt spaces

By a theorem of Banagl-Chriestenson, intersection spaces of depth one pseudomanifolds exhibit generalized Poincar\'{e} duality of Betti numbers, provided that certain characteristic classes of the link bundles vanish. In this paper, we show that the middle-perversity intersection space of a depth one Witt space can be completed to a rational Poincar\'{e} duality space by means of a single cell attachment, provided that a certain rational Hurewicz homomorphism associated to the link bundles is surjective. Our approach continues previous work of Klimczak covering the case of isolated singularities with simply connected links. For every singular stratum, we show that our condition on the rational Hurewicz homomorphism implies that the Banagl-Chriestenson characteristic classes of the link bundle vanish. Moreover, using Sullivan minimal models, we show that the converse implication holds at least in the case that twice the dimension of the singular stratum is bounded by the dimension of the link. As an application, we compare the signature of our rational Poincar\'{e} duality space to the Goresky-MacPherson intersection homology signature of the given Witt space. We discuss our results for a class of Witt spaces having circles as their singular strata.


Introduction
The method of intersection spaces has been introduced by Banagl in [2,3] to provide a spatial perspective on Poincaré duality for singular spaces. Following Banagl's original idea, such a theory should assign to a given singular space X a family of intersection spaces -namely, spaces I p X parametrized by a so-called perversity function p -in such a way that, when X is a closed, oriented n-dimensional pseudomanifold, generalized Poincaré duality H * (I p X; Q) ∼ = H n− * (I q X; Q) holds for the reduced singular (co)homology groups across complementary perversities p and q. Recall that this generalized form of Poincaré duality involving perversity functions originates from the well-established intersection homology theory IH p * (X; Q) of Goresky-MacPherson [18,19].
The purpose of this paper is to upgrade the middle perversity intersection space of a depth one Witt space to a rational Poincaré duality space. The fundamental class will be constructed by a single cell attachment in the top degree. Before discussing our results (see Section 2), which build on work of Klimczak [22] and Banagl-Chriestenson [7], we give in the following an outline of several existing results in the theory of intersection spaces.
In comparison with Banagl's intersection space homology theory HI p * (X; Q) = H * (I p X; Q) one can observe that Goresky-MacPherson's intersection homology, and also Cheeger's L 2 cohomology of Riemannian pseudomanifolds [12,13,14], arise from certain intermediate algebraic chain complexes rather than from spatial modifications. The intersection space construction itself modifies a space only near its singular strata: Loosely speaking, each singularity link is replaced by a spatial approximation that truncates its homology in a degree dictated by the perversity function. Motivation for using such spatial homology truncations (Moore approximations) of the links comes from a similar behavior of intersection homology groups in the case of isolated singularities (see Section 2 in [2]). As it turns out, the homology theories HI p * and IH p * (as well as their corresponding cohomology theories) are in general not isomorphic. However, at least in the case of singular Calabi-Yau 3-folds, they are related via mirror symmetry (see [3]).
Whenever intersection spaces exists, they serve as a source of desirable features that are not available in the context of intersection homology. For instance, intersection space cohomology comes automatically equipped with a perversity internal cup product. Moreover, addressing a problem suggested by Goresky and MacPherson in [20], intersection spaces provide an approach to construct generalized homology theories for singular spaces, like intersection K-theory (see Chapter 2.8 in [3], as well as [25]). Naturally, the advantages of the theory HI p * over IH p * come at the cost that the existence of intersection spaces which satisfy generalized Poincaré duality is far from granted. For pseudomanifolds with isolated singularities, intersection spaces do always exist, and their duality theory is well-studied [3]. However, for pseudomanifolds with more complicated singularities, the implementation of intersection space theory becomes rapidly more involved. This is already evident in the case of arbitrary two strata pseudomanifolds: Surprisingly, even if an intersection spaces can be constructed, the existence of a generalized Poincaré duality isomorphism turns out to be obstructed in general. As discovered by Banagl and Chriestenson [7], the failure of duality is precisely measured by local duality obstructions, which are certain characteristic classes associated to the link bundle over the singular stratum of the pseudomanifold. These obstruction classes are abstractly definable for fiberwise truncatable fiber bundles, and they vanish for product bundles and certain flat bundles, but not for generally twisted bundles. For some specific three strata pseudomanifolds with bottom stratum a point, a duality result for intersection spaces has been established in [4]. By developing an inductive method of intersection space pairs, Agustín and de Bobadilla have recently proposed in [1] a quite general construction of intersection spaces for pseudomanifolds of arbitrary stratification depth, at least when the link bundles can be compatibly trivialized. However, an obstruction theory for generalized Poincaré duality of intersection space pairs is not known (compare problem (6) in Section 2.6 in [1]).
In view of the difficulties that arise in constructing intersection spaces, it seems beneficial to study intersection space homology by means of techniques that avoid constructing the intersection space itself. Notable alternative approaches are via L 2 theory [8], via linear algebra [17], via sheaf theory [1], and via differential forms [5,15]. The approach via L 2 theory applies to two strata pseudomanifolds having trivial link bundle. As for the linear algebra approach, Geske [17] constructs socalled algebraic intersection spaces on the chain level. His construction is based on a generalization of Moore approximations to multiple degrees that might in general not be realizable as a spatial modification of a tubular neighborhood of the singular set. While algebraic intersection spaces that satisfy generalized Poincaré duality exist for a large class of Whitney stratified pseudomanifolds, they are remote from the spatial concept in that they do not exhibit a natural cup product on cohomology, and the generalized local duality obstructions of Banagl-Chriestenson can be shown to vanish for an appropriate choice of the local intersection approximation (see Theorem 4.10 in [17]). On the level of homology, algebraic intersection spaces turn out to be non-isomorphic to the intersection space pair approach of Agustín and de Bobadilla (see Section 6 in [17]). Note that in [1], Agustín and de Bobadilla pursue a sheaf theoretic approach that is inspired by work of Banagl, Maxim and Budur [9,10,6,23]. Namely, they associate to intersection space pairs certain constructible sheaf complexes on the original pseudomanifold satisfying axioms analogous to those of the intersection chain complex in intersection homology theory [19]. Then, in Theorem 10.6 in [1] they show that so-called general intersection space complexes give rise to generalized Poincaré duality for two strata pseudomanifolds. Finally, concerning the differential form approach, we note the special and important feature that wedge product of forms followed by integration induces a canonical non-degenerate intersection pairing on cohomology in analogy with ordinary de Rham cohomology.
Returning to the original spatial approach, Klimczak [22] pursues the idea to realize Poincaré duality for intersection spaces by cup product followed by evaluation with a fundamental class rather than only showing equality of complementary Betti numbers. Let us consider the important case of a Witt space X with isolated singularities. In this case, the intersection spaces associated to the lower middle and upper middle perversities m and n exist, and can be chosen to be equal, IX = I m X = I n X. By a Klimczak completion of IX we shall mean a rational Poincaré duality space of the form IX = IX ∪ e n , where n denotes the dimension of X. If IX admits a Klimczak completion, then the fundamental class in H n ( IX) arises from the newly attached top-dimensional cell e n , and an easy Mayer-Vietoris argument implies that the Betti numbers of IX and IX agree in degrees 1, . . . , n−1.
In [22] it is shown that Klimczak completions exist for compact Witt spaces having isolated singularities with simply connected links. Moreover, the rational homotopy type of a Klimczak completion is determined by the intersection space whenever a theorem of Stasheff [26] is applicable. In view of future applications it seems interesting to invoke rational surgery theory to realize Klimczak completions by manifolds.
In this paper we study Klimczak completions for middle perversity intersection spaces of compact depth one Witt spaces. Future study will have to clarify the obstructions to the existence of Klimczak completions in the case of higher stratification depth.
The paper is structured as follows. Section 2 presents our main results in case of a two strata Witt space. In Section 3 we list some notation that will be used throughout the paper. Section 4 and Section 5 contain the proofs of our main technical results. Finally, in Section 6, we prove our main results for depth one Witt spaces, and illustrate them in an example.
Acknowledgements We are grateful to Markus Banagl and Timo Essig for several discussions. Also, we thank Manuel Amann for a brief correspondence leading to Example 5.3.
At the time this work was completed, the author was a JSPS International Research Fellow (Postdoctoral Fellowships for Research in Japan (Standard)).

Statement of results
In this paper we study Klimczak completions for middle perversity intersection spaces of compact depth one Witt spaces. For this purpose, we adopt the framework of Thom-Mather stratified spaces as presented in Section 8 of [7]. For simplicity, we consider in the following a two strata Witt space X with singular stratum B. Then, the Thom-Mather control data induce a possibly twisted smooth fiber bundle E → B with fiber the link L of B such that the complement of a suitable tubular neighborhood of B in X is a smooth n-manifold M with boundary ∂M = E. In this setting, as explained in Section 10 of [7], the intersection spaces I m X and I n X associated to the lower middle and upper middle perversities m and n exist and can be chosen to be equal, IX = I m X = I n X, provided that the fiber L admits an equivariant Moore approximation f < : L < → L of degree ⌊ 1 2 (dim L + 1)⌋ (with respect to a suitable structure group for E → B). In view of Theorem 9.5 in [7] one might speculate that vanishing of Banagl-Chriestenson's local duality obstructions for E → B is in some way related to existence of a Klimczak completion for IX because both assumptions imply Poincaré duality for the Betti numbers of IX. In this context, a central role is played by the truncation cone, cone(F < ), the mapping cone of the fiberwise truncation F < : ft < E → E induced by f < . Namely, the local duality obstructions of the bundle E → B vanish if and only if all (n − 1)-complementary cup products in H * (cone(F < )) vanish, where n denotes the dimension of X. Moreover, when B is a point and L is simply connected, then the construction of the Klimczak completion for the intersection space IX = cone(F < )∪ ∂M M is implemented in [22] (see Section 3.2.1 and also Proposition 3.11 therein) as follows. In a first local step, an n-cell e n is attached to the truncation cone to produce a Poincaré duality pair (cone(F < ) ∪ e n , E), which is then in a second global step glued to the regular part (M, ∂M ) of X to yield the desired rational Poincaré duality space IX. Generalizing to an arbitrary singular stratum B and arbitrary link L, our Theorem 4.1 states that a Poincaré duality pair of the form (cone(F < ) ∪ e n , E) exists if and only if the rational Hurewicz homomorphism of the truncation cone in degree n − 1, is surjective. Moreover, we show that the local duality obstructions for E → B vanish necessarily in that case, which reveals some part of their homotopy theoretic nature. As a counterpart of Theorem 9.5 in [7], our Theorem 4.1 implies the following Theorem 2.1. Let (X, B) be a compact two strata Witt space of dimension n ≥ 3. Assume that the link L admits an equivariant Moore approximation f < : L < → L of degree ⌊ 1 2 (dim L + 1)⌋. If the rational Hurewicz homomorphism of the associated truncation cone (see (1)) is surjective in degree n − 1, then the middle perversity intersection space IX admits a Klimczak completion.
More generally, our method applies to depth one Witt spaces with more than one singular stratum (see Theorem 6.3(a)). Note that when B is a point and L is simply connected, we recover Klimczak's original result. Recall that the argument in [22] uses the rational Hurewicz theorem to show that the rational Hurewicz homomorphism of the truncation cone is always surjective in the relevant degree. We do not need to assume that the link is simply connected by employing the results of [27] for constructing Moore approximations for arbitrary path connected cell complexes.
According to Theorem 4.1, our surjectivity condition on the rational Hurewicz homomorphism is sufficient for the local duality obstructions to vanish. However, even in the case of a globally trivial link bundle, we do in general not know whether the converse implication is also true. Nevertheless, under the additional assumption that the truncation cone is simply connected, the converse implication can be analyzed further by means of minimal Sullivan models from rational homotopy theory (see Corollary 5.7). In particular, in view of Theorem 4.1, an important consequence of our Theorem 5.1 is the following Theorem 2.2 (compare Theorem 6.3(b)). Let (X, B) be a compact two strata Witt space of dimension n ≥ 3. Assume that the link L is path connected, and admits an equivariant Moore approximation f < : L < → L of degree ⌊ 1 2 (dim L + 1)⌋. Suppose that cone(F < ), the associated truncation cone, is simply connected. If The assumption that the truncation cone is simply connected is valid in many cases of practical interest -for instance, whenever the link has abelian fundamental group as pointed out in Example 4.15, or when the link bundle is trivial, see Example 4.16.
If the dimension of the Witt space X is of the form n = 4d, then it is natural to study the symmetric intersection form H 2d ( IX) × H 2d ( IX) → Q of a Klimczak completion IX. In accordance with the results of Section 11 in [7], we can compare it to the Goresky-MacPherson-Siegel intersection form IH 2d (X) × IH 2d (X) → Q on middle-perversity intersection homology (see Section I.4.1 in [24]) as follows.
Theorem 2.3 (compare Theorem 6.3(c)). Let (X, B) be a compact two strata Witt space of dimension n = 4d. Suppose that the rational Hurewicz homomorphism of the truncation cone (see (1)) is surjective in degree n − 1. Then, the Witt element w HI ∈ W (Q) induced by the symmetric intersection form H 2d ( IX)×H 2d ( IX) → Q of the Poincaré duality space IX equals the Witt element w IH ∈ W (Q) induced by the Goresky-MacPherson-Siegel intersection form IH 2d (X) × IH 2d (X) → Q on middle-perversity intersection homology. In particular, it follows that the two intersection forms have equal signatures.
We point out that our proof of Theorem 6.3(c) exploits massively the existence of a fundamental class for IX, which enables us to invoke Novikov additivity for Poincaré duality pairs (see Lemma 3.4 in [22]). On the other hand, lacking the existence of a fundamental class under the assumption that the local duality obstructions vanish, the argument of Banagl-Chriestenson in Section 11 in [7] requires an involved construction of an abstract, non-canonical symmetric intersection form for IX. It seems to be an interesting problem to compare intersection forms of Klimczak completions to intersection forms that arise from the differential form approach [5].
In Section 6.2 we provide a class of examples of depth one Witt spaces with twisted link bundles for which our Theorem 6.3 applies. For further examples concerning the existence of equivariant Moore approximations in general we refer to Sections 3 and 12 in [7].

General notation
We collect some general notation that will be used throughout the paper. By a pair of spaces (X, A) we mean a topological space X together with a subspace A ⊂ X. A pointed pair of spaces (X, A, x 0 ) is a pair of spaces (X, A) together with a basepoint Let D p = {x ∈ R p ; x 2 1 + · · · + x 2 p ≤ 1} denote the closed unit ball in Euclidean p-space R p , and S p−1 := ∂D p the standard (p − 1)-sphere. We also fix s 0 = 1 ∈ S 0 ⊂ S p−1 as a basepoint.
Given a pointed pair of spaces (X, A, x 0 ), the Hurewicz map in degree n ≥ 1 is For a pair of spaces (X, A), we will denote by H i (X, A) and H i (X, A) the ith homology and cohomology groups with rational coefficients, respectively. Using the canonical identifications H i (X, A) = H i (X, A; Z) ⊗ Z Q and H i (X, A) = Hom Z (H i (X, A; Z), Q), we will also write Hur n * = Hur n ⊗ Z Q : π n (X, A, x 0 ) ⊗ Z Q → H n (X, A) and Hur * n = Hom Z (Hur n , Q) : H n (X, A) → Hom Z (π n (X, A, x 0 ) , Q).
between topological spaces, we define the homotopy pushout of f and g (see Section 2 in [7]) to be the topological space X ∪ A Y defined as the quotient of the disjoint union A × [0, 1] ⊔ X ⊔ Y by the smallest equivalence relation generated by {(a, 0) ∼ f (a)| a ∈ A} ∪ {(a, 1) ∼ g(a)| a ∈ A}. In particular, if X = pt is the one-point space, then pt ∪ A Y = cone(g : A → Y ) = cone(g) is just the mapping cone (the homotopy cofiber) of A

Trucation cones
Before stating Theorem 4.1, the main result of this section (see Section 4.3 for the proof), we explain the necessary notation taken from [7]. Throughout this section, let π : E → B be a (locally trivial) fiber bundle of closed manifolds with closed manifold fiber L and structure group G such that B, E and L are compatibly oriented. In our applications in Section 6, π will arise as a link bundle of a depth one pseudomanifold X, where we utilize the setting of Thom-Mather stratified pseudomanifolds that is considered in the work of Banagl and Chriestenson (see Section 8 in [7]).
Recall from Definition 3.2 in [7] that a G-equivariant Moore approximation to L of degree r is a G-space L <r together with a G-equivariant map L <r → L that induces isomorphisms H i (L <r ) ∼ = H i (L) in degrees i < r, and such that H i (L <r ) = 0 in degrees i ≥ r. We assume that H i (L) = 0 for i = min{k, l}, . . . , max{k, l} − 1, and that the fiber L possesses a G-equivariant Moore approximation of degree k. Equivalently, there is a map f < : L < → L which is a G-equivariant Moore approximation to L both of degree k and of degree l. This situation is of interest in the important case that π is the link bundle of a two strata Witt space, and p = m and q = n are the lower middle and upper middle perversities defined by m(s) = ⌊s/2⌋ − 1 and n(s) = ⌈s/2⌉ − 1 for all s ∈ {2, 3, . . . }, respectively (see Section 10 of [7]). In this case it follows that k = l = (c + 1)/2 for c odd, and min{k, l} = c/2 and max{k, l} = c/2 + 1 for c even.
Following the discussion leading to Definition 6.1 in [7], we can consider the induced fiberwise truncation (both of degree k and of degree l) Recall that ft < E is the total space of the fiber bundle π < : ft < E → B obtained by replacing the fiber L of π with the fiber L < (by means of the G-action), and The mapping cone of the fiberwise truncation F < plays a central role in the theory. In fact, according to Definition 9.1 in [7] the perversity p and perversity q intersection spaces of a two strata pseudomanifold X n are given by where M is the n-dimensional manifold with boundary ∂M = E that arises as the complement of a suitable tubular neighborhood of B in X. (Note that IX is actually defined as the mapping cone of the composition of F < with the inclusion E ⊂ M , but this space can be seen to be homeomorphic to cone(F < ) ∪ E M .) Furthermore, we note that in our setting, the general definition of the local duality obstructions O * (π, k, l) (see Definition 6.8 in [7]) reduces in degree i to which is a linear subspace of H n−1 (cone(F < )). In particular, note that we have O i (π, k, l) = 0 if and only if x ∪ y = 0 in H n−1 (cone(F < )) for all x ∈ H i (cone(F < )) and y ∈ H n−1−i (cone(F < )). Since the truncation cone cone(F < ) is path connected (compare Lemma 4.14), we have O i (π, k, l) = 0 for all i / ∈ {1, . . . , n − 2}.
Theorem 4.1. Let n ≥ 3 be an integer. Let π : E n−1 → B be a fiber bundle of closed manifolds with closed manifold fiber L and structure group G such that B, E and L are compatibly oriented. Suppose that B admits a good open cover (this holds whenever B is smooth or at least PL). Let p and q be complementary perversities, and set k = c − p(c + 1) and l = c − q(c + 1), where c = dim L. Suppose that the fiber L possesses a G-equivariant Moore approximation f < : L < → L which is both of degree k and of degree l. Let F < : ft < E → E denote the induced fiberwise truncation of π. Then, the following statements are equivalent: (i) There exists an attaching map induced by the inclusion E ⊂ cone(F < ) is contained in the image of the rational Hurewicz homomorphism Furthermore, if either of the above statements holds, then O i (π, k, l) = 0 for all i.
does not depend on the choice of the attaching map φ in the following sense. In degrees r = 0 and such that for every attaching map φ there is the following commutative diagram: To achieve this, we assume that the spaces B, L, and L < carry CW structures, and that the map f < : L < → L is cellular. Then, note that E and ft < E inherit natural CW structures in such a way that the induced fiberwise truncation F < : ft < E → E is cellular. Thus, the truncation cone cone(F < ) has a cell structure. Finally, by applying the cellular approximation theorem to the map φ in Lemma 4.4, we may without loss of generality assume that the map φ in statement (i) of Theorem 4.1 is cellular. Hence, it follows that (cone(F < ) ∪ φ D n , E) is a CW pair.

4.1.
The rational Hurewicz homomorphism. The following lemma is a slight modification of Lemma 3.8 in [22, p. 248], and will be used in the proofs of Theorem 4.1 (see Section 4.3) and Theorem 6.3.
Lemma 4.4. Let n ≥ 3 be an integer. Let φ : (S n−1 , s 0 ) → (X, x 0 ) be a map of pointed spaces, and let X φ = cone(φ) denote the associated mapping cone. Then for any class x ∈ H n−1 (X) the following statements are equivalent: (ii) The class x ∈ H n−1 (X) lies in the image of the connecting homomorphism Proof. Consider the commutative diagram Hur n * ∂ n Hur n−1 * ∂ n By construction of X φ we have a map of pointed pairs and statement (ii) follows. Conversely, if statement (ii) holds, then there exists q ∈ Q such that and statement (i) follows.
Remark 4.5. By passing to the abelianization π 1 (X, x 0 ) → π 1 (X, x 0 ) ab , we can state Lemma 4.4 as well for n = 2. However, the integer n will arise in our applications in Section 6 as the dimension of depth one pseudomanifolds. As their singular strata are required to have codimension at least 2 (see Section 6.1) and the case of point strata is covered by [22], we will generally assume that n ≥ 3 throughout the paper.

4.2.
Rational Poincaré duality pairs. We recall the fundamental concept of Poincaré duality pairs of spaces (compare Section 3.1 in [22] and Section II.2 in [11]). In this paper we do not require cell structures on our spaces, but see Remark 4.3. is an isomorphism for all r ∈ Z. Any such class a ∈ H n (A, B) will be called an orientation class for (A, B). In the following, a Poincaré duality pair of the form A = (A, ∅) will be called a Poincaré space.
We state our main technical result, which is a careful extension of Lemma 3.7 in [22, p. 247]. Our purpose is to cover also the case of depth one pseudomanifolds having non-isolated singular strata as considered in [7].
is bijective, and let X = cone(f ) denote the mapping cone of f . Suppose that for every r ∈ Z the following conditions hold: (1) The inclusion Z ⊂ X induces a surjective homomorphism H r (Z) → H r (X).
(2) The rational vector spaces H r (Y ) and H n−r−1 (X) have the same rank. Then, for any space X φ = cone(φ) given by the mapping cone of some map φ : S n−1 → X the following statements are equivalent: Let us show that η and ξ are isomorphisms. First, note that η and ξ are surjective by assumption (1) applied for r = n and r = n − 1, respectively. Next, observe (2) applied for r = 0. All in all, we have shown that η and ξ are isomorphisms. Finally, the five lemma implies that the map ζ in the above diagram is an isomorphism as well, and the equivalence (ii) ⇔ (iii) follows.
Remark 4.9. For future reference we note that assuming either (ii) or (iii) in Proposition 4.8, we can show that the connecting homomorphisms in the diagram Here, the isomorphism H n (X φ , X) ∼ = H n (X φ /X) holds by Proposition 2.22 in [21, p. 124]. Note that for any map α : C → D, the pair (cone(α), D) is a good pair, that is, D is a nonempty closed subset of cone(α) that is a deformation retract of some neighborhood in cone(α).) is an isomorphism for every r ∈ Z. (It is clear by assumption (1) that all rational homology groups of X φ have finite rank, so that (X φ , Z) will then be a Poincaré duality pair of dimension n according to Definition 4.6.) In general, observe that every element α ∈ H n (X φ , Z) gives by Proposition 1.1.4(ii) in [11, p. 4] rise to a commutative diagram If we specialize to α = [e φ ], then the lower horizontal homomorphism − ∩ [Z] is an isomorphism because Z is a Poincaré space of dimension n − 1. Thus, in order to show that the upper horizontal row of the above diagram is an isomorphism, it suffices to verify that for every r ∈ Z, the following two assertions hold: In view of assumption (1), our claim (a) is in fact equivalent to showing that the inclusion X ⊂ X φ induces a surjective homomorphism H r (X) → H r (X φ ). We compute Thus, for r = n, the claim follows from the exactness of For r = n we consider the exact sequence Since the connecting homomorphism ∂ n is injective by Remark 4.9, we conclude that the homomorphism H n (X) → H n (X φ ) is surjective.
Remark 4.10. Suppose that either (ii) or (iii) holds in Proposition 4.8. Then, for future reference, we note that an inspection of the homology long exact sequence of the pair (X φ , X) combined with the above information implies that the inclusion X ⊂ X φ induces an isomorphism H r (X) rank.
In view of assumption (2) and Remark 4.10, it suffices to show that Observe that for all r ∈ Z, −→ H r (X φ ) for r = n − 1, the claim follows for r / ∈ {n − 1, n} from the five lemma applied to the above diagram. We check the remaining cases: • In the case r = n−1, we note that in the above diagram the homomorphism H n−1 (Z) → H n−1 (X) is surjective by assumption (1), and the homomorphism H n−1 (X) → H n−1 (X φ ) is surjective by Remark 4.10. Thus, we obtain the following simplified commutative diagram with exact rows: Finally, as the right vertical arrow H n−2 (X) → H n−2 (X φ ) is an isomorphism by Remark 4.10, the five lemma yields • In the case r = n, we consider the following portion of the homology exact sequence of the pair (X, Z): Note that H n−1 (X) ∼ = H 0 (Y ) and H n (X) ∼ = 0 by assumption (2). As the homomorphism H n−1 (Z) → H n−1 (X) is surjective by assumption (1), we obtain On the other hand, H n (X φ , Z) ∼ = Q according to Remark 4.9.
Then, for future reference, we note that the inclusion of pairs (X, Z) ⊂ (X φ , Z) induces an isomorphism H r (X, Z)

4.3.
Proof of Theorem 4.1. Following Definition 6.2 in [7], we define the fiberwise cotruncation ft ≥ E (both of degree k and of degree l) of the fiber bundle π : E → B as the homotopy pushout of the diagram In Definition 6.3 of [7] an auxiliary space Q ≥ E together with a structure map C ≥ : Q ≥ E → E is introduced. In Section 6 of [7] the following properties of Q ≥ E are derived (the assumption that B admits a good cover is needed to deduce item (a) and (c) because they are both based on Proposition 6.5 in [7]): (a) By Lemma 6.6 in [7], the map C ≥ : E → Q ≥ E induces for every r ∈ Z a surjective homomorphism C ≥ * : H r (E) → H r (Q ≥ E). (b) By the proof of Lemma 6.6 in [7], there exists a diagram E cone(F < ) which commutes on the nose. In particular, Q ≥E and cone(F < ) are homotopy equivalent spaces. (c) According to Proposition 6.7 in [7], the rational vector spaces H r (ft < E) and H n−r−1 (Q ≥ E) have for every r ∈ Z the same rank. Let us prove the equivalence (i) ⇔ (ii) claimed by Theorem 4.1. For this purpose, we first apply the equivalence (i) ⇔ (iii) of Proposition 4.8 to the map f : Y → Z given by F < : ft < E → E. Note that the space Z = E is a Poincaré space with orientation class given by [Z] = [E] ∈ H n−1 (Z) because E is a closed oriented manifold. Moreover, note that the map F < : ft < E → E induces a bijection F < : H 0 (ft < E) → H 0 (E). Taking X = cone(F < ) to be the mapping cone of the map f = F < , we see the condition (1)  We continue to use the notation f : Y → Z for the map F < : ft < E → E, and write X = cone(F < ) for the mapping cone of f = F < . From now on we suppose that either of the equivalent statements of Theorem 4.1 holds. Fix r ∈ {1, . . . , n − 2}. We have to show that the local duality obstruction O n−1−r (π, k, l) vanishes. By statement (i) there is a map φ : S n−1 → X and a lift [e φ ] ∈ H n (X φ , Z) of the orientation class [Z] ∈ H n−1 (Z) such that (X φ , Z) is a rational Poincaré duality pair of dimension n with orientation class [e φ ]. By Theorem I.2.2 in [11, p. 8], there is the following commutative diagram: Factoring the inclusion Z ⊂ X φ as Z ⊂ X ⊂ X φ , and using the inclusion of pairs (X, Z) ⊂ (X φ , Z), we can extend the diagram to a commutative diagram as follows: We claim that the previous diagram is part of a commutative diagram of the form In fact, the commutative square (I) exists by property (b) of Q ≥ E, and the commutative square (II) is given by the square (II) ′ in the following commutative diagram, which is induced by the canonical map of pairs ( f , f ) : (cone(Y ), Y ) → (X, Z): is an isomorphism because cone(Y ) is contractible, and r = n − 1.) All in all, we obtain a commutative diagram f * After applying the functor Hom Q (−, Q), we conclude from Proposition 6.9 in [7] that O n−1−r (π, k, l) vanishes. (Moreover, by the same proposition, the isomorphism H r (Q ≥ E) → H n−r−1 (Y ) is uniquely determined by commutativity of the above diagram. Consequently, the isomorphism D φ r does actually not depend on φ, and is thus the desired isomorphism D φ r = D r used in Remark 4.2.) This completes the proof of Theorem 4.1.

4.4.
Connectivity of truncation cones. In order to be able to apply rational homotopy theory to truncation cones in Section 5, we study the connectivity properties of truncation cones in the present section.
The following result shows that the reduced homology of truncation cones vanishes in low degrees.
Lemma 4.12. If B admits a good cover (e.g., if B is smooth or at least PL), then the homology of cone(F < ) satisfies H i (cone(F < )) = 0 for i < max{k, l}.
Proof. We employ the local to global technique based on precosheaves as presented in Section 4 of [7], and assume familiarity with the notation and definitions used therein.
Our argument requires a slight modification of Proposition 4.4 of [7] that applies to finite sequences of δ-compatible morphisms between precosheaves instead of inifinite sequences (see Proposition 4.13 below). For this purpose, consider an open cover U of the topological space B. Let τ U denote the category whose objects are unions of finite intersections of open sets in U, and whose morphisms are inclusions. On the product category τ U × τ U, consider the functors ∩, ∪ : τ U × τ U → τ U induced by intersection and union of open sets, respectively. For i = 1, 2, projection to the i-th factor determines a projection functor p i : τ U × τ U → τ U. We also need for i = 1, 2 the natural transformations j i : p i → ∪ and ι i : ∩ → p i induced by the inclusions U, V ⊂ U ∪ V and U ∩ V ⊂ U, V , respectively. Now consider a finite sequence F i = F 0 , . . . , F N of precosheaves on B. Slightly modifying Definition 4.3 in [7], we say that the finite sequence F i satisfies the U-Mayer-Vietoris property if there are natural transformations of functors on τ U × τ U, such that for every pair of open sets U, V ∈ τ U the following sequence is exact: (Note that the only difference to Definition 4.3 in [7] is that our sequence ends to the left with the term Next we state our adaption of Proposition 4.4 in [7] to finite sequences of δcompatible morphisms between precosheaves. (The only change in the proof is that the five lemma is applied to a commutative ladder that ends on the left with the homomorphism f N (V j ) : Recall that we have a fiber bundle π < : ft < E → B with fiver L < . Consider also the fiber bundle ρ : cyl(F < ) → B with fiber cyl(f < ), and note that there are morphisms of fiber bundles ft < E → cyl(F < ) → E restricting to fiberwise maps Fix a good cover U of B, and set N = max{k, l} − 1. We apply Proposition 4.13 to the precosheaves F i and G i , 0 ≤ i ≤ N , given on open sets U ∈ τ U by F i (U ) = H i (ρ −1 (U ), π −1 < (U )) and G i (U ) = 0, and the morphisms of precosheaves f i : The following result provides a proof of simple connectivity of truncation cones, at least under a mild hypothesis on the underlying Moore approximation. Simple connectivity of the truncation cone will enable us to invoke the machinery of rational homotopy theory in Section 5.
Proof. Since f < : L < → L is a Moore approximation of positive degree, the induced map H 0 (L < ) → H 0 (L) is an isomorphism. Hence, cone(f < ), the mapping cone of f < , is path connected. As the bundle morphism F < : ft < E → E restricts on fibers to copies of the map f < , it follows that cone(F < ) is path connected.
Example 4.15. If L is path connected and has abelian fundamental group, then any equivariant Moore approximation f < : L < → L to L of degree ≥ 2 induces automatically a surjection on fundamental groups. In fact, by naturality of the Hurewicz homomorphism we have for any basepoint x 0 ∈ L < the commutative diagram Hur f < * Hur f < * Note that the left vertical map is surjective and the right vertical map is an isomorphism because in degree one, Hurewicz maps are abelianization maps. Hence, the claim follows because f < * : Example 4.16. If L is a path connected cell complex and the fiber bundle E → B is trivial, then there exists for any d ≥ 2 an equivariant Moore approximation f < : L < → L to L of degree d which induces a surjection on fundamental groups. Indeed, since the bundle E → B is trivial, the structure group G can be chosen to be trivial. Thus, by the results of [27], L admits a cellular Moore approximation f < : L < → L of degree d ≥ 2 in such a way that f < restricts to the identity on (d − 1)-skeleta. In particular, the map induced by f < on fundamental groups is surjective.

Rational homotopy theory
Theorem 4.1 reveals a natural relation between the rational Hurewicz homomorphism of the truncation cone and the condition that in some degree all complementary cup products in the reduced cohomology ring of the truncation cone vanish. In this section we study the homotopy theoretic ramifications of this cohomological vanishing condition (see e.g. Corollary 5.7).
The main purpose of this section is prove (see Section 5.5) the following Theorem 5.1. Let n ≥ 3 be an integer. Let π : E n−1 → B be a fiber bundle of connected closed manifolds with connected closed manifold fiber L and structure group G such that B, E and L are compatibly oriented. Suppose that B admits a good open cover (this holds whenever B is smooth or at least PL). Let p and q be complementary perversities, and set k = c − p(c + 1) and l = c − q(c + 1), where c = dim L. Suppose that the fiber L possesses a G-equivariant Moore approximation f < : L < → L both of degree k and of degree l such that f < induces a surjection on fundamental groups. Furthermore, we suppose that (1) n ≤ 3 · max{k, l} − 1, and (2) O i (π, k, l) = 0 for all i. Then, the rational Hurewicz homomorphism of the associated truncation cone, Hur n−1 * : π n−1 (cone(F < ), pt) ⊗ Z Q → H n−1 (cone(F < )), is surjective.

Minimal Sullivan algebras.
Assuming the ground field to be the rationals, we provide some necessary notation from [16].
By a graded (rational) vector space we mean a collection V = {V p } p≥1 of rational vector spaces (see [16, p. 40] and [16, p. 138]). We say that v ∈ V p is an element of V of degree p, and write v ∈ V and |v| = p. As in [16, p. 42], we use the notation As in Example 6 of [16, p. 45], the free graded commutative algebra of V is the quotient algebra ΛV = T V /I, where T V denotes the tensor algebra of V (see Example 4 in [16,p. 45]), and I ⊂ T V denotes the ideal generated by all elements of the form v ⊗ w − (−1) |v|·|w| w ⊗ v, where v, w ∈ V . Following [16, p. 140], we also write ΛV ≥p = Λ(V ≥p ), ΛV <p = Λ(V <p ), etc.
Note that if a Sullivan algebra (ΛV, d) is minimal, then the projection induces according to the discussion in [16, p. 173] a homomorphism ζ : H + (ΛV ) = ker(d : Lemma 5.2. Fix integers r, s, t ≥ 1. Suppose that (ΛV, d) is a minimal Sullivan algebra whose underlying graded vector space V is of the form V = {V p } p≥r , and whose differential d vanishes on all elements of degree ≤ s. Furthermore, suppose that all t-complementary products in H + (ΛV ) vanish, that is, for all classes α ∈ H i (ΛV ), β ∈ H t−i (ΛV ), 1 ≤ i ≤ t − 1, we have α ∧ β = 0 in H t (ΛV ). If t ≤ r + s, then the associated homomorphism ζ : H + (ΛV ) → V is injective in degree t.
Proof. Suppose that [z] ∈ H t (ΛV ) is a class such that ζ([z]) = 0. In particular, z is a cocycle of degree t in (ΛV, d) satisfying z ∈ im d. Thus, since im d ⊂ Λ ≥2 V , z can be written as a rational linear combination of elements of the form v 1 ∧ · · · ∧ v m , where m ≥ 2, and v 1 , . . . , v m ∈ V are nonzero elements such that |v 1 | + · · · + |v m | = t. Since m ≥ 2, it follows from |v 1 |, . . . , |v m | ≥ r and t ≤ r + s that The following example shows that the assumption t ≤ r+s is in general necessary in Lemma 5.2. Then, it is easy to check that (ΛV, d) is a minimal Sullivan algebra with differential d determined by dx = dy = 0 and dz = xy. By construction, the only non-trivial cohomology groups of H + (ΛV ) are H u (ΛV ) (generated by the classes of x and y), H 3u−1 (ΛV ) (generated by the classes of xz and yz), and H 4u−1 (ΛV ) (generated by the class of xyz). Now let r = u, s = 2u − 2, and t = r + s + 1 = 3u − 1. Then, the minimal Sullivan algebra (ΛV, d) satisfies all assumptions of Lemma 5.2 except for t ≤ r + s. Furthermore, the homomorphism ζ : H + (ΛV ) → V is clearly not injective in degree t because V 3u−1 = 0, whereas H 3u−1 (ΛV ) = 0.  , d), the next step of the induction is as follows. Choose cocycles a α ∈ A k+1 and z β ∈ (ΛV ≤k ) k+2 such that Let V k+1 be a vector space with basis {v ′ α } ∪ {v ′′ β } in correspondence with the elements {a α } ∪ {z β }. The (derivational) differential d is extended from ΛV ≤k to ΛV ≤k+1 = ΛV ≤k ⊗ ΛV k+1 by setting dv ′ α = 0 and dv ′′ β = z β . Finally, the map m k : (ΛV ≤k , d) → (A, d) is extended to a cochain algebra morphism has been chosen in such a way that db β = m k z β . Proof. The claim V = {V p } p≥r is part of Proposition 12.2(ii) in [16, p. 145].
In order to show that the differential d vanishes on all elements of degree ≤ 2r−2, it suffices by the inductive construction of d to show that ker H k+2 (m k ) = 0 for k = 2, . . . , 2r − 3. (For r = 2 there is nothing to show since d vanishes on all elements of degree ≤ 2 by construction.) Fix k = 2, . . . , 2r − 3. If k < r, then V = {V p } p≥r implies that (ΛV ≤k ) 0 = Q and (ΛV ≤k ) p = 0 for p > 0. If, however, k ≥ r, then V = {V p } p≥r implies that In any case, we see that (ΛV ≤k ) k+2 = 0, and, in particular, ker H k+2 (m k ) = 0.

5.3.
Commutative cochain algebras for spaces. Recall from Section 10 in [16, p. 115 ff.] that Sullivan has constructed a contravariant functor A P L from the category of topological spaces and continuous maps to the category of commutative cochain algebras and cochain algebra morphisms. Furthermore, the functor A P L has the important property that for any topological space X, the graded algebras H * (X) and H(A P L (X)) are naturally isomorphic (and can hence be identified). Given a path connected topological space X, we can in particular consider the minimal Sullivan model for X, that is, the (unique) minimal Sullivan model m X : (ΛV X , d) ≃ → A P L (X) for the commutative cochain algebra (A, d) = A P L (X). Moreover, recall from Section 5.1 that we can associate to the minimal Sullivan algebra (ΛV X , d) a homomorphism ζ X : In Proposition 5.5 below, we characterize the non-vanishing of the rational Hurewicz homomorphism in terms of rational homotopy theory (compare the proof of Proposition 3.14 in [22, p. 250]).
Proposition 5.5. Let (X, x 0 ) be a simply connected pointed space, and suppose that H * (X) is of finite type (i.e., the rational vector space H r (X) has finite dimension for all r ∈ Z). Then for any integer n ≥ 3 the following statements are equivalent: (i) The Hurewicz homomorphism Moreover, if either of the above statements is satisfied for X, then all (n − 1)complementary cup products in H * (X) vanish.
Proof. We fix a minimal Sullivan model m X : (ΛV X , d) Since (X, x 0 ) is simply connected and H * (X) is of finite type, the corollary to Theorem 15.11 in [16,p. 210] implies that in degree n − 1 > 1 the homomorphism can be identified with the dual Hur * n−1 : H n−1 (X) → Hom Z (π n−1 (X), Q) of the Hurewicz map Hur n−1 : π n−1 (X) → H n−1 (X; Z). Since the extension of scalars functor − ⊗ Z Q is left adjoint to the restriction of scalars functor (along Z ⊂ Q), the latter homomorphism can furthermore be identified with Hom Q (Hur n−1 * (−), Q) : Hom Q (H n−1 (X) ⊗ Z Q, Q) → Hom Q (π n−1 (X) ⊗ Z Q, Q).
Thus, the equivalence (i) ⇔ (ii) follows from the observation that a homomorphism V → W of rational vector spaces is surjective if and only if its dual Hom Q (W, Q) → Hom Q (V, Q) is injective.
Let us suppose that there is a non-vanishing (n − 1)-complementary cup product in H * (X), that is, for some i ∈ {1, . . . , n − 2}, there exist classes α ∈ H i (X) and β ∈ H n−1−i (X) such that 0 = α ∪ β ∈ H n−1 (X). As H * (A P L (X)) ∼ = H * (X) (as graded algebras) and m X : (ΛV X , d) ≃ −→ A P L (X) is a quasi-isomorphism of commutative differential graded algebras, there exist cocycles a ∈ ΛV X of degree i and b ∈ ΛV X of degree n − 1 − i such that 0 = [a ∧ b] ∈ H n−1 (ΛV X ). But then Hence, we have shown that all (n − 1)-complementary cup products in H * (X) must vanish if X satisfies either of the statements (i) and (ii).
Remark 5.6. In the context of Theorem 4.1 we have exploited statement (i) in our proof that the local duality obstructions vanish. Under the assumption that X = cone(F < ) is simply connected (compare Lemma 4.14), Proposition 5.5 provides another proof by means of the minimal Sullivan model of X, based on statement (ii).

5.4.
A rational Hurewicz theorem. One part of Proposition 5.5 states that in a given degree, surjectivity of the rational Hurewicz homomorphism implies that complementary cup products in the reduced cohomology ring vanish. Our purpose is to prove the converse implication under reasonable hypotheses. This is addressed in Corollary 5.7, where the additional assumption is that the considered degree n−1 should not be too large compared to the rational connectedness r of the space X. Note that for n−1 = 2r−1, Corollary 5.7 specializes to a part of the classical rational Hurewicz theorem (for spaces with homology of finite type) because the vanishing condition for complementary cup products is trivially satisfied. This is exactly the implication that is employed in [22] in the context of isolated singularities.
Corollary 5.7. Let (X, x 0 ) be a simply connected pointed space, and suppose that H * (X) is of finite type (i.e., the rational vector space H r (X) has finite dimension for all r ∈ Z). Let n ≥ 3 be an integer such that the following assumptions hold: (1) If r ≥ 2 denotes the smallest positive integer such that H r (X) = 0, then n ≤ 3r − 1.
Then, the rational Hurewicz homomorphism is surjective.
Proof. We consider the commutative cochain algebra (A, d) = A P L (X) and its minimal Sullivan model m X : (ΛV X , d) ≃ −→ A P L (X). Using that the graded algebras H * (X) and H * (A P L (X)) are isomorphic, we obtain from Lemma 5.4 that V X = {V p X } p≥r , and that the differential d vanishes on all elements of degree ≤ 2r − 2, where r ≥ 2 denotes the smallest positive integer such that H r (X) = 0. Next, we apply Lemma 5.2 to the minimal Sullivan algebra (ΛV X , d) and the integers r, s = 2r − 2 and t = n − 1. Note that t ≤ r + s by assumption (1), and all t-complementary cup products in H + (ΛV X ) vanish by assumption (2) because m X is a quasi-isomorphism. Hence, we conclude that the homomorphism ζ X : H + (ΛV X ) → V X is injective in degree t = n − 1. Finally, the claim that the Hurewicz homomorphism Hur n−1 * : π n−1 (X, x 0 ) ⊗ Z Q → H n−1 (X) is surjective follows from implication (ii) ⇒ (i) of Proposition 5.5. 5.5. Proof of Theorem 5.1. Since the manifolds B and L are path connected and the G-equivariant Moore approximation f < : L < → L induces by assumption a surjection on fundamental groups, we conclude from Lemma 4.14 that the truncation cone cone(F < ) is simply connected. Next, we wish to apply Corollary 5.7 to the simply connected pointed space (X, x 0 ) = (cone(F < ), pt). Note that H * (X) is of finite type because the inclusion E ⊂ cone(F < ) induces a surjective homomorphism H r (E) → H r (cone(F < )) for all r ∈ Z (see properties Finally, application of Corollary 5.7 completes the proof of Theorem 5.1.

Intersection spaces and the signature
In Section 6.1 we apply the results of the previous sections to study middle perversity intersection spaces of depth one Witt spaces within the framework of Thom-Mather stratified spaces. In particular, given a depth one Witt space X (see Definition 6.2), Theorem 6.3(a) provides our Hurewicz criterion for the existence of a Klimczak completion IX = IX ∪ e n of the middle perversity intersection space IX. Then, in part (b) of Theorem 6.3, we relate the Hurewicz criterion to the vanishing of the local duality obstructions to Banagl-Chriestenson by showing that they are equivalent if the dimensions of the singular strata are not too big. Moreover, when the dimension n of X is of the form n = 4d, Theorem 6.3(c) implies that the signature of the symmetric intersection form H 2d ( IX) × H 2d ( IX) → Q equals the signature of the Goresky-MacPherson-Siegel intersection form IH 2d (X)× IH 2d (X) → Q on middle-perversity intersection homology. Finally, we illustrate our results by a concrete example in Section 6.2.
6.1. Depth one Witt spaces. Before focusing on depth one Witt spaces and their middle perversity intersection spaces (see Theorem 6.3), we discuss more generally intersection spaces of stratified pseudomanifold of depth 1 along the lines of [7].
First of all, an n-dimensional two strata pseudomanifold is a pair (X, Σ) consisting of a locally compact, second countable Hausdorff space X and a closed connected subspace Σ such that Σ ⊂ X is equipped with a Thom-Mather C ∞ -stratification of X as follows. The regular stratum X \ Σ is a smooth n-manifold that is dense in X, and the singular stratum Σ is a smooth manifold whose codimension in X is at least 2. Moreover, Σ is required to be equipped with so-called Thom-Mather control data (for details, see Section 8 of [7]).
Suppose that (X, Σ) is an n-dimensional two strata pseudomanifold with nonempty singular stratum Σ. Then, as explained at the beginning of Section 9 in [7], the Thom-Mather control data of Σ ⊂ X can be used to construct an open neighborhood U of Σ in X and a smooth (locally trivial) fiber bundle π : E → B with the following properties. The complement M = X \ U is a smooth n-manifold with boundary ∂M = E, and there exists a homeomorphism from the closure We assume that the data (U, π : E → B) have been fixed, and call U a regular neighborhood, and the fiber bundle π : E → B the link bundle of the singular stratum Σ = B. The (non-empty) fiber L of π is called the link of the singular stratum Σ = B.
More generally (see Definition 8.3 in [7]), an n-dimensional stratified pseudomanifold of depth 1 is a tuple (X, Σ 1 , . . . , Σ r ), where the Σ i are mutually disjoint subspaces of X such that for every i = 1, . . . , r, the pair (X \ j =i Σ j , Σ i ) is a two strata pseudomanifold whose singular stratum Σ i has regular neighborhood U (i) , and link bundle π (i) : E (i) → B (i) having link L (i) . The stratified depth 1 pseudomanifold (X, Σ 1 , . . . , Σ r ) is called oriented if the top stratum X \ i Σ i is equipped with an orientation.
Consider an n-dimensional stratified pseudomanifold X = (X, Σ 1 , . . . , Σ r ) of depth 1. Given a perversity p, we proceed to explain the construction of the p-intersection space I p X, which will depend on the choice of equivariant Moore approximations of the link bundles of the singular strata. We denote by c i the dimension of the link L (i) of the singular stratum Σ i , and set k i = c i − p(c i + 1), which is a positive integer. For every i we assume that the link L (i) possesses for some choice of structure group G (i) of the link bundle π (i) a G (i) -equivariant Moore approximation of degree k i , say As explained in the beginning of Section 4, the Moore approximations f Definition 6.1 (compare Definition 9.1 in [7]). The perversity p intersection space I p X of the depth 1 pseudomanifold X = (X, Σ 1 , . . . , Σ r ) is defined as the homotopy cofiber of the composition In other words, From now on, we are concerned with Witt spaces, which are an important class of stratified pseudomanifolds defined by Siegel [24]. Definition 6.2 (see Definition 8.3 in [7]). An oriented depth 1 stratified pseudomanifold (X, Σ 1 , . . . , Σ r ) is called Witt space if the following condition is satisfied. For each 1 ≤ i ≤ r such that the dimension c i of the link L (i) of the singular stratum Σ i is even, we have H c i 2 (L (i) ) = 0.
We specialize to the case of the lower middle perversity p = m and the upper middle perversity q = n. Consider a compact depth one Witt space (X, Σ 1 , . . . , Σ r ). Suppose that f (i) Then, it follows from the homology vanishing condition of Definition 6.2 that f (i) < is also a G (i) -equivariant Moore approximation of L (i) of complementary degree l i = c i − m(c i + 1) = ⌈ 1 2 (c i + 1)⌉. Thus, the resulting intersection spaces I m X and I n X of Definition 6.1 can be chosen to be equal, I m X = I n X.
Let us state the main result of this section. Hur n−1 * : π n−1 (cone(F < )) is surjective for every i, then the middle perversity intersection space IX admits a completion IX = IX ∪ e n to a rational Poincaré duality space by attaching a single n-cell. The rational homotopy type of IX is determined by the intersection space IX whenever a theorem of Stasheff [26] is applicable. (b) Fix an index i ∈ {1, . . . , r}. If the rational Hurewicz homomorphism in part (a) is surjective, then the local duality obstructions O * (π (i) , k i , l i ) vanish. The converse implication holds at least when the truncation cone, cone(F (i) < ), is simply connected (see Lemma 4.14), and (c) Suppose that the dimension of X is of the form n = 4d. Furthermore, suppose that the rational Hurewicz homomorphism in part (a) is surjective for every i, and let IX = IX∪e n be a completion to a Poincaré duality space as provided by par (a). Then, the Witt element w HI ∈ W (Q) induced by the symmetric intersection form H 2d ( IX) × H 2d ( IX) → Q of the Poincaré duality space IX equals the Witt element w IH ∈ W (Q) induced by the Goresky-MacPherson-Siegel intersection form IH 2d (X) × IH 2d (X) → Q on middle-perversity intersection homology (see Section I.4.1 in [24]). In particular, the two intersection forms have equal signatures.
Proof. (a). Fix i ∈ {1, . . . , r}. In the following, we let f (i) : Y (i) → Z (i) denote the map F (i) < : ft < E (i) → E (i) , and set X (i) = cone(f (i) ). By the same argument as in the first half of the proof of Theorem 4.1 (see Section 4.3), we can show that (c). In view of Corollary 10.2 in [7] it suffices to show that in the Witt group of Q, W (Q), the Witt element induced by the intersection form of the Poincaré space IX equals the Witt element induced by the intersection form of the manifold with boundary (M, ∂M ). The proof is analogous to the proof of Corollary 3.10 in [22], which covers the case of isolated singularities. Generalizing to singular strata of arbitrary dimension, we still apply Novikov additivity of Witt elements under gluing of Poincaré duality pairs (see Lemma 3.4 in [22]), where we now have to use the fact that the inclusion Z ⊂ X induces a surjective homomorphism H 2d (Z) → H 2d (X), which holds by condition (1) of Proposition 4.8.
Remark 6.4. Passing from Witt elements to signatures in our proof of Theorem 6.3(c), we recover the statement of Theorem 11.3 in [7], saying that the signature of the intersection form on H 2d ( IX) = H 2d (IX) equals the so-called Novikov signature of the manifold-with-boundary (M, ∂M ). Note that the latter is shown in [7] for the intersection space of a closed oriented two strata Witt space under the assumption that the local duality obstructions of the link bundle vanish. The proof given in [7] is involved, and requires to construct an intersection form of the intersection space IX that is symmetric. As in our proof of Theorem 6.3(c), one also exploits surjectivity of the map H 2d (Z) → H 2d (X), namely in diagram (11.2) in [7], where the map takes the form C ≥k * : H m (∂M ) → H m (Q ≥k E).

An example.
To illustrate a non-trivial case in which Theorem 6.3 applies, we discuss a class of examples of Witt spaces having as singular strata a finite number of circles with twisted link bundles. As input data for our construction, we employ commutative diagrams of the form • L is a closed oriented smooth manifold of dimension c > 0, • f < : L < → L is a Moore approximation of L of degree ⌊ 1 2 (c + 1)⌋, • λ : L → L is an orientation preserving diffeomorphism, • λ < : L < → L < is a homeomorphism, such that • H c 2 (L) = 0 when c is even, • f < induces a surjection π 1 (L < , x 0 ) → π 1 (L, f < (x 0 )) for every x ∈ L < , and • λ N = id L and λ N L = id L for some integer N > 0. Given a commutative diagram as above, we consider the mapping torus E = (L × [0, 1])/(x, 0) ∼ (λ(x), 1) as the total space of a smooth fiber bundle π : E → S 1 with fiber L and structure group G = Z/N Z acting on L via i+N Z → λ i . The bundle π is flat in the sense that we can choose G-valued transition functions that are locally constant. Recall that the integers k = ⌊ 1 2 (c + 1)⌋ and l = ⌈ 1 2 (c + 1)⌉ are associated to the upper-middle perversity n and the lower-middle perversity m, which are complementary. By the properties of the above diagram, the map f < : L < → L is a G-equivariant Moore approximation both of degree k and of degree l. It can be shown that the local duality obstructions O * (π, k, l) vanish. (In fact, we can follow the proof of Theorem 7.1 in [7], which applies literally when we replace the universal cover B → B by the N -sheeted cover S 1 → S 1 having the finite group π 1 = Z/N as its group of deck transformations.) Let F < : ft < E → E denote the fiberwise truncation induced by the G-equivariant Moore approximation f < : L < → L. The truncation cone of f < , cone(F < ), is simply connected by Lemma 4.14. Hence, Theorem 6.3(b) implies that the rational Hurewicz homomorphism Hur n−1 * : π n−1 (cone(F < ), pt) → H n−1 (cone(F < )) is surjective.
We choose a suitable finite disjoint union i E (i) of mapping tori E (i) coming from the previous construction that can be realized as the boundary of a compact oriented smooth manifold M of dimension n = c + 2. For instance, when c = 6, then we can use a single mapping torus because Ω SO 7 = 0. Denoting the homotopy pushout of S 1 π ←− E (i) id E (i) −→ E (i) by DE (i) , we thus obtain a compact depth one Witt space X = M ∪ ∂M i DE (i) of dimension n ≥ 3. Its middle perversity intersection space is given by and according to Theorem 6.3(a), there is a completion IX = IX ∪ e n to a rational Poincaré duality space by attaching a single n-cell. The rational homotopy type of IX is determined by the intersection space IX whenever a theorem of Stasheff [26] is applicable. By Theorem 6.3(c), the signature of the Poincaré duality space IX agrees with the signature of the Goresky-MacPherson-Siegel intersection form on middle-perversity intersection homology of X.
In conclusion, let us discuss an example for a commutative diagram as above.
Example 6.5. Consider a closed smooth manifold K of dimension c ≥ 3 such that H c 2 (K) = 0 when c is even. We fix a c-dimensional triangulation K 0 ⊂ · · · ⊂ K c of K. Set k = ⌊ 1 2 (c + 1)⌋. The (k − 1)-skeleton K k−1 can be extended to a k-dimensional CW complex K < in such a way that there exists a Moore approximation g < : K < → K k of degree k which restricts to the identity map on K k−1 . (When K is simply connected and k ≥ 3, this can be achieved by using Proposition 1.6 of [3]. More generally, note that Proposition 1.3 in [27] can be applied whenever k ≥ 2, and without assuming K to be simply connected.) The connected sum L = K♯K is a closed smooth c-manifold which admits an orientation preserving diffeomorphism λ : L → L interchanging the two summands in such a way that λ 2 = id L . We may equip L with a c-dimensional CW structure L 0 ⊂ · · · ⊂ L c in such a way that L c−1 = K c−1 ∨ x0 K c−1 for some basepoint x 0 ∈ K 0 , and such that λ restricts to the homeomorphism L c−1 → L c−1 that interchanges the two copies of L c−1 . (To achieve this, we fix a c-simplex ∆ c of K, choose an embedded closed unit ball B c in the interior of ∆ c , and delete the interior U c of B c . Then, we form the connected sum L = K♯K by gluing two copies of K \ U c via the identity map on ∂B c . In order to find the desired CW structure on L, we modify B c by moving one point of ∂B c to a 0-simplex {x 0 } ⊂ ∂∆ c . Then, we see that L c−1 = K c−1 ∨ x0 K c−1 , and the c-cells of L are given by the c-simplices of the two copies of K that are different from ∆ c , plus one new c-cell whose attaching map arises from gluing the two copies of ∆ c along the modified ∂B c .) We define a degree k Moore approximation f < : L < → L by taking the composition Finally, we define λ < : L < → L < to be the homeomorphism that interchanges the two copies of K < in the bouquet L < = K < ∨ x0 K < . Then, the desired diagram commutes by construction.