Multiplicative de Rham Theorems for Relative and Intersection Space Cohomology
aa r X i v : . [ m a t h . A T ] J a n MULTIPLICATIVE DE RHAM THEOREMS FOR RELATIVEAND INTERSECTION SPACE COHOMOLOGY
FRANZ WILHELM SCHL ¨ODER AND J. TIMO ESSIG
Abstract.
We construct an explicit de Rham isomorphism relating the co-homology rings of Banagl’s de Rham and spatial approach to intersectionspace cohomology for stratified pseudomanifolds with isolated singularities.Intersection space (co-)homology is a modified (co-)homology theory extend-ing Poincar´e Duality to stratified pseudomanifolds. The novelty of our resultcompared to the de Rham isomorphism given previously by Banagl is, that weindeed have an isomorphism of rings and not just of graded vector spaces. Wealso provide a proof of the de Rham Theorem for cohomology rings of pairs ofsmooth manifolds which we use in the proof of our main result.
Contents
1. Introduction 22. A Multiplicative Relative de Rham Theorem 52.1. Sheaf Theory 52.2. Relative Singular Cohomology 62.3. Relative de Rham Cohomology 82.4. Relative de Rham Map 103. The cohomology ring of I ¯ p X φ ∗ is a ring isomorphism 164. A Multiplicative de Rham Theorem for HI 224.1. de Rham Map to Cellular Cochains 224.2. Multiplicativity of ˜ ρ φ φ φ is a Quasi-Isomorphism 285. Compatibility with Banagl’s de Rham Theorem for HI 31Acknowledgements 35References 35 Date : March, 2019.2010
Mathematics Subject Classification.
Primary: 55N33, 55N30, 14J17, 58A10, 58A12; sec-ondary: 57P10, 81T3, 14J33.
Key words and phrases.
Singularity, Stratified Space, Pseudomanifold, Poincar´e Duality, Inter-section Space Cohomology, Intersection Cohomology, Sheaf Theory, De Rham Theorem, RelativeDe Rham Theorem, Differential Forms, Cellular Cup Products, Cup Products on Cochains. Introduction
We prove that the de Rham approach to intersection space cohomology yields thesame cohomology ring as the spatial approach in analogy to ordinary cohomologyon smooth manifolds. We give an explicit ring isomorphism that integrates smoothforms on the top stratum over smooth cycles.In Section 2, we use classical sheaf theory to prove that integration of differentialforms on a smooth manifold over smooth cycles induces a ring isomorphism betweenthe relative de Rham and singular cohomology rings. To prove the multiplicativitywith respect to a cup product ∪ : H p ( M, L ) × H q ( M, F ) → H p + q ( M, L ∪ F )induced by the wedge product of forms we need submanifolds L, F ⊂ M whichsatisfy the restrictive condition that their union L ∪ F ⊂ M is also a submanifold.This is trivially fulfilled for L = F ,though, and we use the corresponding relativede Rham result in the second part of the paper, where we prove the existence of amultiplicative de Rham isomorphism for intersection space cohomology.Intersection space cohomology is a method, introduced by Banagl in [1], to re-establish Poincar´e duality for singular spaces by assigning a family of so-calledintersection spaces I ¯ p X indexed by Goresky-MacPherson perversity functions ¯ p toan n -dimensional stratified pseudomanifold X . The intersection space cohomology HI • ¯ p ( X ) of X is defined to be the reduced singular cohomology of I ¯ p X with co-efficients in Q or, as in our case, in R . If ¯ q is the complementary perversity of ¯ p ,Poincar´e duality holds in the sense that HI • ¯ p ( X ) ∼ = HI n −• ¯ q ( X ).The same duality statement is true for intersection cohomology introduced in[12, 13]. Intersection cohomology is Goresky and MacPherson’s original theory tore-establish Poincar´e duality on singular spaces. Note, that intersection cohomologyand intersection space cohomology are not isomorphic but tend to be interchangedby mirror symmetry. The former can be tied up to type IIA string theory whilethe latter relates to type IIB.In [2], Banagl introduces a description of intersection space cohomology for pseu-domanifolds of stratification depth 1 and with geometrically flat link bundle as thecohomology of a complex of smooth differential forms on the top stratum or theblowup of X . This enlarges the class of pseudomanifolds to which intersection spacecohomology is applicable. In [3], Banagl and Hunsicker give a L − description ofintersection space cohomology in the case of stratification depth 1 and product linkbundle. In [10] the second author uses the differential form approach to define in-tersection space cohomology for pseudomanifolds of stratification depth 2 with zerodimensional bottom stratum and geometrically flat link bundle for the intermediatestratum. De Rham theorems for intersection space cohomology are given in [2] forpseudomanifolds with isolated singularities and in [9] for pseudomanifolds of depthone with product link bundles. In both cases, the de Rham isomorphisms are givenby integrating differential forms over certain smooth cycles.A description of intersection cohomology via smooth differential forms was pro-vided in [8]. A different approach to intersection cohomology is pursued by Brasseletand Legrand in [5] and [6], using a complex of differential forms with coefficients inthe module of poles. De Rham theorems similar to the ones for intersection spacecohomology are given by Brasselet, Hector and Saralegi in [4] and [17].In contrast to intersection cohomology, both approaches to intersection spacecohomology naturally come with a perversity internal cup product. Neither of theabove de Rham theorems clarifies whether the constructed isomorphisms respect ULTIPLICATIVE DE RHAM THEOREMS FOR RELATIVE COHOM. AND HI 3 this multiplicative structure. This is the topic of the main part of this paper. Weestablish an isomorphism of the cohomology rings in the case of isolated singulari-ties.As an application of our result, note that intersection space cohomology providesthe correct count of massless 3-branes in type IIB string theory on a conifold [1].The de Rham description allows to represent those branes as differential forms andour result represents the intersection product of the branes as the wedge productof these forms.For a space X ′ with only isolated singularities the intersection space cohomologycoincides by construction with the intersection space cohomology of the space X obtained by collapsing all the singularities into a single one. Therefore we only con-sider the case of one isolated singularity and can think of a stratified pseudomanifold X of dimension n as X = cone ( i ∂ ) := ( ¯ X ∪ cone ( L )) / ∼ . Here ¯ X is a smooth manifold of dimension n with boundary L and i ∂ the inclusionof this boundary. The relation “ ∼ ” glues the bottom of the cone to the boundaryof ¯ X, identifying the cone coordinate with the collar coordinate of a smooth collarof the boundary. In the more general context, ¯ X is the blowup of the singular space X and L is the link of the singularity. Let us briefly describe the two approachesto intersection space cohomology.The spatial approach uses Moore approximation to truncate the links. This tech-nique is also referred to as spatial homology truncation in [1] and is Eckmann-Hiltondual to Postnikov approximation. In this process we associate to the link L its de-gree k spatial (co-)homology truncation t 1. This con-struction involves a choice of a splitting of the boundary map ∂ k : C k ( L ) → im ( ∂ k ),where C • here and in the rest of the paper denotes the cellular chains and cellularcochains are written as C • analogously. Importantly the intersection space coho-mology, constructed in this way is independent of the choice of the splitting. Theintersection space is defined as the homotopy cofiber of the composition g := i ∂ ◦ f, with i ∂ : ∂ ¯ X ֒ → ¯ X the inclusion of the boundary, i.e. I ¯ p X := cone ( g ) = ( ¯ X ∪ cone ( t 1. The choiceof the metric does not affect the cohomology groups we obtain, as demonstratedin [2]. τ ≥ k Ω • ( L ) cotruncates the cohomology of Ω • ( L ) in the sense that the sub-complex inclusion induces an isomorphism on cohomology in degrees ≥ k whereasthe cohomology of τ ≥ k Ω • ( L ) is zero in degrees smaller than k. Rather than the original definition as in [2], we adopt the definition in [3] andset Ω I • ¯ p ( ¯ X ) := { ω ∈ Ω • ( ¯ X ) | i ∂ ( ω ) ∈ τ ≥ k Ω • ( L ) } where k = n − − ¯ p ( n ) as above and i ∂ denotes the pullback of differential formsalong i ∂ . We use i ∗ ∂ for the induced map on cohomology. This distinction ismore relevant in our work. Note, that Ω I • ¯ p ( ¯ X ) with the restricted wedge productis a sub-DGA of Ω • ( ¯ X ). This product turns H • (Ω I • ¯ p ( ¯ X )) into a ring. The finalresult of this paper is Theorem 6 (Multiplicative Ω I • ¯ p ( ¯ X ) de Rham Theorem) . The cohomology rings H • (Ω I • ¯ p ( ¯ X )) and H • ( e C • ( I ¯ p X )) are isomorphic. To show this, we construct a de Rham map φ, which is different from the oneprovided by Banagl in [2] on cochain level. However, both are related on cohomologyas we prove in Section 5. Observe, that I ¯ p X is a pushout by construction. InSection 3, we establish that in the category of cochain complexes, the reducedcochain complex of I ¯ p X fits into the pullback diagram e C • ( I ¯ p X ) e C • ( cone ( t In this section, we introduce relative de Rham cohomology groups via sheafcohomology and then prove that the multiplicative de Rham isomorphism betweenabsolute de Rham and singular cohomology groups descends to a multiplicativeisomorphism between relative groups. This fact is then used to prove that there isa multiplicative de Rham isomorphism between spatial and de Rham descriptionof intersection space cohomology.2.1. Sheaf Theory. We use sheaf cohomology to prove a result about ordinaryrelative singular and de Rham cohomology. Basics about sheaves and sheaf coho-mology can be found in [7]. We recall only the notion of supports: Definition 2.1. (see [7, Def. I-6.1])Let X be a topological space. A family of supports on X is a family Φ of closedsubsets of X such that(1) A closed subset of an element of Φ is an element of Φ ;(2) Φ is closed under finite unions. Φ is a paracompactifying family of supports if in addition(3) each element of Φ is paracompact.(4) each element of Φ has a (closed) neighbourhood also contained in Φ . Examples of supports are the family of all closed subsets of X , and the familyconsisting of the empty set. The first is paracompactifying if X is paracompact.If s ∈ A ( X ) is a global section of a sheaf A on X , then | s | = { x ∈ X | s ( x ) = 0 } denotes its support. The sections of A with supports in Φ are defined byΓ Φ ( A ) := { s ∈ A ( X ) | | s | ∈ Φ } . In the same way one defines A Φ ( X ) := { s ∈ A ( X ) | | s | ∈ Φ } for the presheavesof differential forms A = Ω • and singular cochains with values in some locally FRANZ WILHELM SCHL ¨ODER AND J. TIMO ESSIG constant sheaf A on X , A = S • ( − ; A ). The de Rham and singular cohomologywith supports in Φ is then defined by taking the cohomology groups H p (Ω • Φ ( X ))and H p ( S • Φ ( X ; A )). The sheaf cohomology groups with supports in Φ for the sheaf A are defined by taking any injective resolution A → J • of A and setting H r Φ ( X ; A ) := H r (Γ Φ ( J • )) . Relative Singular Cohomology. Before explaining the notions of relativede Rham cohomology, we recall the results of [7, Chapter III-1] about relativesingular cohomology. For our purpose, it is sufficient to consider the reals R asbase ring for our singular cohomology groups and therefore all sheaves are sheavesof real vector spaces and all tensor products are taken over the reals. In this section,let X denote an arbitrary topological space. Later, we specify X to be a smoothmanifold. Let A be a sheaf on X and let Φ be a paracompactifying family ofsupports on X. The singular cohomology groups of X with coefficients in A andsupport in the paracompactifying family Φ are then defined by S H p Φ ( X ; A ) := H p (Γ Φ ( S • ⊗ A )) , where S • = S • ( X ; R ) is the sheafification of singular cochains. Note that thesecohomology groups agree with the regular singular cohomology groups with realcoefficients H • S ( X ; R ) for A = R the constant sheaf and Φ the family of all closedsubsets of X. The ordinary, singular cup product induces a homomorphism ∪ : S H p Φ ( X ; A ) ⊗ S H q Ψ ( X ; B ) → S H p + q Φ ∩ Ψ ( X ; A ⊗ B )with the usual properties (see [7, Theorem II-7.1]). If X is HLC (= singular ho-mology locally connected), e.g. X a manifold or more generally a CW complex,then there is a multiplicative isomorphism between sheaf cohomology and singularcohomology groups:(1) θ : H • Φ ( X ; A ) ∼ = −→ S H • Φ ( X ; A ) , (see [7, pp. 180-181] for a more detailed explanation).To define relative singular cohomology with coefficients in the sheaf A , let F ⊂ X be a closed subspace and consider the homomorphism Γ Φ ( S • ( X ; R ) ⊗ A ) ։ Γ Φ | F ( S • ( F ; R ) ⊗ A | F ) . It is a surjection, since the kernel of the epimorphism( S • ( X ; R ) ⊗ A ) | F ։ S • ( F ; R ) ⊗ A , which is induced by a restriction morphism,is an S ( X ; R ) | F -module and hence Φ | F -soft. Therefore, [7, Theorem II 9.9] isapplicable.Let K • Φ ( X, F ; A ) denote the kernel of this map and define the relative singularcohomology groups of the pair ( X, F ) with coefficients in A and supports Φ by S H • Φ ( X, F ; A ) := H • ( K • Φ ( X, F ; A )) . By definition, one gets the usual long exact sequence of a pair. . . . → S H p Φ ( X, F ; A ) → S H p Φ ( X ; A ) → S H p Φ | F ( F ; A | F ) +1 −−→ . . . Let U F = X − F denote the complement of the closed set F and A U F the extensionby zero to X of the restriction A | U F , see [7, I 2.6]. The morphismΓ Φ ( S • ( X ; R ) ⊗ A U F ) → K • Φ ( X, F ; A ) ULTIPLICATIVE DE RHAM THEOREMS FOR RELATIVE COHOM. AND HI 7 induces an isomorphism on cohomology, which follows by a 5-Lemma argument(see [7, p. 183] for details). By this isomorphism we can introduce a relative cupproduct given by the following composition S H p Φ ( X, F ; A ) ⊗ S H q Ψ ( X, L ; B ) S H p + q Φ ∩ Ψ ( X, F ∪ L ; A ⊗ B ) S H p Φ ( X ; A U F ) ⊗ S H q Ψ ( X ; B U L ) S H p + q Φ ∩ Ψ ( X, A U F ⊗ B U L ) ∪∼ = ∪ ∼ = The vertical map on the right is induced by the inclusion A U F ⊗ B U L ֒ → A ⊗ B ,F, L ⊂ X are closed, and U F = X − F, U L = X − L. This coincides with theordinary relative cup product in singular cohomology for A = B = R and Φ , Ψ thefamilies of all closed subsets of X. Also, together with the long exact cohomologysequence for sheaf cohomology of pairs and the maps θ of (1), this definition givesa multiplicative isomorphism θ : H • Φ ( X, F ; A ) ∼ = −→ S H Φ ( X, F ; A )for X, F both HLC .Note that if X, F are smooth manifolds, one can use smooth singular cochainsinstead of continuous ones. To see this, let S •∞ ( X ; R ) denote the complex of smoothsingular cochains with coefficients in R and let ρ : S • ( X ; R ) ։ S •∞ ( X ; R ) denotethe restriction of the complex of all singular cochains to smooth ones.Then, one gets a map of sheaves ρ : S • ( X ; R ) ⊗ A → S •∞ ( X ; R ) ⊗ A , whichwe also denote by ρ. Further, for any sheaf A on X and any family of supportsΦ, one gets a map ρ : Γ Φ ( S • ( X ; R ) ⊗ A ) → Γ Φ ( S •∞ ( X ; R ) ⊗ A ) . Note, thatfor Φ paracompactifying, all the sheaves S r ( X ; R ) and S r ∞ ( X ; R ) are Φ-soft asmodules over the sheaf of continuous respectively smooth real valued functions,which are Φ-soft by a standard partition of unity argument. Hence, the sheaves S • ( X ; R ) ⊗ A and S •∞ ( X ; R ) ⊗ A are resolutions of A by Φ-soft sheaves and ρ induces an isomorphism on cohomology groups by [7, II-4.2], ρ ∗ : S H • Φ ( X ; A ) ∼ = −→ ∞ S H • Φ ( X ; A ) . Here, ∞ S H • Φ ( X ; A ) = H • (Γ Φ ( S •∞ ( X ; R ) ⊗ A )) . Let K • Φ , ∞ ( X, F ; A ) denote thekernel of the surjectionΓ Φ ( S •∞ ( X ; R ) ⊗ A ) ։ Γ Φ | F ( S •∞ ( F ; R ) ⊗ A | F )and define the smooth relative singular cohomology groups with values in A by ∞ S H • Φ ( X, F ; A ) := H • (cid:0) K • Φ , ∞ ( X, F ; A ) (cid:1) . Again, there is a long exact sequence of the pair ( X, F ) and we get a cup producton the relative smooth singular cohomology groups, that coincides with the regularone for A = R and Φ the family of all closed subsets.Restriction of singular cochains to smooth chains induces a multiplicative iso-morphism on cohomology as follows from the following commutative diagram S H • Φ ( X ; A U F ) ∞ S H • Φ ( X ; A U F ) S H • Φ ( X, F ; A ) ∞ S H • Φ ( X, F ; A ) , ∼ =mult ρ ∗ ∼ = , mult ∼ =mult ρ ∗ FRANZ WILHELM SCHL ¨ODER AND J. TIMO ESSIG where the vertical map on the right is a multiplicative isomorphism analogously tothe non-smooth case.2.3. Relative de Rham Cohomology. To consider de Rham cohomology, weneed smooth manifolds. We prove a relative version of de Rham’s Theorem for thefollowing pairs of smooth manifolds (possibly with boundary). Let M n be a smoothmanifold and F m ⊂ M a submanifold of dimension m which is closed as a subspace(not necessarily as a manifold). The pair ( M, F ) might be compact (or M open and F compact, or both non-compact manifolds). We only consider submanifolds thatare closed subsets, since then the relative sheaf cohomology groups can be replacedby absolute cohomology groups of the complement.If two different submanifolds F m , L m ⊂ M occur, we demand that their union F ∪ L ⊂ M is also a submanifold. This is relevant later to insure that relative cupproducts on de Rham cohomology actually have a well-defined target.Let A be a sheaf of R − modules on M . The de Rham presheaves on M aregiven by the assignments U Ω r ( U ), where Ω r ( U ) is the set of smooth differential r -forms on the open set U (of M respectively F ). This gives conjunctive mono-presheaves and hence sheaves. Let Ω r ( M ) denote the so defined sheaf on M andΩ r ( F ) the corresponding sheaf on the manifold F . In contrast, Ω r ( M ) | F denotesthe restriction of the sheaf Ω r ( M ) to the subspace F . The de Rham cohomologywith coefficients in A is defined as Ω H • Φ ( M ; A ) := H • (Γ Φ (Ω • ⊗ A )) . The wedge product ∧ : Ω p ( U ) ⊗ Ω q ( U ) → Ω p + q ( U ) , U ⊂ M open, induces a cupproduct on Ω H • Φ ( M ; A ) . To define the relative de Rham cohomology groups, wenote that the restriction homomorphism i ∗ : Ω • ( U ) → Ω • ( U ∩ F ) is surjectiveand hence induces an epimorphism (Ω • ( M ) ⊗ A ) | F ։ Ω • ( F ) ⊗ A | F of sheaveson F . Since the kernel of this homomorphism is an Ω ( M ) | F − module and henceΦ | F − soft by [7, Theorem II 9.16], we get an epimorphismΓ Φ (Ω • ( M ) ⊗ A F ) = Γ Φ | F ((Ω • ( M ) ⊗ A ) | F ) ։ Γ Φ | F (Ω • ( F ) ⊗ A | F )of chain complexes by [7, Theorem II 9.9]. The kernel of Ω • ( M ) ⊗ A → Ω • ( M ) ⊗ A F is Ω • ( M ) ⊗ A U , a Φ-soft sheaf, because of [7, II 9.18] and the fact that Ω • ( M ) isΦ-fine and hence Φ-soft (for Φ paracompactifying). Then, again by [7, Theorem II9.9], the map Γ Φ (Ω • ( M ) ⊗ A ) → Γ Φ (Ω • ( M ) ⊗ A F ) is also onto. Both epimorphismscombine to an epimorphismΓ Φ (Ω • ( M ) ⊗ A ) ։ Γ Φ | F (Ω • ( F ) ⊗ A | F ) . We let Q • Φ ( M, F ; A ) denote the kernel of this epimorphism and define the relativede Rham cohomology with coefficients in the sheaf A as follows. Definition 2.2 (Relative de Rham Cohomology) . The relative de Rham coho-mology of the pair ( M, F ) of smooth manifolds, F ⊂ M closed as a subset, withcoefficients in the sheaf A , is defined by Ω H • Φ ( M, F ; A ) := H • ( Q • Φ ( M, F ; A )) . As for the relative singular cohomology groups, we want to relate these groupsto the absolute groups and the sheaf-theoretic cohomology groups. To do so, we ULTIPLICATIVE DE RHAM THEOREMS FOR RELATIVE COHOM. AND HI 9 note that for Φ paracompactifying Ω • ⊗ A is a resolution of A by Φ − fine sheavesand hence there is a natural isomorphism ρ : Ω H • Φ ( M ; A ) → H • Φ ( M ; A ) , which preserves cup products (see [7, II-5.15 and II-7.1] for details). If f ∈ C ∞ ( F, M ) is a smooth map between the smooth manifolds F, M and Φ , Ψ areparacompactifying families on M and F respectively, such that f − Ψ ⊂ Φ and A is a sheaf on F , we get a commutative diagram(2) Ω H • Φ ( M ; A ) H • Φ ( M ; A ) Ω H • Ψ ( F ; f ∗ A ) H • Ψ ( F ; f ∗ A ) f ∗ f ∗ in analogy to singular cohomology (compare to the similar diagram for singularcohomology on [7, p. 182]). Lemma 2.1. Let j : F ֒ → M be the inclusion of the submanifold F ⊂ M , whichis closed as a subspace. Let Φ be a paracompactifying family of supports on M andlet A be a sheaf on M . Then the map j ∗ : Ω H • Φ ( M ; A F ) → Ω H • Φ | F ( F ; A | F ) is an isomorphism and preserves cup products.Proof. For arbitrary smooth manifolds M and arbitrary sheaves A on M the map ρ : Ω H • Φ ( M ; A ) → H • Φ ( M ; A ) is an isomorphism that preserves cup products.Since Diagram (2) commutes, it suffices to show that the map j ∗ : H • Φ ( M ; A F ) → H • Φ | F ( F ; A | F ) is an isomorphism of sheaf cohomology groups and preserves cupproducts. But this is essentially the statement of [7, Corollary II-10.2]. (cid:3) The exact sequence 0 → A U F → A → A F → → Γ Φ (Ω • ( X ) ⊗ A U F ) → Γ Φ (Ω • ( X ) ⊗ A ) → Γ Φ (Ω • ( X ) ⊗ A F ) . Hence,the first morphism gives rise to a (trivially multiplicative) mapΓ Φ (Ω • ( X ) ⊗ A U F ) → Q • Φ ( M, F ; A ) . Together with the (multiplicative) map j ∗ : Γ Φ (Ω • ( X ) ⊗ A F ) → Γ Φ | F (Ω • ( F ) ⊗ A | F ) , induced by the submanifold inclusion j : F ֒ → M, we get a commutative diagramΓ Φ (Ω • ( M ) ⊗ A U F ) Γ Φ (Ω • ( M ) ⊗ A ) Γ Φ (Ω • ( M ) ⊗ A F ) Q • Φ ( M, F ; A ) Γ Φ (Ω • ( M ) ⊗ A ) Γ Φ | F (Ω • ( F ) ⊗ A | F ) = j ∗ We consider the induced diagram on cohomology: . . . Ω H p Φ ( M ; A U F ) Ω H p Φ ( M ; A ) Ω H p Φ ( M ; A F ) . . .. . . Ω H p Φ ( M, F ; A ) Ω H p Φ ( M ; A ) Ω H p Φ | F ( F ; A | F ) . . . = +1 j ∗ ∼ = +10 FRANZ WILHELM SCHL ¨ODER AND J. TIMO ESSIG By the statement of the last lemma, the last vertical map is an isomorphism. The5-Lemma implies that the first vertical map is also an isomorphism, which leavesus with the following result. Proposition 2.2. Let ( M, F ) be any pair of smooth manifolds, possibly with bound-ary, where F ⊂ M is closed as a subset, and let Φ be any paracompactifying familyof supports and A any sheaf. Then, there is an isomorphism Ω H p Φ ( M ; A U F ) ∼ = −→ Ω H p Φ ( M, F ; A ) . In particular, this induces a multiplicative structure on Ω H p Φ ( M, F ; A ) , which co-incides with the multiplicative structure induced by the relative wedge product ∧ : Ω p ( U, U ∩ L ) ⊗ Ω q ( U, U ∩ F ) → Ω p + q ( U, U ∩ ( L ∪ F )) for A = R , provided L, F ⊂ M are two smooth submanifolds that are closed assubsets and such that L ∪ F ⊂ M is also a smooth submanifold.Proof. What is left is a proof for the last part of the statement. So let A = R and L, F ⊂ M be as in the proposition. Then the map Γ Φ (Ω • ( M ) ⊗ A ) ։ Γ Φ | F (Ω • ( F ) ⊗ A | F ) coincides with the pullback map j ∗ : Ω • Φ ( M ) ։ Ω • Φ | F ( F ) . The kernel of this map is the complex Ω • Φ ( M, F ) of the forms on M that vanish on F. The maps R U F → R and R U L → R of sheaves on M induce maps of complexesof sheaves Ω • ( M ) ⊗ R U L → Ω • ( M ) ⊗ R and the same with F instead of L. Thesemaps induce chain maps Γ Φ (Ω • ( M ) ⊗ R U F ) → Γ Φ (Ω • ( M ) ⊗ R ) ∼ = Ω • Φ ( M ) and thesame map for L instead of F . Since they factor through Ω • Φ ( M, F ), respectivelyΩ • Φ ( M, L ) , and the differential of these complexes comes from the differential onthe total de Rham complexes, we get the following induced maps. H (Γ Φ (Ω • ( M ) ⊗ R U F )) = ker d → ker d | Ω • Φ ( M, F ) = Ω H ( M, F ) ,H (Γ Φ (Ω • ( M ) ⊗ R U L )) = ker d → ker d | Ω • Φ ( M, L ) = Ω H ( M, L ) . Since the cup product ∪ : Ω H p Φ ( M ; R U F ) ⊗ Ω H q Φ ( M, R U L ) → H p + q Φ ( M, R U F ⊗ R U L )is induced by the wedge product of forms, we get a commutative diagram Ω H ( M ; R U F ) ⊗ Ω H ( M ; R U L ) Ω H ( M ; R U F ⊗ R U L ) Ω H ( M, F ) ⊗ Ω H ( M, L ) Ω H ( M, F ∪ L ) . ∪ ρ ρ ∪ This allows us to apply [7, Theorem II-6.2] to the two natural transformations α ⊗ β ρ ( α ∪ β ) and α ⊗ β ρ ( α ) ∪ ρ ( β ) from the top left corner of the diagramto the bottom right corner. (cid:3) Relative de Rham Map. In [7, Chapter III-3], Bredon proves that the clas-sical de Rham map k : Ω • ( M ) → S •∞ ( M ; R ) , defined by integrating forms oversmooth chains, induces a multiplicative homomorphism k ∗ : Ω H • Φ ( M ; A ) → ∞ S H • Φ ( M ; A ) , which is an isomorphism for Φ paracompactifying and coincides with the usual deRham isomorphism for A = R . Let F ⊂ M be a smooth submanifold, closed as ULTIPLICATIVE DE RHAM THEOREMS FOR RELATIVE COHOM. AND HI 11 a subspace as above, and let j : F ֒ → M. For each open set U ⊂ M we get acommutative diagram Ω • ( U ) Ω • ( U ∩ F ) S •∞ ( U ; R ) S •∞ ( U ∩ F ; R ) j | U k M k F j | U where all the maps commute with the corresponding restriction maps. Hence, forany sheaf A on M this induces a commutative diagramΓ Φ (Ω • ⊗ A ) Γ Φ | F (Ω • ⊗ A | F )Γ Φ ( S •∞ ( M ; R ) ⊗ A ) Γ Φ | F ( S •∞ ( F ; R ) ⊗ A | F ) . k M k F This implies that k M factors through the kernels of the horizontal maps. Thatmeans it induces a relative de Rham morphism(3) k : Q Φ ( M, F ; A ) → K Φ , ∞ ( M, F ; A ) . Theorem 1 (Relative de Rham Theorem) . Let ( M, F ) be a pair of smooth mani-folds, F ⊂ M closed as a subset, let A be a sheaf on M and Φ a paracompactifyingfamily of supports and let k : Q Φ ( M, F ; A ) → K Φ , ∞ ( M, F ; A ) denote the relativede Rham map. Then, the induced map on cohomology k ∗ : Ω H • Φ ( M, F ; A ) → ∞ S H • Φ ( M, F ; A ) is a multiplicative isomorphism.Proof. As before, let U F := M − F ⊂ M, which is an open subset of M. We provethat the following diagram commutes.(4) Ω H • Φ ( M ; A U F ) ∞ S H • Φ ( M ; A U F ) Ω H • Φ ( M, F ; A ) ∞ S H • Φ ( M, F ; A ) k ∼ = ∼ = ∼ = k This will complete the proof, since the vertical maps are clearly multiplicative andthe isomorphism k on the top is multiplicative by [7, Theorem III 3.1].Since the absolute de Rham morphism k : Γ Φ (Ω • ( M ) ⊗ A ) → Γ Φ ( S •∞ ( M ; R ) ⊗ A )is natural, the following diagram commutes.Γ Φ (Ω • ( M ) ⊗ A U F ) Γ Φ ( S •∞ ( M ; R ) ⊗ A U F )Γ Φ (Ω • ( M ) ⊗ A ) Γ Φ ( S •∞ ( M ; R ) ⊗ A ) . kτ τk By definition of the relative de Rham morphism, the following diagram also com-mutes. Q • Φ ( M, F ; A ) K • Φ , ∞ ( M, F ; A )Γ Φ (Ω • ( M ) ⊗ A ) Γ Φ ( S •∞ ( M ; R ) ⊗ A ) . kσ σk The diagrams above can be combined as follows.Γ Φ (Ω • ( M ) ⊗ A U F ) Γ Φ ( S •∞ ( M ; R ) ⊗ A U F ) Q • Φ ( M, F ; A ) K • Φ , ∞ ( M, F ; A )Γ Φ (Ω • ( M ) ⊗ A ) Γ Φ ( S •∞ ( M ; R ) ⊗ A ) . kτ τkσ σk By the previous statements, the bottom square and the exterior big square com-mute. Since the σ ’s are subcomplex inclusions by definition, this gives the desiredcommutativity of the top square, which induces Diagram (4) on cohomology. (cid:3) Corollary. The composition of the classical relative de Rham map k : Ω • ( M, F ) → S •∞ ( M, F ; R ) , defined by integration of relative forms over smooth chains, and Lee’ssmoothing operator s ∗ : S •∞ ( M, F ; R ) → S • ( M, F ; R ) , see [14, pp. 474 ff], inducesa multiplicative isomorphism on cohomology. s ∗ ◦ k ∗ : H rDR ( M, F ) ∼ = −→ H rS, ∞ ( M, F ; R ) ∼ = −→ H rS ( M, F ; R ) . Proof. This follows from Theorem 1 with A = R and Φ the family of all closed sub-sets of M and the results of Section 2.2. In this setting, the relative de Rham map(3) coincides with the classical relative de Rham map k : Ω • ( M, F ) → S •∞ ( M, F )defined by k ( ω )( σ ) = Z ∆ p σ ∗ ω, for ω ∈ Ω r ( M, F ) and any smooth p − simplex σ . Since Lee’s smoothing operatoris a chain homotopy inverse of the restriction map ρ : S • ( X ; R ) ։ S •∞ ( X ; R ) , which induces a multiplicative isomorphism on cohomology, s ∗ = ( ρ ∗ ) − is alsomultiplicative on cohomology. (cid:3) The cohomology ring of I ¯ p X In this section we prove Theorem 2. The reduced cohomology ring of I ¯ p X is isomorphic to the cohomologyring of the pullback Q • in the following diagram. (5) Q • e C • ( cone ( t The universal property of the pullback is later used to construct the map φ :¯ C • → Q • which is the middle part of our intersection space cohomology de Rhammap.Recall that in the spatial picture of intersection space cohomology we are workingwith cellular cochains and accordingly the various cochain complexes of topologicalspaces here are their cellular cochain complexes. Note that g and i are cellu-lar maps. Therefore they induce cochain maps, and Q • is a cochain complex byconstruction. On the other side it is in general false that cellular maps inducemultiplicative maps on cochain level and we have to clarify how the product on Q • and hence H • ( Q • ) arises. The cup product of cellular cohomology is induced oncochain level by an Eilenberg-Zilber type map and a cellular approximation to thediagonal, called cellular diagonal approximation in the following. In Section 3.1we demonstrate that i ˜ and g are DGA homomorphisms for the right choice ofproducts on cochain level. This upgrades the construction above from the categoryof cochain complexes to the category of DGAs and Q • is naturally equipped withan appropriate product.After this we turn to the proof of Theorem 2. An explicit quasi-isomorphism isgiven by the map φ in the diagram(6) e C • ( I ¯ p X ) Q • e C • ( cone ( t As explained above, we need to establishthat we can choose products on cochain level such that i ˜ and g are DGA homo-morphisms. With this choice of products, the pullback is a pullback in the categoryof DGAs and Q • is a DGA by construction. Note that every choice of graded prod-uct on C • ( cone ( t For a CW subcomplex inclusion i : Y ֒ −→ Z and a given cellulardiagonal approximation ˜∆ Y on Y , we can choose a cellular diagonal approximation ˜∆ Z on Z such that i : C • ( Z ) → C • ( Y ) is a DGA homomorphism with respect tothe cup products induced by those diagonal approximations.Proof. Choosing the canonical cell structure on the CW complexes Y × Y and Z × Z ensures that for given cells a and b their product a × b is a cell again. Thus theEilenberg-Zilber type maps θ Y : C • ( Y ) ⊗ C • ( Y ) → C • ( Y × Y )and θ Z : C • ( Z ) ⊗ C • ( Z ) → C • ( Z × Z )are given by mapping a † ⊗ b † bijectively to ( a × b ) † . The daggered objects are therespective cochains in the basis dual to the basis of chains given by the cells. Wecalculate( i × i ) ◦ θ Z ( a † ⊗ b † ) = ( i × i ) ( a × b ) † = ( i ( a ) × i ( b )) † = θ Y ( i ( a ) † ⊗ i ( b ) † ) = θ Y ◦ ( i ⊗ i )( a † ⊗ b † )so ( i × i ) ◦ θ Z = θ Y ◦ ( i ⊗ i ) . The relative version of the Cellular Approximation Theorem (cf. [16, p. 76]) assuresthat we can extend a cellular diagonal approximation ˜∆ Y of Y to a cellular diagonalapproximation ˜∆ Z of Z , i.e. ( i × i ) ◦ ˜∆ Y = ˜∆ Z ◦ i. In conclusion i ◦ ∪ Z = i ◦ ˜∆ Z ◦ θ Z = ˜∆ Y ◦ ( i × i ) ◦ θ Z = ˜∆ Y ◦ θ Y ◦ ( i ⊗ i )= ∪ Y ◦ ( i ⊗ i ) . So, i induces a DGA homomorphism on cochain level if we define the cup productvia the specific diagonal approximations ˜∆ Y and ˜∆ Z . (cid:3) Next we establish that there are DGA-structures on C • ( ¯ X ) and C • ( t Theorem 3. Given a cellular diagonal approximation ˜∆ t 3. Then g = i ∂ ◦ f with f defined as composition of the subcomplexinclusion i L/k ◦ θ L/k , with θ t L/k is a k dimensional CWcomplex with the same ( k − L k . The k cells are glued in such a waythat they correspond to a spacification of a base change in the k -th chain group of L k . The map h is constructed as the homotopy inverse of the map h ′ : L k → L/k relative to the ( k − h ′ is defined such that it spatially realizesthe aforementioned base change. In particular, it induces an isomorphism on the k -th chain and cochain group and is the identity on all cochain groups of lower(and trivially also all other) cochain groups. The latter can be formulated as thecommutativity of the following diagram. L k L/kL k − L k − h ′ id i k − Note, that it is not obvious that the cellular cochain map induced by h is alsoan isomorphism a priori. In this setting, it is true, though, as we outline in thefollowing. In formulas, we know that h ′ ◦ h ≃ id rel L k − . Thus, there is a cochain homotopy operator s : C • ( L/k ) → C •− ( L/k ) such that(7) h ◦ h ′ = id + sd + ds and i k − ◦ s = 0 . Since ( i k − ) r : C r ( L/k ) → C r ( L k − ) is an isomorphism for r < k and ( i k − ) r − ◦ s r = 0 for all r ∈ Z , we get that s r = 0 for r ≤ k. Since L/k is a k -dimensionalCW-complex, C r ( L/k ) = 0 for r > k and hence, s r = 0 for all r ∈ Z . That impliesthat h ◦ h ′ = id. By the same argument, h ′ ◦ h = id also holds and therefore h : C • ( L k ) → C • ( L/k ) is an isomorphism with inverse ( h ) − = h ′ . Let ˜∆ L/k be the cellular approximation used above and set ∇ L k := ( h × h ) ◦ ˜∆ L/k ◦ h ′ . Since ∇ L k is cellular, the following calculation shows that ∇ L k is a cellular approx-imation of the diagonal ∆ L k of L k , ∇ L k = ( h × h ) ◦ ˜∆ L/k ◦ h ′ ≃ ( h × h ) ◦ ∆ L/k ◦ h ′ = ∆ L k ◦ h ◦ h ′ ≃ ∆ L k . Let ∪ L k = ∇ L k ◦ θ L k : C r ( L k ) ⊗ C s ( L k ) → C r + s ( L k ) be the resulting cup producton L k , with θ L k the Eilenberg-Zilber map as before. By this definition, h becomesa DGA homomorphism since h ◦ h ′ = id, h ◦ ∪ L k = h ◦ h ′ ◦ ˜∆ L/k ◦ ( h × h ) ◦ θ L k = ∪ L/k ◦ ( h ⊗ h ) . Finally apply the homotopy extension and lifting property (cf. [16, p. 75]) to ( i ∂ ◦ i k × i ∂ ◦ i k ) ◦ ∇ L k and ∆ ¯ X to obtain a map ∇ ′ ¯ X such that ∇ ′ ¯ X | L k = ∇ L k and ∇ ′ ¯ X ≃ ∆ ¯ X . Note that ∇ L k is a composition of cellular maps. So, the cellular approximationtheorem relative to L k yields a cellular map ∇ ¯ X such that ∇ ¯ X | L k = ∇ L k and ∇ ¯ X ≃ ∆ ¯ X . The first equation can be re-written as ∇ ¯ X ◦ ( i ∂ ◦ i k ) = { ( i ∂ ◦ i k ) × ( i ∂ ◦ i k ) } ◦ ∇ L k . Setting ∪ ¯ X := ∇ X ◦ θ ¯ X , where once again θ ¯ X is the Eilenberg-Zilber map on ¯ X ,( i ∂ ◦ i k ) is an DGA homomorphism with respect to the cup products ∪ ¯ X and ∪ L k .( i ∂ ◦ i k ) ◦ ∪ ¯ X = ∪ L k ◦ (cid:0) ( i ∂ ◦ i k ) ⊗ ( i ∂ ◦ i k ) (cid:1) We combine the previous results in the following equation. g ◦ ∪ ¯ X = i Note that after the initial cellular approximation to the diagonal mapof t The map (8) p ∗ : H • ( I ¯ p X, { c } ) ∼ = −→ H • ( M ( g ) , t The map r | ker ( g ) : ker ( g ) → C • ( M ( g ) , t We apply the 5-Lemma to the pair of long exact sequences on cohomologyinduced by the following diagram.0 C • ( M ( g ) , t The map r | ker ( g ) : ker ( g ) → C • ( M ( g ) , t Let ϕ ∈ ker ( g ) l and ψ ∈ ker ( g ) m be closed. Then the second of the aboverelations implies that r ◦ D X ( ϕ × ψ ) = D M ( g ) ◦ ( r × r ) ( ϕ × ψ ) + d M ( g ) s ( ϕ × ψ ) + s d ¯ X × ¯ X ( ϕ × ψ )= D M ( g ) ◦ ( r × r ) ( ϕ × ψ ) + d M ( g ) s ( ϕ × ψ ) . The last summand in the first line vanishes since ϕ × ψ is closed as the crossproduct of two closed forms. In the following, we show that there is a cochain α ∈ C l + m − ( M ( g ) , t The map incl : ker ( g ) ֒ → Q • , ϕ ( ϕ, induces a ring isomorphism on cohomology.Proof. Since Q • was defined as the pullback in diagram (5), it can be explicitelywritten as Q • = n ( ϕ, ψ ) ∈ C • ( ¯ X ) ⊕ e C • ( cone ( t The map F sends ( ϕ, ψ ) g ϕ − i ψ and is surjective, since g is surjective aswas established in the proof of Proposition 3.3. The lower sequence also appearedalready in this proof. The vertical map in the middle is the inclusion as the firstfactor. It is a quasi-isomorphism, since all the reduced cohomology groups of thecone of t Theorem 4. The map φ induced by the universal property of the pullback indiagram e C • ( I ¯ p X ) Q • e C • ( cone ( t 0) = ( dβ, b ) = ( dβ, g β ) = d ( β, . (cid:3) A Multiplicative de Rham Theorem for HI The goal of this section is to prove that the cohomology rings H • (Ω I • ¯ p ( ¯ X )) and H • ( C • ( I ¯ p X )) are isomorphic. In Section 4.1, we extend the de Rham map tomap from singular cochains to cellular cochains since we are forced to use cellularcochains in Section 4.3. This is established for both the absolute and relative caseand is then applied in Section 4.2 to construct the first part ˜ ρ of our eventualintersection space cohomology de Rham map φ . For each case we demonstrate thatthe maps induce multiplicative maps on cohomology. In Section 3 we establishedthat the cellular cochains of I ¯ p X fit up to the isomorphism φ into a pullback square.This property is now used to construct a map φ in Section 4.3 that combines with ˜ ρ and φ − into φ . We take care to construct φ as DGA homomorphisms. Accordinglythe induced map on cohomology is multiplicative and so is the induced map of φ .Section 4.4 gives the explicit form of φ on cochain level and Section 4.5 establishesthat φ is a quasi-isomorphism. Accordingly the induced map is a ring isomorphismand we have proven our result.4.1. de Rham Map to Cellular Cochains. Let C • ( M ) denote the cellular, S • ( M ) the singular and b S • ( M ) the normalized singular chain complex of M , C • ( M ) ,S • ( M ) and b S • ( M ) the corresponding cochains complexes and define relative chainsand cochains as usual.In the rest of this article, we work with a de Rham type map ρ : Ω • ( M ) → C • ( M )and the corresponding relative morphism. M is a smooth manifold, possibly openor with boundary, with some CW decomposition. In the relative setting, L is asubmanifold of M and we choose CW decompositions such that L ⊂ M is a CW-subcomplex. We outline the construction of the cellular de Rham maps ρ and ρ rel and explain why they induce multiplicative isomorphisms on cohomology.We can restrict a singular cochain to the nondegenerate simplices and get anelement of b S n ( M ) . This defines a restriction operatorrestr : S n ( M ) ։ b S n ( M ) , which is clearly multiplicative. The geometric realization Γ M of the non-degeneratesingular simplices of M is a CW complex with one n − cell for each non-degeneratesingular n − simplex and with cellular chain complex C • (Γ M ) naturally isomorphicto b S • ( M ) . Further, there is a weak equivalence γ : Γ M → M . In our settingthe Whitehead Theorem implies that γ actually is a homotopy equivalence. Tak-ing a CW-approximation of the homotopy inverse of γ , which is still a homotopyequivalence and which we denote by δ : M → Γ M , this gives a cochain homotopyequivalence δ : b S • ( M ) = C • (Γ M ) → C • ( M ) that induces a ring isomorphism oncohomology. This construction also carries over to the relative case.The last component of the multiplicative de Rham isomorphism we use hence-forth is Lee’s smoothing operator (see [14, pp. 474 ff]). It is a chain homotopyequivalence s : S • ( M ) → S ∞• ( M ), where S ∞• ( M ) is the chain complex of smoothsingular chains. Given a submanifold L ⊂ M, the operator can be defined in such ULTIPLICATIVE DE RHAM THEOREMS FOR RELATIVE COHOM. AND HI 23 a way that it commutes with the inclusion of this submanifold in M . Hence, s induces a quasi-isomorphism of relative chain complexes s : S • ( M, L ) qis −−→ S ∞• ( M, L ) . The induced maps on the absolute and relative cochain complexes are also quasi-isomorphisms and therefore induce isomorphisms, i.e. s ∗ : H • sing, ∞ ( M ) ∼ = −→ H • sing ( M ) ,s ∗ : H • sing, ∞ ( M, L ) ∼ = −→ H • sing ( M, L ) . Since the inverses are induced by the restrictions S • ( M ) → S •∞ ( M ) and S • ( M, L ) → S •∞ ( M, L ), which are clearly multiplicative, the induced maps s ∗ on cohomologyare ring isomorphisms.Let us denote the linear dual to s by s † . We define the de Rham maps ρ : Ω • ( M ) → C • ( M ) and ρ rel : Ω • ( M, L ) → C • ( M, L ) by the following commutative diagrams.Ω • ( M ) S •∞ ( M ) S • ( M ) b S • ( M ) C • ( M ) , ρ s ρ s † restrδ where ρ s is the classical de Rham map andΩ • ( M, L ) S •∞ ( M, L ) S • ( M, L ) b S • ( M, L ) C • ( M, L ) , ρ s,rel ρ rel s † restrδ rel where ρ s,rel is the relative de Rham map of Theorem 1. All the maps we usedto define the de Rham maps ρ and ρ rel are DGA morphisms, or at least inducering isomorphisms on cohomology. For the classical de Rham maps this followsby [7, Theorem III-3.1] in the absolute case and by the relative de Rham Theorem inSection 2.4 in the relative case. Hence the maps ρ and ρ rel induce ring isomorphismson cohomology. This proves the following theorem. Theorem 5 (Cellular multiplicative relative de Rham Theorem) . Let L ⊂ M bea smooth submanifold of the smooth manifold M . Choose a CW-structure on Msuch that L ⊂ M is also a CW-subcomplex. Then, the map ρ rel induces a ringisomorphism ρ ∗ rel : H • dR ( M, L ) → H • ( C • ( M, L )) . Multiplicativity of ˜ ρ . In this section, we construct a map ˜ ρ that translatesthe construction of Ω I • ¯ p ( ¯ X ) to cochains of CW complexes. We then use Theorem5 to prove that ˜ ρ induces a multiplicative map on cohomology.Recall that Ω I • ¯ p ( ¯ X ) := { ω ∈ Ω • ( ¯ X ) | i ∗ ∂ ( ω ) ∈ τ ≥ k Ω • ( L ) } , so the de Rham map ρ ¯ X on Ω • ( ¯ X ) restricts to a map ρ ¯ X | on Ω I • ¯ p ( ¯ X ). By thenaturality of the de Rham map its restriction ρ ¯ X | factors over¯ C • := { x ∈ C • ( ¯ X ) | i ∂ ( x ) ∈ T ≥ k C • ( L ) } with T ≥ k C • ( L ) the naive cotruncation (i.e. 0 in degrees lower than k and C • ( L )in degrees greater than or equal to k ). We define ˜ ρ to be this factor. So with incl. : ¯ C • → C • ( ¯ X ) the sub-complex inclusion we have ρ ¯ X | = incl. ◦ ˜ ρ .Note that with the definition C • ( ¯ X, L ) := { x ∈ C • ( ¯ X ) | i ∂ ( x ) = 0 } we have sub-complex inclusions C • ( ¯ X, L ) ⊂ ¯ C • ⊂ C • ( ¯ X ), analogously to the in-clusions Ω • ( ¯ X, L ) ⊂ Ω I • ¯ p ( ¯ X ) ⊂ Ω • ( ¯ X ). Recall that in the proof of Theorem 3 weestablished that i ∂ is multiplicative and the product ∪ ¯ X on C • ( ¯ X ) from Section3 restricts to C • ( ¯ X, L ). The restriction of ∪ ¯ X to ¯ C • is also well-defined due tothe construction of ¯ C • via cotruncation together with the graded nature of the cupproduct. Therefore the inclusions above are sub-DGA inclusions. Proposition 4.1. The map ˜ ρ induces a multiplicative map on cohomology.Proof. Let ω ∈ Ω I q ¯ p ( ¯ X ) and η ∈ Ω I r ¯ p ( ¯ X ) be two closed forms. Let us consider thecase q = 0 and r arbitrary. The case q arbitrary and r = 0 is analogous. For q = 0, the closed form ω is a constant function. Recall that k = n − − ¯ p ( n ) with¯ p a Goresky-MacPherson perversity function. The definition of these perversityfunctions directly implies k ≥ 1. Therefore, Ω I p ( ¯ X ) = Ω ( ¯ X, L ) and ω has tovanish on the boundary. We conclude that ω = 0 and˜ ρ ( ω ) ∪ ˜ ρ ( η ) = 0 ∪ ˜ ρ ( η )= 0= ˜ ρ (0 ∧ η )= ˜ ρ ( ω ∧ η ) . Thus the multiplicativity already holds on cochain level.Next, consider the case q and r < k . Here, we have ω ∈ Ω q ( ¯ X, L ) and η ∈ Ω r ( ¯ X, L ). Using that ρ rel is a restriction of ˜ ρ we calculate˜ ρ ( ω ) ∪ ˜ ρ ( η ) = ρ rel ( ω ) ∪ ρ rel ( η )= ρ rel ( ω ∧ η ) + dx for some x ∈ C q + r − ( ¯ X, L ) since ρ rel induces a multiplicative map on cohomologyif we take the relative de Rham theorem into account. Since C q + r − ( ¯ X, L ) ⊂ ¯ C q + r − , we have x ∈ ¯ C q + r − . Further, we observe that since ω and η vanish onthe boundary, their wedge product does so, too. So, ρ rel ( ω ∧ η ) = ˜ ρ ( ω ∧ η ) and,combing all these facts, we see[˜ ρ ( ω ) ∪ ˜ ρ ( η )] = [˜ ρ ( ω ∧ η ) + dx ] = [˜ ρ ( ω ∧ η )] ULTIPLICATIVE DE RHAM THEOREMS FOR RELATIVE COHOM. AND HI 25 with [ . . . ] denoting the cohomology class in the cohomology of ¯ C • .Finally, let q ≥ k and r ≥ 1, and in particular q + r ≥ k + 1. We use Ω I • ¯ p ( ¯ X ) ⊂ Ω • ( ¯ X ) and that ˜ ρ is a restriction of ρ ¯ X to calculate˜ ρ ( ω ) ∪ ˜ ρ ( η ) = ρ ¯ X ( ω ) ∪ ρ ¯ X ( η )= ρ ¯ X ( ω ∧ η ) + dx = ˜ ρ ( ω ∧ η ) + dx for some x ∈ C q + r − ( ¯ X ). The last line follows because the classical de Rham map ismutliplicative on cohomology. We used that if q + r ≥ k + 1, then ˜ ρ = ρ ¯ X . Further, C q + r − ( ¯ X ) = ¯ C q + r − and the multiplicativity also holds in the cohomology of ¯ C • ,analogously as for the case above. (cid:3) Multiplicativity of φ . In Section 3 and 4.2 we obtained the maps φ : C • ( I ¯ p X ) → Q • and ˜ ρ : Ω I • ¯ p ( ¯ X ) → ¯ C • which induce ring isomorphisms on coho-mology. To complete our intersection space cohomology de Rham map, we constructthe connecting piece φ : ¯ C • → Q • via the universal property of the pullback Q • applied to the diagram(11) ¯ C • T ≥ k C • ( L ) Q • e C • ( cone ( t There is a DGA homomorphism X : T ≥ k C • ( L ) → e C • ( cone ( t 1. In conclusion,the diagram commutes for all degrees q and r independent of the cut-off value k .Thus X is not only a cochain map but a DGA homomorphism for all k .Finally, recall that i ˜ ( b , b ) = b , implying i ˜ ◦ X = f | T ≥ k C • ( L ) . (cid:3) ULTIPLICATIVE DE RHAM THEOREMS FOR RELATIVE COHOM. AND HI 27 Let us point out that the proof above and especially the part concerning themultiplicativity of X abused the fact that the cellular cochain complexes vanishabove the dimension of the space. This is the primary motivation to work withcellular cochains in this article. In order to justify the construction of φ via theuniversal property of the pullback we prove Lemma 4.3. The diagram ¯ C • T ≥ k C • ( L ) e C • ( cone ( t Note that changing the sign of the first map preserves the exactness, therefore wehave the short exact sequence(13) 0 C •− ( t 7→ − (0 , b ) . We introduce the sign here since it will be necessary to have commutativity later.Let us remark that while π is a cochain map, I is only a cochain map up to sign,however this is enough to induce a long exact sequence on cohomology.This sequences combines with the long exact sequence induced from the shortexact sequences of Lemma 4.4 into the following diagram.(14) H q − (Ω • ( ¯ X )) H q − ( τ Lemma 4.5. Diagram (14) commutes at least up to a sign and has exact rows.Proof. The top and bottom rows are exact since they are part of the long exactsequence induced from the short exact sequence from Lemma 4.4 and the Sequence13, respectively. In the following we prove the commutativity of this diagram.Let us start with the commutativity of the left square. If q − k, the cohomology group H q − ( C • ( t ULTIPLICATIVE DE RHAM THEOREMS FOR RELATIVE COHOM. AND HI 31 Finally, we consider the right square. π ◦ φ ( ω ) = ( π ◦ ( ρ ¯ X | ( ω ) , deg ( ω ) = kπ ◦ ( ρ ¯ X | ( ω ) , x ◦ f ,k | T ≥ k C • ( L ) ◦ ˜ ρ L ◦ i ∂ ( ω )) deg ( ω ) = k = ρ ¯ X | ( ω )= ρ ¯ X ◦ incl. ( ω )This proves the commutativity of the square already on cochain level. (cid:3) Furthermore, ρ L is a quasi-isomorphism and f induces an isomorphism on co-homology in degrees lower than k . Thus their composition also induces an isomor-phism in degrees lower than k . Restricting to the truncated complex τ In [2, Section 9], Banagl constructs an alternative de Rham map, which is definedby integrating forms in Ω I • ¯ p ( ¯ X ) over smooth cycles on the blowup ¯ X of X . We recallhis construction and show that his de Rham map is compatible with the de Rhamring isomorphism of Section 4.Banagl uses a partial smooth model ( S ∝• ( g ) , ∂ ) for the mapping cone of the map g : t This map is well defined by the same arguments as in [2, Prop. 9.8] and fits intothe following commutative diagram of isomorphisms(16) H • (cid:0) Ω I • ¯ p ( ¯ X ) (cid:1) H • ( S ∝• ( g )) † H • (cid:0) ¯ S • ( g ) (cid:1) † . φ B ¯ φ B ( b s ⊕ id) † The choice made to define the partial smooth complex S ∝• ( g ) corresponds to achoice for the map x : C k ( t B ∪ J ∪ { y , · · · , y s } to a basis of C k − ( t Before stating the compatibility theorem, note that we do not write allthe decorations of the different geometric realizations γ in the following. We do soto make the theorem and proof more readable. We encourage the reader to checkwhich γ is used in the respective situation. For the sake of readability we will in thefollowing also write φ for the map φ ∗ . Theorem 7. Let the map x : C k ( t Let ω ∈ Ω I q ¯ p ( ¯ X ) and let ( v, x ) ∈ ¯ S p ( g ) be closed. Then¯ φ B ([ ω ]) ([( v, x )]) = Z sv ω. On the other hand, equation (12) yields φ ([ ω ]) ([( γ v, γ qx )])= ( ρ ¯ X | ( ω )( γ v ) , p = k (cid:16) x (cid:16) f k | T ≥ k C • ( L ) ◦ ˜ ρ L ◦ i ∂ ( ω ) (cid:17)(cid:17) ( γ qx ) + ( ρ ¯ X | ( ω )) ( γ v ) , p = k . To be precise, the map φ denoted here is the composition of the actual de Rhammap φ with the isomorphism e H • ( I ¯ p X ) ∼ = −→ e H • ( I ¯ p X ) † . Note that (cid:16) x (cid:16) f k | T ≥ k C • ( L ) ◦ ˜ ρ L ◦ i ∂ ( ω ) (cid:17)(cid:17) ∈ span (cid:16)n ( ∂ k x ) † , · · · , ( ∂ k x r ) † o(cid:17) , by our choice for the map x . Our choice of basis for C k − ( t Note, that only H • (cid:0) Ω I • ¯ p ( ¯ X ) (cid:1) and e H • ( I ¯ p X ) are naturally equipped witha cup product. The question, if any of the maps besides φ are ring isomorphisms,therefore depends on our choice of product on H • (cid:0) ¯ S • ( g ) (cid:1) † and H • ( S ∝• ( g )) † . Sinceall the maps in the diagram are isomorphisms, they are multiplicative if and onlyif we choose the products that are obtained by transporting the naturally definedproducts via those isomorphisms. All products defined in this way are consistentdue to the commutativity of the diagram.This means that Banagl’s de Rham isomorphism φ B is a ring isomorphism ifand only if we define the cup product ∪ ∝ on H • ( S ∝• ( g )) as the transport of the cupproduct ∪ I ¯ p X of e H • ( I ¯ p X ) via the isomorphism I : H • ( S ∝• ( g )) † ∼ = −→ e H • ( I ¯ p X ) , i.e. α ∪ ∝ β := I − ( I ( α ) ∪ I ¯ p X I ( β )) for α, β ∈ H • ( S ∝• ( g )) † . It is unclear though if this cup product on cohomology level comes from a cup producton the cochain complex S •∝ ( g ) , defined in analogy to S ∝• ( g ) . A cup product on thestandard mapping cone S • ( g ) can be defined by ( ψ, µ ) ∪ ( ξ, ν ) := ( ψ ∪ ξ, µ ∪ g ξ ) . The definition of S •∝ ( g ) uses the map q , however, which depends on choices andcannot be expected to be natural. Therefore, it is open whether this constructionalso gives a cup product on the cochain complex S •∝ ( g ) . Acknowledgements. We thank Prof. Dr. Markus Banagl (Universit¨at Heidel-berg) for his input as supervisor of the master thesis of the first author on whichparts of this article are based. The second author wants to thank the Canon Foun-dation, that supported him during his stay at the Hokkaido University, Japan, andProf. Toru Ohmoto for being a generous host. References [1] Markus Banagl. Intersection Spaces, Spatial Homology Truncation, and String Theory , vol-ume 1997 of Lecture Notes in Mathematics . Springer-Verlag, Berlin, 2010.[2] Markus Banagl. Foliated stratified spaces and a De Rham complex describing intersectionspace cohomology. J. Differential Geom. , 104(1):1–58, 2016. [3] Markus Banagl and Eugenie Hunsicker. Hodge Theory for Intersection Space Cohomology. arXiv e-prints , page arXiv:1502.03960, Feb 2015. to appear in Geom. Topol.[4] J.-P. Brasselet, G. Hector, and M. Saralegi. Th´eor`eme de de Rham pour les vari´et´es stratifi´ees. Ann. Global Anal. Geom. , 9(3):211–243, 1991.[5] J. P. Brasselet and A. Legrand. Differential forms on singular varieties and cyclic homology.In Singularity theory (Liverpool, 1996) , volume 263 of London Math. Soc. Lecture Note Ser. ,pages xviii, 175–187. Cambridge Univ. Press, Cambridge, 1999.[6] Jean-Paul Brasselet and Andr´e Legrand. Un complexe de formes diff´erentielles `a croissanceborn´ee sur une vari´et´e stratifi´ee. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) , 21(2):213–234,1994.[7] Glen E. Bredon. Sheaf theory , volume 170 of Graduate Texts in Mathematics . Springer-Verlag,New York, second edition, 1997.[8] Jean-Luc Brylinski. Equivariant intersection cohomology. In Kazhdan-Lusztig theory and re-lated topics (Chicago, IL, 1989) , volume 139 of Contemp. Math. , pages 5–32. Amer. Math.Soc., Providence, RI, 1992.[9] J. Timo Essig. About a de rham complex discribing intersection space cohomology in a non-isolated singularity case. Master’s thesis, University of Heidelberg, 2012.[10] J. Timo Essig. Intersection Space Cohomology of Three-Strata Pseudomanifolds. arXiv e-prints , page arXiv:1804.06690, Apr 2018. to appear in J. Topol. Anal.[11] Rudolf Fritsch and Renzo A. Piccinini. Cellular structures in topology , volume 19 of Cam-bridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1990.[12] M. Goresky and R.D. MacPherson. Intersection homology theory. Topology , 19(2):135–162,1980.[13] M. Goresky and R.D. MacPherson. Intersection homology ii. Invent. Math. , 72(1):77–129,1983.[14] John M. Lee. Introduction to Smooth Manifolds , volume 218 of Graduate Texts in Mathe-matics . Springer, New York, second edition, 2013.[15] William S. Massey. Singular homology theory , volume 70 of Graduate Texts in Mathematics .Springer-Verlag, New York-Berlin, 1980.[16] J. P. May. A Concise Course in Algebraic Topology . Chicago Lectures in Mathematics. Uni-versity of Chicago Press, Chicago, IL, 1999.[17] Martin Saralegi. Homological properties of stratified spaces. Illinois J. Math. , 38(1):47–70,1994. Department of Mathematics and its Applications, University of Milano-Bicocca, ViaCozzi 55, 20125 Milano, Italy E-mail address : [email protected] Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060-0810, Japan E-mail address ::