Alternative to Morse-Novikov Theory for a closed 1-form (II)
aa r X i v : . [ m a t h . A T ] O c t Alternative to Morse-Novikov Theory for a closed 1-form (II)
Dan Burghelea * Abstract
This paper is a continuation of [1] and establishes:– a refinement of Poincar´e duality to an equality between the configurations BM δ ω and δ ω resp. BM γ ω and γ ω in complementary dimensions,– the stability property for the configurations δ ωr , – a collection of additional results needed for the proof of Theorems 1.2 and 1.3 in [1] . Contents
The reader is supposed to be familiar with the notations and the main definitions in [1]. This paper is acontinuation of [1] and establishes:– a refinement of Poincar´e duality to an equality between the configurations BM δ ω and δ ω resp. BM γ ω and γ ω in complementary dimensions– the stability property for the configurations δ ωr , – a collection of additional results needed for the proof of the announced Theorems 1.2 and 1.3 in [1] .Since we are interested in Poincar´e duality which has to be considered for non compact manifolds, it isnecessary to involve the functor BM H r , the Borel-Moore homology, abbreviated BM-homology and thenatural transformation θ r : H r → BM H r . We refer to [7] chapter 5 for reference to Borel Moore homology.Recall that Borel-Moore homology with coefficients in a field κ is a collection BM H r of κ − vector spacevalued functors defined on the category of pairs ( X, Y ) , X and Y locally compact spaces, with Y ⊆ X closed subset of X and proper continuous maps of pairs, which are homotopic functors (i.e. properhomotopic maps induce equal linear maps) and have excision and the long exact sequence property for any * Department of Mathematics, The Ohio State University, Columbus, OH 43210,USA. Email: [email protected] ( X, Y ) . When restricted to the subcategory of pairs of compact spaces they coincide with thestandard homology. The reader unfamiliar with Borel-Moore homology should know that for ( M, N ) a pairof f.d. manifolds with N a closed subset, BM H r ( M, N ) = H r ( ˆ M , ˆ N ) and BM H r ( M ) = H r ( ˆ M , ∗ ) where ˆ X denotes the one point compactification of the locally compact space X. This explains the natural maps H r ( · · · ) → BM H r ( · · · ) from the singular homology (= standard homology) to the Borel- Moore homologyas well as the contravariant behavior of Borel-Moore homology when restricted to open sets of a locallycompact space.The Borel-Moore homology is not a priory defined for a pair ( X, Y ) when both are locally compact but Y is not closed, however a vector space BM H fr ( X, Y ) can be introduced in the particular case Y is open in X and appears as the interior of a sub-level for a tame map as described in section 3.By passing to direct limit when ǫ → the long exact sequence for the pair ( ˜ X a , ˜ X a − ǫ ) leads to the longexact sequence · · · / / BM H r ( ˜ X
Suppose M is a closed topological manifold and ω ∈ Z t ( X ; ξ ) . Then the following holds true.1. BM δ ωr ( t ) = δ ωn − r ( − t ) , BM γ ωr ( t ) = γ − ωn − r − ( t ) . Theorem 1.2 The assignment δ r : Z t ( X ; ξ ) Conf β Nr ( X : ξ ) ( R ) is continuous and extends to a continuous assign-ment on the entire Z ( X ; ξ ) , when the source Z t ( X ; ξ ) is equipped with the compact-open topology andthe target Conf β Nr ( X : ξ ) ( R ) with the collision topology. We will also establish the following technical results essential for the proofs in Theorems 1.2 and 1.3 in[1]. If f : ˜ X → R is a lift of a tame TC1-form ω on a closed compact ANR then for both the standard andthe Borel-Moore homology one has the following equalities. the Novikov-Betti number of ( X, ξ ) , ξ ∈ H ( X : κ ) roposition 1.3 dim( H r ( ˜ X a , ˜ X a γ fr ( a, t ) + P γ fr − ( t, a ) . dim( BM H r ( ˜ X a , ˜ X a BM γ fr ( a, t ) + P t
Proposition 1.4 If f : ˜ X → R is a lift of a tame TC1-form ω on a closed topological manifold X, then the linear maps θ r ( a ) : H r ( ˜ X a , ˜ X
The canonical isomorphisms θ extend to the canonical isomorphisms θ ( D ) : (c okerD ) ∗ → ker( D ∗ ) and θ ( D ) : (ker D ) ∗ → c oker ( D ∗ ) . Proof: : To check the statements observe that the diagram D can be completed to the diagrams D and DD := A α (cid:15) (cid:15) α $ $ ❏❏❏❏❏❏❏❏❏❏ α / / B i z z tttttttttt β (cid:15) (cid:15) α ∪ A α β $ $ ❏❏❏❏❏❏❏❏❏❏ B i : : ttttttttt β / / C D := A α (cid:15) (cid:15) α / / α $ $ ■■■■■■■■■■ B β (cid:15) (cid:15) β × C β p z z ✉✉✉✉✉✉✉✉✉ p : : ✉✉✉✉✉✉✉✉✉ β $ $ ■■■■■■■■■■ B β / / C and notice that ( D ) ∗ identifies to ( D ∗ ) and ( D ) ∗ identifies to ( D ∗ ) which imply the statements.q.e.d.Let α : A → B, β : B → C, γ : C → D be linear maps. To these three maps we associate thediagrams, b D and D : b D ( α, β, γ ) ≡ ker( γβα ) j / / ker( γβ )ker( βα ) j / / i O O k rrrrrrrrrr ker( β ) i O O D ( α, β, γ ) ≡ c oker ( γβα ) j ′ / / c oker ( γβ )c oker ( βα ) j ′ / / i ′ O O k ′ ♦♦♦♦♦♦♦♦♦♦♦ c oker ( β ) i ′ O O . In the diagram ˆ D (a) i and i are injective, hence one can write ker β ⊆ ker( γβ ) , (b) i : ker j → ker j is an isomorphism,(c) i mg k = i mg j ∩ i mg i and in the diagram D (a) j ′ and j ′ are surjective,(b) c okeri ′ → c okeri ′ is an isomorphism,(c) ker( j ′ · i ′ = i ′ · j ′ 1) = ker i ′ + ker j ′ . 4n view of Observation 2.1 one has Observation 2.2 ( ˆ D ( α, β, γ )) ∗ = D ( γ ∗ , β ∗ , α ∗ ) and ( D ( α, β, γ )) ∗ = ˆ D ( γ ∗ , β ∗ , α ∗ ) . One defines the vector space ˆ ω ( α, β, γ ) = c oker ˆ D ( α, β, γ ) := c oker ( j ∪ ker( βα ) i → ker( γβ ))= ker( γβ ) / ( j (ker( γβα )) + i (ker β )) , (2)a quotient space of ker( γβ ) , and the vector space ω ( α, β, γ ) = ker D ( α, β, γ ) := ker(c oker ( βα ) → i ′ × c oker ( γβ ) j ′ ) , a subspace of c oker ( βα ) . Note that the assignments ( α, β, γ ) b ω ( α, β, γ ) and ( α, β, γ ) ω ( α, β, γ ) are functorial and in viewof the definitions above if α is surjective or if γ is injective then ˆ ω ( α, β, γ ) = 0 . Theorem 2.3 For α, β, γ linear maps, the isomorphisms θ extend to the canonical isomorphism θ : ˆ ω ( α, β, γ ) ∗ → ω ( γ ∗ , β ∗ , α ∗ ) θ : ω ( α, β, γ ) ∗ → ˆ ω ( γ ∗ , β ∗ , α ∗ ) . For α, β, γ and α ′ , β ′ , γ ′ linear maps consider the diagram M = / / M = / / M = / / MA α / / λ A O O B β / / λ B O O C γ / / λ C O O D d O O A ′ α ′ / / a O O B ′ β ′ / / b O O C ′ γ ′ / / c O O D ′ d O O N θ A O O = / / N θ B O O = / / N θ C O O = / / N. θ D O O (3) If the columns are exact sequences then the linear maps b ω ( α ′ , β ′ , γ ′ ) → b ω ( α, β, γ ) and ω ( α, β, γ ) → ω ( α, β, γ ) are isomorphisms. For α, β, γ linear maps the following holds true. (i) A factorization α = α · α , with α : A → A ′ and α : A ′ → B linear maps, induces the short exactsequences → b ω ( α , βα , γ ) → b ω ( α, β, γ ) → b ω ( α , β, γ ) → , → ω ( α , βα , γ ) → ω ( α, β, γ ) → ω ( α , β, γ ) → . (4)5ii) A factorization β = β · β , with β : B → B ′ and β : B ′ → B linear maps, induces the short exactsequences → b ω ( α, β , β ) → b ω ( α, β , γβ ) → b ω ( α, β, γ ) → , → ω ( α, β , β ) → ω ( α, β , γβ ) → ω ( α, β, γ ) → . (5)(iii) A factorization γ = γ · γ , with γ : C → C ′ and γ : C ′ → D maps, induces the short exactsequences → b ω ( α, β, γ ) → b ω ( α, β, γ ) → b ω ( α, γ β, γ ) → , → ω ( α, β, γ ) → ω ( α, β, γ ) → ω ( α, γ β, γ ) → . (6) Proof: Item 1. follows from (2.1) and (2.2).To prove Item 2. notice the following.i. If N = 0 , then ker( β ) ≃ ker( β ′ ) , ker( βγ ) ≃ ker( β ′ γ ′ )ker( αβ ) ≃ ker( α ′ β ′ ) , ker( αβγ ) ≃ ker( α ′ β ′ γ ′ ) . hence ˆ D ( α ′ , β ′ , γ ′ ) = ˆ D ( α, β, γ ) and then ˆ ω ( α ′ , β ′ , γ ′ ) = ˆ ω ( α, β, γ ) . ii. If M = 0 then the obvious induced maps ker( β ′ α ′ ) → ker( βα )ker( γ ′ β ′ α ′ ) → ker( γβα )ker( γ ′ β ′ ) → ker( γβ )ker( β ′ ) → ker( β ) are surjective and then in view of the definition of ˆ ω the induced linear map ker( β ′ , γ ′ ) / ( i (ker( γβ, α ) + ker( β )) → ker β, γ/ ( i (ker( γβ, α ) + ker( β )) is surjective.To show that it is equally injective one has to verify that any x ′ ∈ ker( γ ′ β ′ ) with b ( x ′ ) = α ( z ) + y,z ∈ ker( γβα ) and y ∈ ker( β ) , can be written as x ′ = α ( z ′ ) + y ′ with z ′ ∈ ker( γ ′ β ′ α ′ ) and y ∈ ker( β ′ ) . In view of the above mentioned surjectivity one chooses z ′ ∈ ker( γ ′ β ′ α ) and y ′ ∈ ker( β ′ ) , with a ( z ′ ) = z and b ( y ′ ) = y, and one considers x ′ − α ′ ( z ′ ) − y ′ ∈ ker( γ ′ β ′ ) which by b is sent tozero, hence is of the form θ B ( u ) , u ∈ N. Clearly γ ′ β ′ ( θ B ( u )) = 0 , therefore one can correct z ′ into z ′ = z ′ + θ A ( u ) , and one obtains x ′ = α ′ ( z ′ ) + y ′ with z ′ ∈ ker( γ ′ β ′ α ′ ) . Consequently, the induced map ˆ ω ( α ′ , β ′ γ ′ ) → ˆ ω ( α, β, γ ) is an isomorphism.iii. To prove the result for M and N arbitrary consider the diagram A α / / B β / / C γ / / D ker a α ′′ / / O O ker b β ′′ / / O O ker c γ ′′ / / O O ker d O O A ′ α ′ / / O O B ′ β ′ / / O O C ′ γ ′ / / O O D ′ O O b ω ( α ′ , β ′ , γ ′ ) → b ω ( α ′′ , β ′′ , γ ′′ ) → b ω ( α, β, γ ) . The first arrow is an isomorphism by (ii) above and the second by (i) above. This establishes Item 2.To prove Item 3 consider the diagram A i A / / i ! ! B i B / / C A j A O O i > > = = ④④④④④④④④ i A / / ♠♠♠♠♠♠♠♠♠♠♠♠♠♠♠ B j B O O > > ⑤⑤⑤⑤⑤⑤⑤⑤ i B / / C j C O O (7)and make the following observation. Observation 2.4 Suppose that each of the three diagrams B , B , B , associated with (7), B with vertices A , A , B , B , B with vertices B , B , C , C and B with vertices A , A , C , C satisfy the properties (a) (b) (c) of the diagram ˆ D. Then (7) induces the exact sequence B / ( i A ( A ) + j B ( B )) i & & ▼▼▼▼▼▼▼▼▼▼▼ O O C / ( i ( A ) + j C ( C )) p & & ▼▼▼▼▼▼▼▼▼▼ C / ( i B ( B ) + j C ( C )) O O with i induced by i B , well defined because i mg ( i B · j B ) ⊆ i mgj C , and p the projection induced by theinclusion ( i ( A ) + j C ( C )) ⊆ ( i B ( B ) + j C ( C )) . Clearly p is surjective and p · i = 0 . Property (c) implies i is injective. Properties (a), (b) (c) imply thatthe sequence is exact. A similar observation holds for the diagram A i / / B A i / / > > ⑤⑤⑤⑤⑤⑤⑤⑤ j A O O B j B O O A j A ? ? j A O O > > ⑤⑤⑤⑤⑤⑤⑤⑤ F F ✌✌✌✌✌✌✌✌✌✌✌✌✌✌✌ i / / B j B _ _ j B O O (8) Observation 2.5 Suppose that each of the three diagrams B , B , B , associated with (8), B with vertices A , A , B , B , B with vertices A , A , B , B and B with vertices A , A , B , B atisfy the properties (a) (b) (c) of the diagram ˆ D Then (8) induces the exact sequence B / ( i ( A ) + j B ( B )) j & & ▼▼▼▼▼▼▼▼▼▼ O O B / ( i ( A ) + j B ( B )) p & & ▼▼▼▼▼▼▼▼▼▼ B / ( i ( A ) + j B ( B )) O O Observation 2.4 applied to diagram (7) with A = ker( βα ) , A = ker( γβα ) ,B = ker( βα ) , B = ker( γβα ) ,C = ker( β ) , C = ker( γβ ) , verifies Item 3. (i).Observation 2.5 applied to diagram (8) with A = ker( β α ) , B = ker( β ) ,A = ker βα ) , B = ker( β ) ,A = ker( γβα ) , B = ker( γβ ) verifies Item 3. (ii) and applied to diagram (8) with A = ker( βα ) , B = ker( β ) ,A = ker( γ βα ) , B = ker( γ β ) ,A = ker( γβα ) , B = ker( γβ ) , verifies Item 3. (iii).The above considerations/proofs were already contained in [6] but under the additional hypothesis thatall the linear maps were Fredholm.As in [8] for a direct sequence of vector spaces { A i , u i } := A u / / A u / / A u / / · · · u i − / / A i u i / / A i +1 u i +1 / / · · · consider Σ : ⊕ i A i → ⊕ A i defined by Σ( a , a , · · · a i , · · · ) = ( a , a − u ( a ) , · · · a i − u i − ( a i − ) , · · · ) . Note that ker Σ = 0 and recall the definition of the direct limit lim −→ i { A i , u i } := c oker Σ . For an inverse sequence of vector spaces { A i , p i } := A A p o o A p o o · · · p o o A ip i o o A i +1 p i +1 o o · · · o o Π : ⊗ i A i → ⊗ A i defined by Π ( a , a , · · · a i , · · · ) = ( a , a − u ( a ) , · · · a i − u i − ( a i − ) , · · · ) and recall the definitions of the inverse limit and the derived inverse limit lim ←− i { A i , p i } := ker Π lim ←− i ′ { A i , p i } := c okerΠ For the proof of Theorem 1.1 one needs some additional elementary linear algebra. Definition 2.6 A quasi-surjection ˜ π : A A ′ is a pair ˜ π := { π : A → P, P ⊃ A ′ } consisting of asurjective linear map π and a subspace A ′ of B. To a directed system of quasi-surjections A π / / /o/o/o A π / / /o/o/o A π / / /o/o/o · · · ˜ π k − / / /o/o/o A k ˜ π k / / /o/o/o A k +1 ˜ π k +1 / / /o/o/o · · · one provides a unique maximal directed system of surjections A A A A k A k +1 A ∞ π ′ / / / / ⊆ O O A ∞ π ′ / / / / ⊆ O O A ∞ π / / / / ⊆ O O · · · ˜ π k − / / / / A ∞ k ˜ π k / / / / ⊆ O O A ∞ k +1 ˜ π k +1 ⊆ ✷✷✷✷✷✷✷✷ / / / / ⊆ O O · · · The construction is rather straightforward and goes as it followsStarting with the right side of the diagram (9) (i.e. the collections of linear maps { π i , ∪} , ) one inductively(from right to left i.e from lower index i to lower index ( i − and from upper index k to upper index ( k +1) )produces the subspace A k +1 i ⊂ A ki , ∞ ≥ i > k and the linear maps shown in the diagram; all horizontalarrows π ki : A ki → A ki +1 surjective and all vertical arrows injective (inclusions), with A k +1 k = ( π k ) − ( A k +1 ) and A ∞ k = ∩ i>k A ki . A k +1 i := ( π ki ) − ( A k +1 i +1 ) . A π / / B A / / O O A π / / ∪ O O B A / / O O A / / O O A π / / ∪ O O B · · · · · · · · · · · · · · · A k / / O O A k / / O O A k / / O O A k / / O O · · · / / A k π k / / ∪ O O B k A k +11 / / O O A k +12 / / O O A k +13 / / O O A k +14 / / O O · · · / / A k +1 k / / O O A k +1 π k +1 / / ∪ O O B k +1 · · · · · · · · · · · · · · · · · · · · · · · · A ∞ / / O O A ∞ / / O O A ∞ / / O O A ∞ / / O O · · · / / A ∞ k / / O O A ∞ k +1 / / O O · · · (9) lim −→ i →∞ ˜ π i : = lim −→ i →∞ π ′ i . (10) Most of the definitions and notations below are the same as in [1] where they were considered for standardhomology only. However, they can be considered for any homology theory, in particular for Borel-Moorehomology of interest in this paper. In this paper a homology theory is a collections of κ − vector spacevalued covariant homotopy functors denoted by H r ( · · · ) defined on the category of pairs of locally compactHausdorff spaces ( X, Y ) , Y closed subset of X, and of proper continuous maps which satisfy the Eilenberg-Steenrod axioms and Milnor continuity axiom. Recall that the continuity axiom as used here states that if X (0) · · · ⊂ X ( i ) ⊆ X ( i + 1) ⊆ · · · X is a filtration of a locally compact ANR X by the ANRs X ( i ) , closedsubsets, with X = ∪ i X ( i ) then H r ( X ) = lim −→ i H r ( X i ) . In this section any such homology theory, in particular the standard and the Borel-Moore homology, will bedenoted by H r . In the next sections the notation H r will be reserved exclusively for singular homology andBorel Moore will acquire the left side exponent (BM) (e.g. BM H r ).In consistency with the notation in [1], for a continuous map f : Y → R one denotes by Y t := f − (( −∞ , t ]) and by Y Proposition 3.1 (cf. [1]) The boxes B , B , B as above induce the commutative diagram whose rows areexact sequences and vertical arrows are isomorphisms. / / F r ( B ) i B B / / θ r ( B ) (cid:15) (cid:15) F r ( B ) π B B / / θ r ( B ) (cid:15) (cid:15) F r ( B ) / / θ r ( B ) (cid:15) (cid:15) / / G r ( B ) i B B / / G r ( B ) π B B / / G r ( B ) / / Observation 3.2 As a consequence if B ⊂ B are boxes with B located in the upper-left corner then theinduced linear map i BB : F r ( B ) → F r ( B ) is injective and if B is located in the down-right corner thenthe induced linear map π B B : F r ( B ) → F r ( B ) is surjective. In Figure 1 below B is in the upper-left corner of B and B is in the lower-right corner of B.b ′′ b ′ b a ′′ a ′ aB B B B Figure 1Define F ˆ δ r ( a, b ) := lim −→ ǫ,ǫ ′ → F r (( a − ǫ, a ] × [ b, b + ǫ ′ )) and G ˆ δ r ( a, b ) := lim −→ ǫ,ǫ ′ → G r (( a − ǫ, a ] × [ b, b + ǫ ′ )) . In view of (11) one has F ˆ δ fr ( a, b ) = G ˆ δ fr ( a, b ) . Denote by ˆ δ fr ( a, b ) := F ˆ δ fr ( a, b ) = G ˆ δ fr ( a, b ) , δ fr ( a, b ) : dim ˆ δ fr ( a, b ) . Note also that ˆ δ r ( a, b ) = F r ( a, b ) F r ( < a, b ) + F r ( a, > b ) (12)and in view of (12) one has the obvious surjective linear map π δa,b ( r ) : F r ( a, b ) → ˆ δ fr ( a, b ) . As in [1] for f : ˜ X → R a tame map and a, b ∈ R with a < b denote by i ba ( r ) : H r ( ˜ X a ) → H r ( ˜ X b ) resp. i Proposition 3.3 (cf. [1]) The boxes B , B , B as above induce the linear maps and the following exactsequence / / T r ( B ) i B B / / T r ( B ) π B B / / T r ( B ) / / . Observation 3.4 As a consequence if B ⊂ B is located as a down-left corner then the induced linear map i BB : T r ( B ) → T r ( B ) is injective and if B ⊂ B is located as the upper-right corner then the inducedlinear map π B B : T r ( B ) → T r ( B ) is surjective. In Figure 2 below B is in the down-left corner and B is in the upper-right corner. bb ′ b ′′ a ′′ a ′ aB B B B Figure 2(All boxes are supposed to be above diagonal and not necessary in the first quadrant as they appear inthis picture.) 13efine ˆ γ fr ( a, b ) := lim −→ ǫ,ǫ ′ → T r (( a − ǫ, a ] × ( b − ǫ ′ , b ]) , γ fr ( a, b ) = dim ˆ γ fr ( a, b ) and observe that ˆ γ fr ( a, b ) = lim −→ ǫ,ǫ ′ → T r ( a, b ) i a,ba − ǫ,b ( T r ( a − ǫ, b )) + T r ( a, b − ǫ ′ ) = T r ( a, b )( i a,b
1. The supports of δ fr and γ fr are subsets of CR ( f ) × CR ( f ) with the following properties:(a) If ( a, b ) ∈ s upp δ fr resp. ( a, b ) ∈ s upp γ fr then for any g ∈ Γ one has ( a + g, b + g ) ∈ s upp δ fr resp. ( a + g, b + g ) ∈ s upp γ fr . (b) For any a ∈ R s uppδ fr ∩ R × a , s uppδ fr ∩ a × R , s uppγ fr ∩ R × a, s uppγ fr ∩ a × R are finitesets, empty if a ∈ R \ Cr ( f ) . (c) There exists a finite set of lines ∆ δt i ( r ) resp. ∆ γt i ( r ) , in the plane R given by the equations y = x + t δi resp. y = x + t γi , i = 1 , , · · · N δr resp. i = 1 , , · · · N γr s.t. s upp δ fr ⊂ ∪ i =1 , ··· ,N δr ∆ δt δi resp. s upp γ fr ⊂ ∪ i =1 , ··· ,N γr ∆ γt δi . As a consequence define the maps δ ωr : R → Z ≥ and γ ω : R > → Z ≥ δ ωr ( t ) := ( δ fr ( a, b ) , t = b − a, ( a, b ) ∈ s upp δ fr , ( a, b ) / ∈ s uppδ fr γ ωr ( t ) := ( γ fr ( a, b ) , t = b − a, ( a, b ) ∈ s upp γ fr , ( a, b ) / ∈ s upp γ fr which are configurations (i.e. maps with finite support) with s upp δ ωr := { t δ , t δ , · · · t δN δr } resp. s upp γ ωr := { t γ , t γ , · · · t γN γr } with −∞ < t δ < t δ , · · · t δN δ ( r ) < ∞ , and < t δ < t γ , · · · t γN γr < ∞ . As in [1] for a ′ < α < a ≤ b one defines: T r (( a ′ , α ] × b ) := lim −→ ǫ → T r ( a ′ , α ] × ( b − ǫ, b ]) , ǫ < b − α , T r (( a ′ , a ) × b ) := lim −→ α
Proposition 3.6 For any a, b ∈ R , a < b (in both cases for standard and Borel-Moore homology) T r (( −∞ , a ] × b ) = T r ( a, b ) / T r ( a, < b ) T r (( −∞ , b ) × b ) = T r ( < b, b ) Proof: As pointed out above ∩ −∞