Iterated Integrals and higher order invariants
aa r X i v : . [ m a t h . DG ] S e p Iterated Integrals and higher order invariants
Anton Deitmar & Ivan HorozovCan. J. Math. 65, 544-552 (2013)
Abstract:
We show that higher order invariants of smooth functions canbe written as linear combinations of full invariants times iterated integrals.The non-uniqueness of such a presentation is captured in the kernel of theensuing map from the tensor product. This kernel is computed explicitly.As a consequence, it turns out that higher order invariants are a free moduleof the algebra of full invariants.MSC: , 11F12, 55D35, 58A10
Contents TERATED INTEGRALS Introduction
Modular forms of higher order have been studied extensively in recent years[CDO02,Dei09,DD09,DD10,DKMO06,DO08,DS06,DK09,DS08,Sre09]. Toconstruct them, one often uses iterated integrals. Form dimension formulae[DS08] it is clear that in the case of holomorphic forms, iterated integrals donot give all higher order forms. But there are strong indicators [DKMO06,DO08, DS06, Sre09] that in the case of smooth functions, all higher orderforms are indeed obtainable via iterated integrals. This has been an implicitopen question for a while which is answered affirmatively in the presentpaper. It is shown that on any smooth manifold, the smooth module ofhigher order invariants is generated by the space of homotopy invariantiterated integrals, thus allowing to deduce structure assertions on higherorder invariants from iterated integrals.In Section 1 we recall the ingredients of the theory of iterated integralsneeded in the sequel. In Section 2 we show that the restriction map fromfree homotopy invariant iterated integrals to loops is a surjective map. Thisassertion is the key to the further sections. In Section 3 we first show thathomotopy invariant iterated integrals always give higher order invariants.For the case of surfaces, this was proven by Sreekantan in [Sre09]. We thenuse the result from Section 3 to finally deduce the main result, saying thathigher order invariants can alway be expressed by iterated integrals. Inthe final section we consider restricted forms of higher order, which comewith triviality assumptions along boundary components. This is typical forautomorphic forms, where these restrictions refer to cusp forms or cuspidalcohomology. We formulate the corresponding theorem asserting that themain result is stable under boundary restrictions.
In this section we fix notations. Let X be a smooth connected manifold and x , x ∈ X points. We write P X for the path space , i.e., the set of all smoothmaps p : [0 , → X . We also write P X x for the subset of all paths thatstart at x and P X x ,x for the subset of all smooth paths from x to x . Thespace LX x = P X x ,x is also called the loop space at x . TERATED INTEGRALS p and 1-forms ω , . . . , ω r we define the iterated integral: Z p ω · · · ω r = Z Z t r · · · Z t p ∗ ω ( t ) p ∗ ω ( t ) . . . p ∗ ω r ( t r ) . For an integer s , let B s ( X ) denote the space of all maps ω : P X → C whichare linear combinations of iterated integrals of length ≤ s . Here we includeconstants as they may be considered as iterated integrals of length zero. Wealso write B ( X ) for the union of all B s ( X ) as s varies. Let T (Ω ( X )) = C ⊕ Ω ( X ) ⊕ (cid:2) Ω ( X ) ⊗ Ω ( X ) (cid:3) ⊕ . . . be the tensorial algebra over the space Ω ( X ) of smooth 1-forms. The mapassigning ω ⊗ · · · ⊗ ω r to the map p R p ω · · · ω r is a linear map from T (Ω ( X )) to B ( X ). This map has a non-trivial kernel which has beendetermined by Chen in [Che71].We denote by B s ( X ) x the set of restrictions of elements of B s ( X ) to P X x and the space B s ( X ) x ,x is defined analogously. Lemma 1.1 (a) If ϕ is an orientation preserving diffeomorphism on [0 , ,then R p ω · · · ω r = R p ◦ ϕ ω · · · ω r . (b) If F is a diffeomorphism on X , then R F ◦ p ω . . . ω r = R p ( F ∗ ω ) . . . ( F ∗ ω r ) . (c) If p and q are composable paths, then Z pq ω · · · ω r = r X j =0 Z p ω · · · ω j Z q ω j +1 · · · ω r . (d) One has (cid:18)Z p ω · · · ω r (cid:19) (cid:18)Z p ω r +1 · · · ω r + s (cid:19) = X σ Z p ω σ (1) · · · ω σ ( r + s ) , where the sum runs over all ( r, s ) -shuffles, i.e., permutations σ on r + s letters with σ − (1) < · · · < σ − ( r ) and σ − ( r + 1) < · · · < σ − ( r + s ) . (e) R p − ω · · · ω r = ( − r R p ω r · · · ω , where p − ( t ) = p (1 − t ) . TERATED INTEGRALS
For given ω ∈ B ( X ) , we extend the map p R p ω to the free abeliangroup Z [ P X ] generated by P X . For given α , . . . , α s ∈ P X x ,x let η = ( α − α − . . . ( α s − ∈ Z [ P X ] . For 1-forms ω , . . . , ω r , r ≤ s we have Z η ω · · · ω r = (Q si =1 R α i ω i r = s, r < s. Note that (d) implies that B ( X ) x = ∞ [ s =0 B s ( X ) x is a filtered algebra and likewise for B ( X ) x ,x . Proof: (a)-(e) are easy exercises. (f) is a result of Hain, see [Hai87], Propo-sition 2.13. (cid:3)
If we replace the tensor product on T ( X ) by the shuffle product ∗ given by ω · · · ω r ∗ ω r +1 · · · ω r + s = X σ ω σ (1) · · · ω σ ( r + s ) , where the sum runs over all ( r, s )-shuffles, we obtain the shuffle algebra Sh( X ). We have shown that the iterated integrals form an algebra homo-morphism Sh( X ) → B ( X ) , where the latter is an algebra under pointwise multiplication.Let B s ( X ) hom denote the space of all elements of B s ( X ) which are invariantunder homotopies with fixed end-points. Similarly define B s ( X ) hom x and B s ( X ) hom x ,x . Theorem 2.1
The restriction map B s ( X ) hom x → B s ( X ) hom x ,x is surjective. Proof:
For a real vector space V , let T ( V ) = R ⊕ V ⊕ V ⊗ ⊕ . . . be the TERATED INTEGRALS D : T (Ω ( X )) → T (Ω( X )) given by D ( ω ⊗ · · · ⊗ ω n ) = n X j =1 ω ⊗ · · · ⊗ ( dω j ) ⊗ · · · ⊗ ω n + n − X j =1 ω ⊗ · · · ⊗ ( ω j ∧ ω j +1 ) ⊗ . . . ω n , and D ( c ) = 0, for c ∈ R . By Proposition 1.5.2 of [Che77] the map R : T (Ω X ) → Ω ( P X ) sending ω ⊗ · · · ⊗ ω s to the map p R p ω · · · ω s extends to a map T (Ω X ) → Ω( P X ) such that for ω ∈ T (Ω ( C )) one has d (cid:18)Z ω (cid:19) = − Z Dω − p ∗ ω Z ω · · · ω s + ( − s − p ∗ ω s Z ω · · · ω s − , where p , p : P X → X are the evaluation maps that map a path to itsstart and end point, respectively. Therefore, the kernel of D is mapped tohomotopy invariant iterated integrals on P X x . Let M be the kernel of D .Then M has a natural filtration by degrees: R = M ⊂ M ⊂ . . . The map I x : ω ⊗ · · · ⊗ ω s R ω . . . ω s maps M s to B s ( X ) hom x . We get acommutative diagram: M s B s ( X ) hom x B s ( X ) hom x ,x . ✲ I x ❅❅❅❅❅❘ I x ,x ❄ res The integral R ω . . . ω s is a function on the path space P X . Its restrictionto the loop space L x X is homotopy invariant, if and only if it is locallyconstant, which is the case if and only if it is anihilated by the differentialof the complex Λ( P M ) as in [Che77]. Now the differential D above alsocoincides with the differential of the bar construction on [Che77], Section4.1. Theorem 4.1.1 of [Che77] states that the iterated integral map is anisomorphism of graded differential algebras from that bar construction tothe iterated integrals on the loop space. Therefore the iterated integral map I x ,x is surjective, hence the restriction map is surjective, too. (cid:3) TERATED INTEGRALS For a group Γ we write its group ring as A = Z Γ. Let J ⊂ Z Γ be theaugmentation ideal, i.e., the span of all elements of the form ( γ − γ ∈ Γ. For any Z Γ-module V we write H s (Γ , V ) for the Z -module of all v ∈ V with J s v = 0. This space can be identified with Hom Z Γ ( Z Γ /J s , V ).The elements of H s (Γ , V ) for varying s are called higher order invariants . If v is in H s (Γ , V ), but not in H s − (Γ , V ), then s is called the order of v .Let X be a connected smooth manifold, x ∈ X a base-point, and Γ = π ( X, x ) the corresponding fundamental group. We consider Γ as group ofdeck transformations on the universal covering ˜ X of X . We also fix a pre-image x in ˜ X , which we will denote by the same symbol x as no confusioncan arise.As ˜ X is simply connected, the iterated integral R p ω for ω ∈ B s ( ˜ X ) hom onlydepends on the endpoints x, y of the path p . We therefore write R yx ω = R p ω .Every γ ∈ Γ can be viewed as a homotopy class of a loop based at x ∈ X . Inthis way we get a map B s ( X ) hom x ,x → Map(Γ , C ) that maps ω ∈ B s ( X ) hom x ,x to the map γ R γ ω . The latter map induces a Z -linear map from the groupring Z Γ to C . It is the content of Chen’s de Rham Theorem for fundamentalgroups (see [Che77], Corollar 1 to Theorem 2.6.1, see also [PS08]) that thismap induces a bijection B s ( X ) hom x ,x ∼ = −→ Hom Z ( Z Γ /J s +1 , C ) . Each ω ∈ B s ( X ) hom x lifts to ˜ X and gives an element of B s ( ˜ X ) hom x . For x ∈ ˜ X we write R xx ω for the iterated integral of this lift over any path joining x and x , or, what amounts to the same, for the integral of ω over the projectionto X of any such path. Theorem 3.1 If ω ∈ B s ( X ) hom x , then the function R xx ω , x ∈ ˜ X , is aninvariant of order at most s + 1 in the Γ -module C ∞ ( ˜ X ) . This defines aninjective linear map Ψ : B s ( X ) hom x ֒ → H s +1 (Γ , C ∞ ( ˜ X )) . The case when X is the hyperbolic plane is in the paper [Sre09]. TERATED INTEGRALS Proof:
Let ω ∈ B s ( X ) hom and for x ∈ ˜ X set f ω ( x ) = R xx ω . We have toshow [( γ − · · · ( γ s +1 − ∗ f ω = 0for any γ , . . . , γ s +1 ∈ Γ. For given x ∈ X and γ ∈ Γ, we choose a path γ x from x to γx . The map γ γ x is extended linearly to a map Z Γ → Z [ P X ]. For every x ∈ X we also fix a smooth path p x from x to x . Let ω ∈ B s ( X ) hom and let η = P γ c γ γ be an arbitrary element of the group ring Z Γ. We have η ∗ f ω ( x ) = X γ c γ γ ∗ f ω ( x ) = X γ c γ Z γxx ω = X γ c γ Z p x γ x ω = Z p x P γ c γ γ x ω = Z p x η x ω. We apply this to the element ( γ − · · · ( γ s +1 −
1) of the group ring andwe look at any monomial ω · · · ω r in ω , where the ω j are 1-forms on X . Wethen have Z p x [( γ − ... ( γ s +1 − x ω . . . ω r = r X k =0 Z p x ω . . . ω k Z [( γ − ... ( γ s +1 − x ω k +1 . . . ω r . Let ¯ x ∈ X be the image of x ∈ ˜ X and let γ ¯ x be the image of γ x in X . Then γ ¯ x is a loop based at ¯ x . As the forms ω j are Γ-invariant, we have Z [( γ − ... ( γ s +1 − x ω k +1 . . . ω r = Z [( γ − ... ( γ s +1 − ¯ x ω k +1 . . . ω r = Z ( γ , ¯ x − ... ( γ s +1 , ¯ x − ω k +1 . . . ω r = 0by Lemma 1.1 (e). This proves the first claim. For the injectivity of theinduced map let ω ∈ B s ( X ) hom x with R xx ω = 0. This just means that ω = 0in B s ( X ) hom x . (cid:3) We formally set H = 0 and B − = 0. Theorem 3.2
Let K be the ideal in the algebra B ( X ) x generated by d ( C ∞ ( X )) and let K hom s = B s ( X ) hom x ∩ K . Then K hom s is the kernel of the restrictionmap B s ( X ) hom x res −→ B s ( X ) hom x ,x , TERATED INTEGRALS and Ψ induces an isomorphism of C ∞ ( X ) -modules C ∞ ( X ) ⊗ (cid:16) B s ( X ) hom x /K hom s (cid:17) ∼ = −→ H s +1 (Γ , C ∞ ( ˜ X )) . It follows that H s +1 (Γ , C ∞ ( ˜ X )) is a free module of the algebra C ∞ ( X ) ofsmooth functions. Proof:
The fact that K hom s is the kernel of the restriction map is a conse-quence of Theorem 4.5 in [Che71]. Write ¯ B s = B s /B s − and let¯ K hom s = ker h B s ( X ) hom x → ¯ B s ( X ) hom x ,x i . Then ¯ K hom s contains B s − ( X ) hom x and our assertion is equivalent to C ∞ ( X ) ⊗ (cid:16) B s ( X ) hom x / ¯ K hom s (cid:17) ∼ = −→ ¯H s +1 (Γ , C ∞ ( ˜ X )) , which is what we prove.By Chen’s de Rham Theorem for fundamental groups (see [Che77], Corollar1 to Theorem 2.6.1, see also [PS08]) the evaluation of iterated integrals givesan isomorphism ¯ B s ( X ) hom x ,x ∼ = −→ Hom Z ( J s /J s +1 , C ) . The right hand side can also be viewed as Hom Z Γ ( J s /J s +1 , C ) and as suchbe embedded intoHom A ( J s /J s +1 , C ∞ ( X )) ∼ = Hom A ( J s /J s +1 , C ∞ ( ˜ X )) , where we have written A = Z Γ. More precisely, the image inHom A ( J s /J s +1 , C ∞ ( X )) ∼ = C ∞ ( X ) ⊗ Hom A ( J s /J s +1 , C )is a basis of this C ∞ ( X )-module, which means that we have an isomorphismof C ∞ ( X )-modules, C ∞ ( X ) ⊗ ¯ B s ( X ) hom x ,x ∼ = −→ Hom A ( J s /J s +1 , C ∞ ( ˜ X )) . Lemma 3.3
The cohomology group H (Γ , C ∞ ( ˜ X )) is trivial. TERATED INTEGRALS Proof:
A 1-cocycle is a map α : Γ → C ∞ ( ˜ X ) such that α ( γτ ) = γα ( τ ) + α ( γ ) holds for all γ, τ ∈ Γ. We have to show that for any given such map α there exists f ∈ C ∞ ( ˜ X ) such that α ( τ ) = τ f − f .Fix a smooth map u : ˜ X → [0 ,
1] such that X τ ∈ Γ u ( τ − x ) ≡ , where we can assume that the sum is locally finite. Set f ( x ) = − X τ ∈ Γ α ( τ )( x ) u ( τ − x ) . Then the function f lies in the space C ∞ ( ˜ X ). We now compute for γ ∈ Γ, γf ( x ) − f ( x ) = f ( γ − x ) − f ( x )= X τ ∈ Γ α ( τ )( x ) u ( τ − x ) − α ( τ )( γ − x ) u ( τ − γ − x )= X τ ∈ Γ α ( τ )( x ) u ( τ − x ) + α ( γ )( x ) X τ ∈ Γ u (( γτ ) − x ) − X τ ∈ Γ α ( γτ )( x ) u (( γτ ) − x )The first and the last sum cancel and the middle sum is α ( γ )( x ). Therefore,the lemma is proven. (cid:3) Since H (Γ , C ∞ ( ˜ X )) = 0, Lemma 2.1 in [Dei11a] implies that the exactsequence 0 → J s /J s +1 → A/J s +1 → A/J s → , induces an isomorphismHom A ( J s /J s +1 , C ∞ ( ˜ X )) ∼ = Hom A ( A/J s +1 , C ∞ ( ˜ X )) / Hom A ( A/J s , C ∞ ( ˜ X )) | {z } def = Hom A ( A/J s +1 ,C ∞ ( ˜ X )) ∼ = ¯H s +1 (Γ , C ∞ ( ˜ X )) . TERATED INTEGRALS B s ( X ) hom x ¯ B s ( X ) hom x ,x Hom A ( A/J s +1 , C ∞ ( ˜ X )) Hom A ( J s /J s +1 , C ∞ ( ˜ X )) ✲ res ❄ ❄✲ ∼ = commutes. This will give the claim, as we already have seen that the rightvertical arrow becomes an isomorphism after tensoring with C ∞ ( X ). Thiscommutativity is a direct consequence of formula (f) in Lemma 1.1. (cid:3) To satisfy needs in the theory of automorphic forms, we also introduce fur-ther structure: Let P ⊂ Γ be a conjugation-invariant subset and let h P i bethe normal subgroup generated by P . Then AJ P is a two-sided ideal of A ,where J P ⊂ Z h P i ⊂ A is the augmentation ideal of h P i . Let J s = J s + AJ P , and let H P,s (Γ , V ) = { v ∈ V : J s v = 0 } . In the theory of automorphic forms, see [Dei09, DD09], P will be the set ofparabolic elements.The P -restriction translates on the side of iterated integrals to the following.Recall Chen’s map B s ( X ) hom x ,x ∼ = −→ Hom Z ( Z Γ /J s +1 , C ) . We define the space B P,s ( X ) hom x ,x to be the inverse image ofHom Z ( Z Γ /J s +1 , C ) = { α ∈ Hom Z ( Z Γ /J s +1 , C ) : α ( p −
1) = 0 ∀ p ∈ P } under this map. Finally, we set B P,s ( X ) hom x to be the inverse image of B P,s ( X ) hom x ,x under the restriction map. So then B P,s ( X ) hom x is the set ofall ω ∈ B s ( X ) hom x with R p − ω = 0 for every p ∈ P . Theorems 3.1 and 3.2generalize to the following. TERATED INTEGRALS Theorem 4.1 If ω ∈ B P,s ( X ) hom x , then the function R xx ω is an invariantof order at most s + 1 in the Γ -module C ∞ ( ˜ X ) . This defines an injectivelinear map Ψ : B P,s ( X ) hom x ֒ → H P,s +1 (Γ , C ∞ ( ˜ X )) . Theorem 4.2
Let K be the ideal in the algebra B ( X ) x generated by d ( C ∞ ( X )) and let K hom P,s = B P,s ( X ) hom x ∩ K . Then K hom P,s is the kernel of the restrictionmap B P,s ( X ) hom x res −→ B P,s ( X ) hom x ,x , and Ψ induces an isomorphism of C ∞ ( X ) -modules C ∞ ( X ) ⊗ (cid:16) B P,s ( X ) hom x /K hom P,s (cid:17) ∼ = −→ H P,s +1 (Γ , C ∞ ( ˜ X )) . Proof:
The proofs are the same as in the unrestricted case. One only hasto check that the restriction conditions match, which is easy to see. (cid:3)
References [Che54] Kuo-Tsai Chen,
Iterated integrals and exponential homomorphisms , Proc. Lon-don Math. Soc. (3) (1954), 502–512.[Che71] Kuo-tsai Chen, Algebras of iterated path integrals and fundamental groups ,Trans. Amer. Math. Soc. (1971), 359–379.[Che77] Kuo Tsai Chen,
Iterated path integrals , Bull. Amer. Math. Soc. (1977),no. 5, 831–879.[CDO02] G. Chinta, N. Diamantis, and C. O’Sullivan, Second order modular forms ,Acta Arith. (2002), no. 3, 209–223, DOI 10.4064/aa103-3-2.[Dei09] Anton Deitmar,
Higher order group cohomology and the Eichler-Shimura map ,J. Reine Angew. Math. (2009), 221–235, DOI 10.1515/CRELLE.2009.032.[Dei11a] ,
Higher order invariants in the case of compact quotients , CentralEuropean Journal of Mathematics (2011), no. 1, 85–101.[Dei11b] , Lewis-Zagier Correspondence for higher order forms , Pacific Journalof Mathematics (2011), no. 1, 11–21.[DD09] Anton Deitmar and Nikolaos Diamantis,
Automorphic forms of higher order ,J. Lond. Math. Soc. (2) (2009), no. 1, 18–34, DOI 10.1112/jlms/jdp015.MR2520375[DD10] , A new multiple Dirichlet series induced by a higher-order form , ActaArith. (2010), no. 4, 303–309, DOI 10.4064/aa142-4-1. MR2640061(2011d:11207)
TERATED INTEGRALS [DKMO06] N. Diamantis, M. Knopp, G. Mason, and C. O’Sullivan, L -functions ofsecond-order cusp forms , Ramanujan J. (2006), no. 3, 327–347, DOI10.1007/s11139-006-0147-2.[DO08] Nikolaos Diamantis and Cormac O’Sullivan, The dimensions of spaces of holo-morphic second-order automorphic forms and their cohomology , Trans. Amer.Math. Soc. (2008), no. 11, 5629–5666, DOI 10.1090/S0002-9947-08-04755-7.[DS06] N. Diamantis and Ramesh Sreekantan,
Iterated integrals and higher order au-tomorphic forms , Comment. Math. Helv. (2006), no. 2, 481–494.[DK09] Nikolaos Diamantis and Peter Kleban, New percolation crossing formulas andsecond-order modular forms , Commun. Number Theory Phys. (2009), no. 4,677–696. MR2610483[DS08] Nikolaos Diamantis and David Sim, The classification of higher-order cusp forms , J. Reine Angew. Math. (2008), 121–153, DOI10.1515/CRELLE.2008.067. MR2433614 (2010a:11067)[Hai87] Richard M. Hain,
The geometry of the mixed Hodge structure on the funda-mental group , Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985),Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987,pp. 247–282.[Kon93] Maxim Kontsevich,
Vassiliev’s knot invariants , I. M. Gel ′ fand Seminar, Adv.Soviet Math., vol. 16, Amer. Math. Soc., Providence, RI, 1993, pp. 137–150.[PS08] Chris A. M. Peters and Joseph H. M. Steenbrink, Mixed Hodge structures ,Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Mod-ern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rdSeries. A Series of Modern Surveys in Mathematics], vol. 52, Springer-Verlag,Berlin, 2008.[Sre09] Ramesh Sreekantan,
Higher order modular forms and mixed Hodge theory ,Acta Arith. (2009), no. 4, 321–340.
Mathematisches Institut, Auf der Morgenstelle 10, 72076 T¨ubingen, Germany, [email protected]@[email protected]@uni-tuebingen.de