Chen’s theory of iterated integrals and the algebraic topology of the based loop space Manuel Rivera

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Iterated integrals and the cohomology of the based loop space

Let M be a smooth manifold, let P M be the space of piecewise smooth free paths and let ΩM be the based loop space of M of piecewise smooth loops at a base point b. Chen’s iterated integral map is a graded vector space map Z : T A(M ) → A(P M ) (1) of degree 0, where T A(M ) denotes the tensor algebra on the desuspension of the cdga A(M ) of differential forms of positive degree on M and A(P M ) denotes the cdga of differential forms on P MR . Chen gives meaning to A(P M ) through the notion of a differentiable space. The n-iterated integral map : A(M )⊗n → A(P M ) is constructed as an integral along the fiber of a trivial bundle having the n-simplex as fibers. The qualification of iterated comes from the way in which the integral is computed. In these notes we will summarize what Chen’s constructions accomplish for the case of based loop space, so we will restrict the range of the map (1) to A(ΩM ). Theorem 1 The image of

R

: T A(M ) → A(ΩM ) is preserved by d.

a) RTheorem 1 follows from Stoke’s theorem for integration along the fiber which reveals that the commutator of : A(M )⊗n → A(ΩM ) with d (where d is extended to A(M )⊗n as a graded derivation) is given by the (n − 1)-iterated integral over the n + 1 faces of the n-simplex. R b) The correction term obtained from the commutator [ , d] can be used to perturb the tensor differential to obtain a new differential D on T A(M ), which one recognizes as the differential of the bar construction of the R dga A(M ), making : (T A(M ), D) → A(ΩM ) into a dg vector space map. c) (T A(M ), D) is a commutative dg Hopf algebra with shuffle product and deconcatenation coproduct. We will write (T c A(M ), D) from now on to refer to such commutative Hopf algebra structure. Theorem 2

R

: (T c A(M ), D) → A(ΩM ) is a cdga map.

a) Theorem 2 implies that the image of

R

is a sub cdga of A(ΩM ). We denote this sub cdga by Chen(ΩM ).

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R b) To prove that is an algebra map, i.e. that a wedge of iterated integrals is the iterated integral of the shuffle product of monomials, uses the decomposition of a prism into simplices. R Theorem 3 The map : T c A(M ) → A(ΩM ) is an acyclic subcomplex of (T c A(M ), D). R kernel of the vector space R c c Moreover, ker is a Hopf ideal of T A(M ), so T A(M )/(ker ) has the structure of a commutative dg Hopf algebra which is quasi-isomorphic to T c A(M ). R R a) Using the dg vector space isomorphism : T cRA(M )/(ker ) → Chen(ΩM ) we can define a coalgebra strucR ture on Chen(ΩM ) such that : T c A(M )/(ker ) → Chen(ΩM ) is an isomorphism of commutative dg Hopf algebras.

Differential forms on ΩM can chains on ΩM . Given Ra smooth real R be integrated over smooth real cubical R cubical chain α on ΩM and w1 ...wn ∈ Chen(ΩM ) denote by < w1 ...wn , α > the integral of w1 ...wn over α. If β is another smooth real cubical chain denote their Pontrjagin product by α × β. Theorem 4 If w1 , ..., wn ∈ A(M ) and α and β are smooth real cubical chains on ΩM then Z Z Z n X < w1 ...wn , α × β >= < w1 ...wi , α >< wi+1 ...wr , β > .

(2)

i=0

a) Theorem 4 implies that the coproduct on Chen(ΩM ) is dual, under integration, to the Pontrjagin product of smooth cubical chains on ΩM b) The proof of Theorem 4 uses a decomposition of a simplex into prisms.

Let H ∗ (ΩM ; R) denote the singular real cohomology Hopf algebra of ΩM . Theorem 5 If M is a simply connected manifold there is an isomorphism of Hopf algebras H ∗ (Chen(ΩM )) ∼ = H ∗ (ΩM ; R). a) Chen constructs a de Rham integration map from Chen(ΩM ) to the dual of Adams cobar construction on singular chains on M which induces the isomorphism of Theorem 5. The fact that this induced map is a coalgebra map follows from Theorem 4. b) Combining Theorem 5 with the remark under isomorphisms of Hopf R Theorem ∗3 we obtain the following algebras: H ∗ (T c A(M ), D) ∼ = H ∗ (T c A(M )/(ker ), D) ∼ = H (Chen(ΩM )) ∼ = H ∗ (ΩM ; R) when M is simply connected. c) Adams original construction is a homological model for ΩM starting with the coalgebra of singular chains on M with Z coefficients. Whereas Adams model only captures the non commutative product on the singular chain complex C∗ (ΩM ; Z), Chen’s model captures the full commutative Hopf algebra structure at the cochain level of real differential forms on the based loop space. 2

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Power series connections and the homology of the based loop space

A power series connection on a manifold M is a pair (ω, ∂) where (1) ∂ is a differential of degree −1 on L = LH∗ (M ; R), the graded free Lie algebra generated by the desuspension of the reduced real homology vector space, which is a derivation of the bracket, and (2) ω =

P

ˆ where ⊗ ˆ denotes the completed tensor products, such that ui ⊗ Ui is a power series in A(M )⊗L,

(i) deg ui = 1 + deg Ui , ˆ (ii) if X1 , X2 , ... is a basis for H∗ (M ; R) and ω ≡ wi ⊗ Xi mod A(M )⊗[L, L] then weach wi is a closed differential ∗ form and the cohomolgy classes {[wi ]} form a basis for H (M ; R) dual to {Xi }, and (iii) ∂ω + dω − 21 [Jω, ω] = 0; i.e. ω is a twisting cochain. Theorem 6 If M is a manifold with finite Betti numbers then M has a power series connection. a) The construction of such power series connection is completely algebraic and obstruction theoretic. In order to construct ω inside the Lie algebra A(M ) ⊗ L we need the graded commutativity of A(M ). ˆ ⊂ b) Let T H∗ denote the tensor algebra on the desuspension of the reduced real homology of M , so ω ∈ A(M )⊗L ˆ H∗ . By duality ω corresponds to a linear map τω : T c H ∗ → A(M ) where T c H ∗ is the free tensor coalA(M )⊗T gebra on the desuspension of the reduced real cohomology of M ; the coproduct in T c H ∗ is deconcatenation. Since ω is a solution for the Maurer Cartan equation (iii), it follows that τω is a twisting cochain in the sense of Brown. c) The differential ∂ : L → L extends to ∂ : T H∗ → T H∗ which makes (T H∗ , ∂) into a cocommutative dg Hopf algebra and (T c H ∗ , ∂ ∗ ) into a commutative dg Hopf algebra. Theorem 7 If (ω, ∂) is a power series connection on a manifold M with finite Betti numbers, then the map τω : T c H ∗ → A(M ) extends to a dg Hopf algebra map τ˜ω : (T c H ∗ , ∂ ∗ ) → (T c A(M ), D) which induces an isomorphism in cohomology. a) In modern language, Theorem 7 says that (ω, ∂) defines a transport of the C∞ -algebra structure on A(M ) to a quasi-isomorphic C∞ -algebra structure on H ∗ (M ; R). Theorem 8 There exists a dga map θ : C∗ (ΩM ) → (T H∗ , ∂), where C∗ (ΩM ) is the dga of smooth real cubical chains on ΩM . If M is simply connected θ induces an isomorphism H∗ (ΩM ; R) ∼ = H∗ (T H∗ , ∂) of Hopf algebras. R R ˆ H∗ over elements of a) The map θ is defined by integrating the element T + 1 + ω + ω 2 + ... ∈ A(ΩM )⊗T C∗ (ΩM ), resembling the theory of connections and holonomy of a smooth vector bundle. b) Theorem 4 implies that θ is an algebra map. Hain shows that the map θ∗ sends the primitive elements of H∗ (ΩM ; R) to H∗ (LH∗ , ∂) which implies that θ∗ is a coalgebra map. c) By a theorem of Milnor and Moore we have an isomorphism of Lie algebras H(L, ∂) ∼ = π∗−1 (M ) ⊗ R. 3

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References

[1] Chen, K.T, Iterated integrals of differential forms and loop space homology, Ann. of Math. (2) 97 (1973), 217-246 [2] Chen,K.T, Iterated path integrals, Bull. Amer. Math. Soc. 83 (1977), 831-879. [3] Hain, R.M., Iterated integrals and homotopy periods, Memoirs of the AMS (1984) vol. 47, number 291 [4] Milnor, J. and Moore, J., On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264

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