Define a category B whose objects are the oriented submanifolds of X, and whose vector space of morphisms from Y to Z is OYZ = Ext∗H∗(X)(H∗(Y), H∗(Z)) – the cohomology, as usual, has complex coefficients, and H∗(Y) and H∗(Z) are regarded as H∗(X)-modules by restriction. The composition of morphisms is given by the Yoneda composition of Ext groups. With this definition, however, it will not be true that OYZ is dual to OZY. (To see this it is enough to consider the case when Y = Z is a point of X, and X is a product of odd-dimensional spheres; then OYZ is a symmetric algebra, and is not self-dual as a vector space.)
We can do better by defining a cochain complex O’YZ of morphisms by
O’YZ = BΩ(X)(Ω(Y), Ω(Z)) —– (1)
where Ω(X) denotes the usual de Rham complex of a manifold X, and BA(B,C), for a differential graded algebra A and differential graded A- modules B and C, is the usual cobar resolution
Hom(B, C) → Hom(A ⊗ B, C) → Hom(A ⊗ A ⊗ B, C) → · · · —– (2)
in which the differential is given by
dƒ(a1 ⊗ · · · ⊗ ak ⊗ b) = a1 ƒ(a2 ⊗ · · · ⊗ ak ⊗ b) + ∑(-1)i ƒ(a1 ⊗ · · · ⊗ aiai+1 ⊗ ak ⊗ b) + (-1)k ƒ(a1 ⊗ · · · ⊗ ak-1 ⊗ akb) —– (3)
whose cohomology is ExtA(B,C). This is different from OYZ = Ext∗H∗(X)(H∗(Y), H∗(Z)), but related to it by a spectral sequence whose E2-term is OYZ and which converges to H∗(O’YZ) = ExtΩ(X)(Ω(Y), Ω(Z)). But more important is that H∗(O’YZ) is the homology of the space PYZ of paths in X which begin in Y and end in Z. To be precise, Hp(O’YZ) ≅ Hp+dZ(PYZ), where dZ is the dimension of Z. On the cochain complexes the Yoneda composition is associative up to cochain homotopy, and defines a structure of an A∞ category B’. The corresponding composition of homology groups
Hi(PYZ) × Hj(PZW) → Hi+j−dZ(PYW) —— (4)
is the composition of the Gysin map associated to the inclusion of the codimension dZ submanifold M of pairs of composable paths in the product PYZ × PZW with the concatenation map M → PYW.
Now let’s attempt to fit the closed string cochain algebra C to this A∞ category. C is equivalent to the usual Hochschild complex of the differential graded algebra Ω(X), whose cohomology is the homology of the free loop space LX with its degrees shifted downwards by the dimension dX of X, so that the cohomology Hi(C) is potentially non-zero for −dX ≤ i < ∞. There is a map Hi(X) → H−i(C) which embeds the ordinary cohomology ring of X to the Pontrjagin ring of the based loop space L0X, based at any chosen point in X.
The structure is, however, not a cochain-level open and closed theory, as we have no trace maps inducing inner products on H∗(O’YZ). When one tries to define operators corresponding to cobordisms it turns out to be possible only when each connected component of the cobordism has non-empty outgoing boundary.