Algebraic Models for the Free Loop Space and Differential Forms of a Manifold.

摘要

Our initial goal is to give a chain level description of the string topology loop product for a large class of spaces. This effort is described in two parts; the first uses Brown's theory of twisting cochains to obtain a model for the free loop space of a manifold and the second constructs a minimal model for the Frobenius algebra of differential forms of a manifold. The first part defines the loop product for closed, oriented manifolds and Poincare Duality spaces. The second part is an attempt to understand the minimal model for the Frobenius algebra of a manifold, with the idea of extending the methods in the first section to define the loop product for open manifolds.;Brown's theory of twisting cochains provides a chain model of a principal G-bundle and its associated bundles. The free loop space is obtained by considering the path space fibration, and taking the associated bundle with the based loop space acting on itself by conjugation. Given a twisting cochain, then, we obtain a chain model of LM using Brown's theory. To describe the chain-level loop product in this setting, we need a model for the intersection product in the chains on M. For this, we use the cyclic commutative infinity algebra structure on the homology of M. Such a description would give a chain level description of the string topology loop product for open manifolds.;Instead of using the cyclic commutative algebra, we could have used the Frobenius algebra structure. One would expect that the Frobenius infinity algebra can be used to show the necessary relations to define the loop product. Then given the Frobenius infinity algebra on the homology of M for an open manifold, we would have a chain level description of the loop product.;The purpose of Section 3 is to gain a better understanding of the Frobenius infinity algebra on the cohomology of M. The Frobenius algebra, induced by the wedge product and Poincare Duality, is well understood; the structure on the level of forms inducing the Frobenius algebra is less well understood. We use the language of operads, dioperads, and properads and Koszul duality to give a definition of Frobenius infinity algebra. We also use descriptions of the transfer of structure using trees and integrating over cells in the moduli space of metrised ribbon graphs. When M is closed and oriented, these tools allow us to build a minimal model for the Frobenius algebra of differential forms on M and to compare it with the cyclic commutative infinity algebra.