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Morse homotopy and topological conformal field theory. (English) Zbl 1373.57047
Two main goals of the article under review are:
1) Presenting a new construction which uses flow graphs to recover interesting information about a manifold.
2) Giving theorems explaining in algebraic language what is recovered.
In this article flow graphs are the graphs analogues to flow trajectories. Flow graphs in manifolds were used to construct invariants in the form of cohomology operations. The operations associated to certain special graphs can be identified explicitly and turn out to correspond to invariants known from classical algebraic topology. Moreover, the operations satisfy a field-theoretic law: there is a compatibility between gluing together graphs and composing the associated operations.
Throughout the article classical Morse theory is translated into the graph theoretic approach. In this approach the Morse complex would become the field theoretic structures. By studying spaces of flow graphs in a closed oriented manifold, the paper under review constructs operations on its cohomology. It is seen that the operations satisfy the gluing axioms of an open homological conformal field theory.
57R19 Algebraic topology on manifolds and differential topology
57R56 Topological quantum field theories (aspects of differential topology)
81T40 Two-dimensional field theories, conformal field theories, etc. in quantum mechanics
53D37 Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category
57R58 Floer homology
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