Morse homotopy and topological conformal field theory.

*(English)*Zbl 1373.57047Two 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.

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.

Reviewer: Vida Milani (North Logan)

##### MSC:

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 |