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Erschienen in:
2018 | OriginalPaper | Buchkapitel
6. Higher Order Multiplications and the \(A_\infty \)-Relations
verfasst von : Stephan Mescher
Erschienen in: Perturbed Gradient Flow Trees and A∞-algebra Structures in Morse Cohomology
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Abstract
We continue by taking a closer look at zero- and one-dimensional moduli spaces of perturbed Morse ribbon trees. The results from Chap. 5 enable us to show that zero-dimensional moduli spaces are in fact finite sets. This basic observation allows us to define homomorphisms \(C^*(\,f)^{\otimes d} \rightarrow C^*(\, f)\) for every \(d \ge 2\) via counting elements of these zero-dimensional moduli spaces. After constructing these higher order multiplications explicitly, we will study the compactifications of one-dimensional moduli spaces of perturbed Morse ribbon trees. The results from Chap. 5 imply that one-dimensional moduli spaces can be compactified to one-dimensional manifolds with boundary, and we will explicitly describe their boundaries. If we impose an additional consistency condition on the chosen perturbations, the boundary spaces will coincide with product of zero-dimensional moduli spaces of perturbed Morse ribbon trees.