OPEN BOOK DECOMPOSITION Meaning and
Definition
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Open book decomposition refers to a mathematical concept used in the study of three-dimensional manifolds. Specifically, it is a way of representing a closed, oriented three-dimensional manifold as the union of two simpler objects: a surface and a one-dimensional curve called the binding.
In an open book decomposition, the surface acts as a cross-section of the manifold, while the binding represents the hinge around which the surface is wrapped. This means that every point on the surface has a unique coordinate with respect to the binding. Additionally, the surface and binding are constructed in such a way that if one moves along the binding, the surface continually changes, allowing for a deformation of the manifold.
The binding serves as a spine for the manifold since the surface can be completely reconstructed by taking a copy of the binding and gluing a copy of the standard surface to each point on it. This construction is known as a fibered link in four-dimensional topology.
Open book decompositions are used to study contact structures, a specific type of geometric structure, on three-dimensional manifolds. Certain properties of contact structures can be easily determined by analyzing the behavior of curves along the binding. Additionally, open book decompositions provide a convenient framework for understanding the topology and geometry of three-dimensional manifolds, leading to advancements in areas such as low-dimensional topology and symplectic geometry.
