Handle Decompositions in Smooth Manifolds
In the study of topology, decomposing a complex space into simpler pieces is a fundamental strategy. While CW-decompositions provide a way to build spaces using cells, they often fall short when dealing with smooth manifolds. The primary issue is that the attaching maps used in CW-complexes are not necessarily smooth, meaning the smooth structure of a manifold—such as an n-sphere—cannot be naturally recovered from the decomposition alone.
To resolve this, mathematicians utilize handle decompositions. Unlike cell complexes, handle decompositions ensure that the gluing process preserves the smooth structure of the manifold, allowing for a rigorous analysis of its geometry and topology.
The Role of the Tubular Neighbourhood Theorem
The transition from cell complexes to handle decompositions relies on the tubular neighbourhood theorem. This theorem states that for any point p in an m-manifold M, there exists a closed tubular neighbourhood Np that is diffeomorphic (smoothly equivalent) to a disk Dm.
By applying this theorem, a manifold can be viewed as a union of pieces glued along their boundaries via diffeomorphisms. For example, if we take a smooth embedded arc in the complement of the point's neighbourhood, its own tubular neighbourhood is diffeomorphic to I × Dm-1. This allows the manifold M to be expressed as the union of three distinct manifolds:
- The disk Dm.
- The product I × Dm-1.
- The complement of the open tubular neighbourhood of the arc within the remaining space.
Because these gluing maps are smooth, the resulting structure maintains the manifold's integrity, overcoming the limitations of standard CW-decompositions.
[ไม่มีภาพประกอบ]Defining the j-Handle
Handle decompositions were pioneered by Stephen Smale. In this framework, the building blocks are called handles. A j-handle (denoted as Hj) is defined as the product of two disks: Dj × Dm-j.
To attach a j-handle to an m-manifold M, one must have a smooth embedding f: Sj-1 × Dm-j → &partial;M. The process involves taking the disjoint union of M and the handle Hj and identifying the region Sj-1 × Dm-j with its image in the boundary of M via the map f.
Mathematically, this is expressed as the quotient space:
M ∪f Hj = (M ⊔ (Dj × Dm-j)) / ∼
where the equivalence relation ∼ is generated by (p, x) ∼ f(p, x) for all points in the attaching region.
Manifolds and Handlebodies
A manifold N is said to be obtained from M by attaching j-handles if the union of M and a finite number of j-handles is diffeomorphic to N. This leads to specific classifications based on the types of handles used:
- 0-handle manifolds: If a manifold has a decomposition consisting only of 0-handles, it is diffeomorphic to a disjoint union of balls.
- Handlebodies: A connected manifold that contains only two types of handles—0-handles and j-handles for a specific fixed j—is referred to as a handlebody.
| Component | Mathematical Definition | Role in Decomposition |
|---|---|---|
| j-Handle (Hj) | Dj × Dm-j | The basic building block used to construct the manifold. |
| Attaching Map (f) | Sj-1 × Dm-j → &partial;M | The smooth embedding used to glue the handle to the boundary. |
| Handlebody | Connected M with 0-handles and j-handles | A specific class of manifold built from two handle types. |
Key Facts
- CW-decompositions are often degenerate for smooth manifolds because their attaching maps are not necessarily smooth.
- The tubular neighbourhood theorem provides the theoretical basis for smooth gluing in handle decompositions.
- Stephen Smale is the inventor of handle decompositions.
- A j-handle is defined as the product of disks Dj × Dm-j.
- A manifold consisting only of 0-handles is a disjoint union of balls.
Frequently Asked Questions
Why are CW-decompositions insufficient for smooth manifolds?
CW-decompositions use attaching maps that do not necessarily live in the world of smooth maps between manifolds. Consequently, the smooth structure of the manifold cannot be naturally determined from the decomposition.
What is a j-handle?
A j-handle is a topological building block defined as the product of two disks, Dj × Dm-j, used to construct smooth manifolds.
How does the tubular neighbourhood theorem help in this process?
It ensures that a point or an arc in a manifold has a neighbourhood diffeomorphic to a disk or a product of an interval and a disk, allowing for gluing maps that are smooth diffeomorphisms.
What is the difference between a general handle decomposition and a handlebody?
A handle decomposition is a general way to build a manifold using various handles. A handlebody is a specific type of connected manifold that uses only 0-handles and j-handles for one fixed value of j.
Who developed the theory of handle decompositions?
The process of attaching handles to manifolds was formulated by the mathematician Stephen Smale.