Glossary

8.2 Handle decompositions of manifolds

3 min read•august 7, 2024

Handle decompositions are a powerful way to build and understand manifolds. Starting with disks, we attach handles of increasing dimension, creating a structured approach to constructing complex shapes. This method bridges smooth structures and topology.

Handle decompositions connect to Morse theory through critical points of Morse functions. Each critical point corresponds to a handle, with the index determining the handle's dimension. This link provides deep insights into manifold structure and topology.

Handles and Handle Attachments

Definition and Components of a Handle

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  • Handle is a copy of Dk×Dn−kD^k \times D^{n-k} attached to the of an nn-manifold MM
  • kk-handle refers to a handle with kk being the dimension of the first factor DkD^k
  • Core of a kk-handle is Dk×{0}D^k \times \{0\}, a kk-dimensional disk
  • Co-core of a kk-handle is {0}×Dn−k\{0\} \times D^{n-k}, an (n−k)(n-k)-dimensional disk

Attaching a Handle to a Manifold

  • Attaching map is a smooth embedding ϕ:Sk−1×Dn−k→∂M\phi: S^{k-1} \times D^{n-k} \to \partial M used to attach a kk-handle to an nn-manifold MM
  • Handle is the process of gluing a kk-handle to MM along the image of the attaching map ϕ\phi
  • Resulting manifold after attaching a kk-handle is M∪ϕ(Dk×Dn−k)M \cup_\phi (D^k \times D^{n-k})
  • Boundary of the resulting manifold changes by replacing Sk−1×Dn−kS^{k-1} \times D^{n-k} with Dk×Sn−k−1D^k \times S^{n-k-1}

Examples of Handle Attachments

  • Attaching a 00-handle to a manifold is equivalent to taking the disjoint union with an nn-disk (DnD^n)
  • Attaching a 11-handle to a manifold can be visualized as connecting two components or creating a tunnel
  • Attaching an (n−1)(n-1)-handle to an nn-manifold corresponds to filling in an (n−1)(n-1)-sphere on the boundary with an nn-disk

Handle Decompositions and Smooth Structures

Handle Decompositions

  • Handle decomposition is a way to build a manifold by starting with a collection of nn-disks and attaching handles of increasing dimension
  • Manifold can be decomposed into handles, M=M0∪(⋃i=1mHi)M = M_0 \cup (\bigcup_{i=1}^m H_i), where M0M_0 is a disjoint union of nn-disks and HiH_i are handles
  • Handles are attached in order of increasing dimension, with each handle attached to the boundary of the union of the previous handles
  • Existence of a handle decomposition for a compact smooth manifold follows from the existence of a self-indexing Morse function

Smooth Structures and Morse Functions

  • Smooth structure on a manifold is a maximal atlas of smoothly compatible charts
  • Morse function is a smooth real-valued function on a manifold with non-degenerate critical points
  • Critical points of a Morse function correspond to handles in a handle decomposition of the manifold
  • Index of a critical point determines the dimension of the corresponding handle

Relationship to CW Complexes

  • CW complex is a topological space constructed by attaching cells of increasing dimension
  • Handle decomposition of a manifold can be viewed as a smooth analogue of a CW decomposition
  • kk-handles in a handle decomposition correspond to kk-cells in a CW decomposition
  • Attaching maps for handles are smooth embeddings, while attaching maps for cells are continuous maps

Key Terms to Review (19)

0-handle: A 0-handle is a basic building block in the handle decomposition of a manifold, representing a topological feature that is homeomorphic to a closed ball in a Euclidean space. It serves as the foundation for constructing higher-dimensional manifolds, as it corresponds to adding a compact region to the manifold without introducing boundaries. The 0-handle plays a crucial role in defining the topology of manifolds and is essential in understanding how more complex handles interact with each other.
1-handle: A 1-handle is a basic building block in the handle decomposition of a manifold, representing a 'thickening' of a 1-dimensional space, like an interval, into a 2-dimensional surface. This concept connects the geometric structure of manifolds to the topology of their critical points, illustrating how these handles are attached and contribute to the overall shape and features of the manifold.
2-handle: A 2-handle is a specific type of handle used in the handle decomposition of 4-dimensional manifolds, where it attaches a 2-dimensional disk to the boundary of a manifold. This operation modifies the topology of the manifold by introducing a new critical point and changing its structure. 2-handles are crucial in understanding how to build complex manifolds and relate to critical points that arise in Morse theory.
Attachment: In the context of handle decompositions of manifolds, attachment refers to the process of connecting lower-dimensional handles to a manifold to construct a more complex topological space. This is done by identifying the boundary of the handle with a specific part of the manifold, allowing for a controlled way to build up the structure of the manifold while preserving its topological properties. Attachment is crucial for understanding how different handles contribute to the overall shape and features of the manifold.
Boundary: In topology and geometry, a boundary refers to the edge or limit of a manifold or a topological space. It essentially separates the inside from the outside and can be thought of as the set of points that do not belong to the interior of a space. This concept is crucial in understanding how spaces are constructed and analyzed, particularly in the context of CW complexes, handle decompositions, and handlebodies.
Cell complex: A cell complex is a type of topological space that is constructed by gluing together cells of various dimensions, such as points (0-cells), line segments (1-cells), disks (2-cells), and so on. This structure allows for a flexible way to study topology, particularly through Morse theory and CW complexes, revealing important properties about spaces and their invariants.
Cohen's Theorem: Cohen's Theorem is a fundamental result in topology that describes how to construct handle decompositions for manifolds. This theorem demonstrates that any smooth, compact manifold can be represented as a union of handles, which are topological building blocks. The connection to handle decompositions highlights the structured way in which manifolds can be analyzed and classified based on their topology.
Collapsing: Collapsing refers to the process of simplifying a topological space by identifying certain subspaces or handles, effectively 'squeezing' the structure down to a more manageable form. This technique is particularly useful in Morse Theory, where it allows for the understanding of the manifold's topology by reducing it to its essential features, making complex spaces easier to analyze and study.
Deformation retract: A deformation retract is a type of continuous mapping that allows one topological space to be 'shrunk' into a subspace, preserving the structure of the original space in a homotopically equivalent way. It shows how complex shapes can be simplified while keeping essential features intact, which connects closely to the study of topological implications, handle decompositions, and handlebodies.
Deformation theory: Deformation theory is a mathematical framework that studies how geometric structures can be continuously transformed or deformed into one another while preserving certain properties. It connects closely with handle decompositions of manifolds, where manifolds can be represented as combinations of simple pieces, or handles, which can be altered to explore the manifold's topological features and relationships.
Homotopy equivalence: Homotopy equivalence is a relation between two topological spaces that indicates they can be transformed into each other through continuous deformations, meaning they have the same 'shape' in a topological sense. This concept is crucial because it implies that if two spaces are homotopy equivalent, they share essential topological properties, making them indistinguishable from a homotopical perspective.
John Milnor: John Milnor is a prominent American mathematician known for his groundbreaking work in differential topology, particularly in the field of smooth manifolds and Morse theory. His contributions have significantly shaped modern mathematics, influencing various concepts related to manifold structures, Morse functions, and cobordism theory.
K-manifold: A k-manifold is a topological space that locally resembles Euclidean space of dimension k, meaning every point has a neighborhood that is homeomorphic to an open subset of $$ extbf{R}^k$$. This concept is foundational in understanding the structure and classification of manifolds, particularly when studying their decomposition into simpler pieces, such as in handle decompositions.
N-manifold: An n-manifold is a topological space that locally resembles Euclidean space of dimension n. This means that for every point in the n-manifold, there exists a neighborhood that is homeomorphic to an open subset of $$ ext{R}^n$$. These structures are fundamental in understanding geometric and topological properties of spaces, especially when discussing their decomposition and relationships in higher dimensions.
Robert G. Bartle: Robert G. Bartle is a renowned mathematician known for his contributions to functional analysis and topology, particularly in the context of Morse Theory. His work has been influential in understanding handle decompositions of manifolds, which are crucial for analyzing the topological properties of these spaces.
Simple Connectivity: Simple connectivity refers to a topological property of a space that indicates it is path-connected and contains no 'holes' that prevent loops from being shrunk to a point. In this context, it plays a vital role in understanding the structure of manifolds, especially when dealing with handle decompositions. Simple connectivity ensures that every loop can be continuously transformed into a single point, which is crucial for the classification and manipulation of manifolds in Morse Theory.
Surgery: In the context of topology and manifold theory, surgery refers to a process that modifies a manifold by removing a portion of it and replacing it with another piece, effectively allowing the construction of new manifolds. This technique is crucial for understanding how different manifolds can be related or transformed into each other, and it plays a key role in handle decompositions and cobordism theory.
Topological classification: Topological classification refers to the method of categorizing topological spaces or manifolds based on their intrinsic properties, such as connectivity, compactness, and the nature of their singularities. This classification enables mathematicians to understand how different spaces relate to each other in terms of deformation, which is fundamental in fields like algebraic topology and differential geometry.
Whitney's Embedding Theorem: Whitney's Embedding Theorem states that any smooth manifold of dimension m can be smoothly embedded into Euclidean space of dimension 2m. This fundamental result highlights the relationship between manifolds and their representations in higher-dimensional spaces, emphasizing the concept of embeddings and their significance in understanding manifold topology.
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