12.2 Morse functions on cobordisms
3 min read•august 7, 2024
Morse functions on cobordisms extend the concept to manifolds with boundaries. They map a compact manifold to [0,1], with critical points in the interior. These functions help us understand the topology of cobordisms through their critical points and gradient-like vector fields.
Handle decompositions express cobordisms as sequences of handle attachments. Each handle corresponds to a critical point of a . By rearranging and canceling critical points, we can simplify the topology and better understand the structure of cobordisms.
Morse Functions and Cobordisms
Defining Morse Functions on Cobordisms
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- Morse function on extends the concept of Morse functions to manifolds with boundary
- Cobordism consists of a compact manifold with boundary
- Morse function on a cobordism satisfies and
- Critical points of lie in the interior of and have distinct critical values
Vector Fields and Critical Points
- Gradient-like vector field associated with a Morse function on a cobordism
- Points outward along and inward along
- Agrees with the gradient of near critical points
- Rearrangement of critical points allows changing the order of critical values while preserving the cobordism structure
- Achieved by modifying the Morse function and gradient-like vector field
- Critical points with different indices can be rearranged freely
- Cancellation of critical points eliminates pairs of critical points with consecutive indices
- Corresponds to simplifying the topology of the cobordism
- Requires the existence of a gradient trajectory connecting the critical points
Handle Decompositions
Constructing Handle Decompositions
- expresses a cobordism as a sequence of handle attachments
- Handles are standard building blocks classified by their index
- Index handle is a product of a -dimensional disc and a -dimensional disc
- Attaching maps specify how handles are glued to the boundary of the existing cobordism
- Attaching maps are embeddings of the boundary of the handle into the boundary of the cobordism
- Handle slides modify the attaching maps while preserving the topology
Elementary Cobordisms and Handle Decompositions
- Elementary cobordism corresponds to attaching a single handle to a trivial cobordism
- Trivial cobordism is the product cobordism
- Attaching an index handle to yields an elementary cobordism
- Every cobordism admits a handle decomposition
- Decomposition is not unique, but different decompositions are related by handle slides and cancellations
- Morse functions induce handle decompositions, with critical points corresponding to handles
Types of Cobordisms
Elementary Cobordisms
- Elementary cobordism is obtained by attaching a single handle to a trivial cobordism
- Index handle attachment corresponds to adding a new connected component (birth)
- Index handle attachment corresponds to filling in a boundary component (death)
- Index handle attachment () modifies the topology of the cobordism
- Composition of elementary cobordisms allows building more complex cobordisms
- Elementary cobordisms can be glued together along their boundaries
- Composition corresponds to concatenating the handle decompositions
Product Cobordisms
- Product cobordism is the trivial cobordism
- Boundaries are two copies of : and
- Morse function is the projection onto the interval
- No critical points in the interior of the cobordism
- Product cobordisms are identity morphisms in the category of cobordisms
- Composition with a product cobordism does not change the topology
- Every cobordism is bordant to a composition of elementary cobordisms and product cobordisms
Key Terms to Review (18)
Cobordism: Cobordism is a concept in topology that relates two manifolds through a higher-dimensional manifold, called a cobordism, that connects them. This idea is fundamental in understanding how manifolds can be transformed into one another and provides a powerful tool for classifying manifolds based on their boundaries and the relationships between them.
Differentiable Manifold: A differentiable manifold is a topological space that locally resembles Euclidean space and has a differentiable structure, allowing for the definition of calculus on it. This concept plays a vital role in various areas of mathematics, as it allows for smooth functions and derivatives to be defined, enabling analysis in more complex spaces that appear non-Euclidean at larger scales.
Gradient Flow: Gradient flow refers to the flow generated by following the negative gradient of a function, effectively describing how a system evolves over time towards its critical points. This concept is crucial in understanding the dynamics of functions, particularly in relation to their critical points, where local minima and maxima exist, and connects deeply with various topological and geometrical properties of manifolds.
Handle Decomposition: Handle decomposition is a process used in topology to describe the structure of manifolds by breaking them down into simpler pieces called handles, which correspond to higher-dimensional analogs of attaching disks. This concept is crucial for understanding how manifolds can be constructed or deconstructed, especially in the context of Morse theory and cobordisms, revealing significant insights into their topological properties.
Homology: Homology is a mathematical concept used to study topological spaces by associating algebraic structures, called homology groups, which capture information about the shape and connectivity of the space. This notion plays a vital role in understanding the properties of manifolds and CW complexes, as it relates to the classification of critical points and provides insights into cobordism theory.
Index of a critical point: The index of a critical point is an integer that reflects the local topology of a manifold at that point, indicating how many directions in which the function decreases or increases. It plays a vital role in Morse Theory, providing insights into the structure of the manifold and its features. The index helps categorize critical points based on their nature, linking these points to local behavior, geometric interpretation, and broader properties of spaces.
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.
Local maximum: A local maximum refers to a point in a function where the value is greater than or equal to the values of the function at nearby points. This concept is crucial in understanding critical points, as it helps classify the behavior of functions and their extrema in various contexts such as differentiable functions, Morse theory, and gradient vector fields.
Marcel Grossmann: Marcel Grossmann was a Swiss mathematician and physicist known for his contributions to the fields of differential geometry and general relativity. He is particularly recognized for his collaboration with Albert Einstein, providing crucial mathematical support that facilitated the formulation of Einstein's theory of general relativity, which has deep connections to various aspects of topology and geometry.
Morse Function: A Morse function is a smooth real-valued function defined on a manifold that has only non-degenerate critical points, where the Hessian matrix at each critical point is non-singular. These functions are crucial because they provide insights into the topology of manifolds, allowing the study of their structure and properties through the behavior of their critical points.
Morse Inequalities: Morse inequalities are mathematical statements that relate the topology of a manifold to the critical points of a Morse function defined on it. They provide a powerful tool to count the number of critical points of various indices and connect these counts to the homology groups of the manifold.
Morse Lemma: The Morse Lemma states that near a non-degenerate critical point of a smooth function, the function can be expressed as a quadratic form up to higher-order terms. This result allows us to understand the local structure of the function around critical points and connects deeply to various concepts in differential geometry and topology.
Morse-theoretic cobordism: Morse-theoretic cobordism is a concept that extends Morse theory to study the relationship between manifolds via cobordisms, focusing on how critical points of Morse functions on these manifolds can reveal topological properties. This approach allows for the exploration of the way one manifold can 'flow' into another through a higher-dimensional space, providing insights into the topology of both the starting and ending manifolds.
Non-degenerate critical point: A non-degenerate critical point of a smooth function is a point where the gradient is zero, and the Hessian matrix at that point is invertible. This condition ensures that the critical point is not flat and allows for a clear classification into local minima, maxima, or saddle points, which connects to many important aspects of manifold theory and Morse theory.
Stable Manifold: A stable manifold is a collection of points in a dynamical system that converge to a particular equilibrium point as time progresses. This concept is essential for understanding the behavior of trajectories near critical points and forms the backbone for analyzing the structure of dynamical systems, especially in relation to Morse functions and their level sets.
Surgery theory: Surgery theory is a mathematical approach that deals with the modification and manipulation of manifolds to understand their structures and relationships. It focuses on how certain surgeries can change the topology of manifolds, providing insights into their properties and classifications. This theory is particularly relevant when studying complex relationships between manifolds, such as in the context of cobordisms and the classification of higher-dimensional spaces.
Topological invariance: Topological invariance refers to the property of certain mathematical structures that remain unchanged under continuous deformations, such as stretching or bending, but not tearing or gluing. This concept is crucial in understanding how certain properties of spaces can be preserved when considering Morse functions on cobordisms, allowing mathematicians to classify and compare different topological spaces effectively.
Unstable Manifold: An unstable manifold is a collection of points in the phase space of a dynamical system that exhibits a tendency to move away from a critical point under the influence of perturbations. This concept is crucial for understanding how systems behave near critical points, especially regarding flow lines and the behavior of trajectories in the vicinity of equilibrium states.