13.1 Statement and proof of the h-cobordism theorem
3 min read•august 7, 2024
The is a powerful result in differential topology. It connects smooth structures on manifolds to algebraic properties of equivalences, providing a way to classify certain types of manifolds up to .
This theorem uses tools from Morse theory, like and critical point cancellation. It also involves topological concepts such as simple connectivity and , bridging different areas of mathematics in a surprising way.
Cobordisms and Diffeomorphisms
Cobordisms and their Properties
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- h-: A cobordism where the inclusions and are homotopy equivalences
- Cobordism: A compact manifold whose boundary is the disjoint union of two closed manifolds and , denoted as
- Provides a smooth transition between two manifolds
- Can be thought of as a "bridge" connecting and
- : A continuous map between two topological spaces that has a homotopy inverse
- If and are homotopy equivalent to , they are homotopy equivalent to each other
- Homotopy equivalence preserves important topological properties (, homotopy groups)
Diffeomorphisms and Smooth Structures
- Diffeomorphism: A smooth bijective map between manifolds whose inverse is also smooth
- Diffeomorphic manifolds have the same smooth structure
- h-cobordisms can be used to study the existence of diffeomorphisms between manifolds
- Example: The torus and the square with opposite sides identified are diffeomorphic
Morse Theory Tools
Morse Functions and Critical Points
- : A smooth real-valued function on a manifold whose are non-degenerate
- Critical points correspond to topological changes in the manifold
- Morse functions can be used to study the topology of a manifold
- Example: The height function on a torus has four critical points (maximum, minimum, and two saddles)
- : A vector field that is compatible with a Morse function
- Integral curves of a gradient-like vector field connect critical points
- Used to define the flow on a manifold associated with a Morse function
Handle Decompositions and Cancellation
- : A way to build a manifold by attaching handles of various indices to a disk
- Each handle corresponds to a critical point of a Morse function
- The index of a handle is the number of "negative" directions of the Hessian at the corresponding critical point
- Example: A 2-dimensional handle decomposition of a torus consists of one 0-handle, two 1-handles, and one 2-handle
- : A process of eliminating pairs of critical points in a Morse function
- Cancellation is possible when the indices of the critical points differ by 1
- Cancellation simplifies the handle decomposition and the topology of the manifold
Topological Conditions
Simple Connectivity and the Fundamental Group
- : A topological space is simply connected if it is path-connected and has a trivial
- The fundamental group measures the number of distinct loops in a space that cannot be continuously deformed into each other
- Simply connected spaces have no non-trivial loops
- Example: The 2-sphere is simply connected, while the torus is not
Whitehead Torsion and h-Cobordisms
- Whitehead torsion: An algebraic invariant associated with a homotopy equivalence between CW complexes
- Measures the "twisting" of the homotopy equivalence
- Vanishing of Whitehead torsion is a necessary condition for a homotopy equivalence to be a simple homotopy equivalence
- In the h-cobordism theorem, the vanishing of Whitehead torsion ensures that the cobordism is trivial (a product cobordism)
- Example: The Whitehead torsion of the identity map on a CW complex is zero
Key Terms to Review (25)
Cancellation of critical points: Cancellation of critical points refers to a process in Morse theory where pairs of critical points are identified and eliminated during a deformation of a function. This concept is crucial in understanding how the topology of a manifold can change through smooth transformations, particularly in the context of the h-cobordism theorem, which relates to the equivalence of certain manifolds under homotopy equivalences.
Cell Complexes: Cell complexes are mathematical structures used in algebraic topology to study topological spaces by breaking them down into simpler pieces called cells. These cells are usually defined in terms of their dimensionality, such as 0-cells (points), 1-cells (lines), and 2-cells (surfaces), allowing for the construction of complex shapes from basic building blocks. Understanding cell complexes is essential when discussing the h-cobordism theorem, as they provide a framework for analyzing the relationships between different topological spaces through their decomposition into cells.
Cobordant manifolds: Cobordant manifolds are pairs of manifolds that represent the same class in a cobordism relation, which intuitively means that they can be 'glued' together to form a higher-dimensional manifold with boundary. This relationship is essential in topology as it allows the study of different manifolds through their connections and transformations, particularly in understanding how they can be related via homotopy and other topological features.
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.
Critical Points: Critical points are locations in the domain of a function where its derivative is zero or undefined. These points are important as they often correspond to local minima, local maxima, or saddle points, influencing the shape and features of the function's graph.
Diffeomorphism: A diffeomorphism is a smooth, bijective mapping between smooth manifolds that has a smooth inverse. It preserves the structure of the manifolds, meaning that both the mapping and its inverse are smooth, allowing for a seamless transition between the two spaces without losing any geometric or topological information.
Dimension condition: The dimension condition refers to a specific requirement in the h-cobordism theorem that states the dimensions of two manifolds involved in a cobordism must differ by one. This is crucial because the theorem discusses the relationships between manifolds and their boundaries, and the dimension condition ensures that the manifolds can be 'glued' together properly to form a cobordism. Without this condition, the structural properties and equivalences needed for the theorem to hold would break down.
Fundamental group: The fundamental group is an algebraic structure that captures the notion of the different ways loops can be drawn in a topological space. It essentially tells us about the 'shape' of the space by analyzing how these loops can be continuously transformed into one another. This concept plays a crucial role in understanding the properties of manifolds and their classifications, especially when examining how spaces can be deformed and related to one another through h-cobordisms.
Gradient-like vector field: A gradient-like vector field is a smooth vector field on a manifold that resembles the behavior of a gradient of a function, guiding trajectories towards critical points while ensuring certain topological properties are maintained. It plays a crucial role in studying the topology of manifolds, especially in relation to the Morse-Smale complex, where it provides a systematic way to analyze the flow of gradients and understand the structure of critical points. This concept is also significant in the context of h-cobordism, as it helps illustrate how manifolds can be transformed and understood through their critical points and flow structures.
H-cobordism theorem: The h-cobordism theorem states that if two compact smooth manifolds have the same homotopy type and are h-cobordant, then they are diffeomorphic if one of them is simply connected. This theorem plays a crucial role in understanding the topology of manifolds and their structures, as it allows mathematicians to classify manifolds based on their homotopy properties and their cobordism relation.
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.
Handle decompositions: Handle decompositions are a method of breaking down a manifold into simpler pieces called handles, which allows for the understanding and classification of its topological structure. This approach is particularly useful in studying high-dimensional manifolds, as it provides a way to visualize and manipulate the manifold by attaching or removing handles of various dimensions. The connection to the h-cobordism theorem arises from the ability to analyze the behavior of these decompositions in terms of homotopy equivalence, which is crucial for establishing results about the manifold's topology.
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.
Homotopy: Homotopy is a concept in topology that describes a continuous deformation of one function or shape into another. It establishes when two functions are considered equivalent if one can be transformed into the other through a continuous path, which is important in studying properties of spaces that remain unchanged under such transformations. This idea connects closely to concepts like differential forms, gradient vector fields, cobordism, and sphere eversion, highlighting how structures can change yet retain their essential characteristics.
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.
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.
Simply Connected: A space is simply connected if it is path-connected and every loop within that space can be continuously contracted to a point without leaving the space. This concept is crucial in understanding the topological properties of spaces, particularly in relation to homotopy and deformation retractions, which influence how we analyze manifolds and their classifications.
Smooth manifold: A smooth manifold is a topological space that locally resembles Euclidean space and is equipped with a differentiable structure, allowing for smooth transitions between coordinate charts. This concept is fundamental in many areas of mathematics and physics, particularly in understanding complex geometric and topological properties through calculus.
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 Manifold: A topological manifold is a topological space that resembles Euclidean space near each point, meaning it is locally homeomorphic to an open subset of a Euclidean space. This concept is crucial in understanding the structure and properties of spaces in higher dimensions, as it allows mathematicians to apply techniques from calculus and analysis to more abstract spaces.
Trivial h-cobordism: A trivial h-cobordism is a specific type of h-cobordism between two manifolds where both manifolds are diffeomorphic to each other and the h-cobordism is contractible. This means that the map between the two manifolds can be continuously transformed into a simpler, trivial form, indicating that they share the same homotopy type. Understanding this concept is essential in the study of the h-cobordism theorem, which addresses when two manifolds can be considered equivalent based on their topological properties.
Whitehead torsion: Whitehead torsion is an algebraic invariant that arises in the study of homotopy theory, particularly in the context of h-cobordisms. It is a measure of the failure of a homotopy equivalence between two spaces to be a homeomorphism, capturing essential information about the topological structure of the space, which becomes significant when considering h-cobordisms and the conditions under which they hold.