12.3 The h-cobordism theorem (preview)
3 min read•august 7, 2024
The bridges the gap between topology and smooth structures on manifolds. It shows that under certain conditions, manifolds that are homotopy equivalent are actually diffeomorphic, providing a powerful tool for classifying high-dimensional manifolds.
This theorem is a cornerstone of Morse theory and , connecting abstract topological properties to concrete geometric structures. It highlights the deep relationship between homotopy theory, differential topology, and the study of manifolds in higher dimensions.
Cobordisms and Diffeomorphisms
Defining Cobordisms and Diffeomorphisms
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- h-cobordism a cobordism where both inclusions and are homotopy equivalences
- a smooth bijective map between manifolds whose inverse is also smooth, capturing the notion of smooth equivalence between manifolds
- states that if is a compact h-cobordism between manifolds and of dimension , then is diffeomorphic to
- a maximal atlas of smoothly compatible coordinate charts on a manifold, enabling the study of differentiable functions and maps between manifolds
Implications and Applications
- h-cobordisms provide a framework for studying the relationship between manifolds that are "almost diffeomorphic"
- Diffeomorphisms allow the classification of manifolds up to smooth equivalence, a fundamental problem in differential topology
- s-cobordism theorem is a powerful tool for understanding the topology of high-dimensional manifolds (dimensions )
- Smooth structures on manifolds are essential for formulating and solving problems in differential geometry and mathematical physics (general relativity, gauge theory)
Simply Connected Manifolds
Definition and Significance
- Simply connected a topological space is simply connected if it is path-connected and its fundamental group is trivial (every loop can be continuously contracted to a point)
- (now a theorem) states that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere ()
- a smooth manifold homeomorphic but not diffeomorphic to the standard -sphere, illustrating the difference between topological and smooth structures
Role in Cobordism Theory
- Simply connected manifolds play a central role in the s-cobordism theorem, which applies specifically to h-cobordisms between simply connected manifolds
- Poincaré conjecture, proven by Perelman in 2003, resolves the classification of simply connected 3-manifolds and has implications for cobordism theory in low dimensions
- Exotic spheres demonstrate the complexity of smooth structures on manifolds and motivate the study of cobordisms and diffeomorphisms to understand these structures
Algebraic Invariants
Whitehead Torsion
- an algebraic invariant associated with a between CW complexes, measuring the "twisting" of the cellular structure
- Defined using the Whitehead group, which is the quotient of the algebraic K-theory group by the subgroup generated by elementary matrices over a ring
- Vanishes for simple homotopy equivalences, which are homotopy equivalences that can be decomposed into a sequence of elementary expansions and collapses of cells
Applications to Cobordism Theory
- Whitehead torsion is a key ingredient in the proof of the s-cobordism theorem, providing an algebraic obstruction to the existence of a diffeomorphism between an h-cobordism and a product cobordism
- For an h-cobordism , the Whitehead torsion of the inclusion vanishes if and only if is diffeomorphic to
- Computing Whitehead torsion allows for the classification of h-cobordisms and the study of the smooth structures on manifolds
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.
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.
Dimensional Condition: The dimensional condition refers to a specific requirement in topology that focuses on the dimensions of manifolds involved in h-cobordisms. It essentially states that for two smooth manifolds to be h-cobordant, they need to have the same dimension, which is crucial in establishing equivalences between them in terms of their homotopy type. This condition highlights the importance of dimensionality in the study of cobordism theory and helps differentiate between manifolds that are homotopically equivalent and those that are not.
Exotic Sphere: An exotic sphere is a type of differentiable manifold that is homeomorphic but not diffeomorphic to the standard sphere. Exotic spheres emerge in differential topology, particularly in the study of manifolds and the h-cobordism theorem, which connects the topology of spheres and their higher-dimensional counterparts.
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.
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.
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-Smale: Morse-Smale refers to a specific class of smooth functions on manifolds that are used in Morse Theory to study the topology of the manifold through critical points. A Morse-Smale function has non-degenerate critical points and satisfies the condition that the stable and unstable manifolds of different critical points intersect transversally. This property is crucial as it ensures that the topology of the manifold can be analyzed by examining the behavior of these critical points, particularly in relation to the h-cobordism theorem.
Poincaré Conjecture: The Poincaré Conjecture is a fundamental statement in topology asserting that every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. This conjecture is pivotal because it links the topology of three-dimensional spaces with their geometric structures, ultimately influencing how manifolds are classified and understood within differential topology and algebraic topology.
S-cobordism theorem: The s-cobordism theorem is a result in differential topology that asserts the equivalence of certain types of manifolds with boundaries, specifically those that are simply connected. It states that if two smooth manifolds with boundary are s-cobordant, then they are homotopy equivalent, which implies they have the same topological structure in a certain sense. This theorem has deep connections to the h-cobordism theorem and is essential for understanding the relationship between cobordism classes and the topology of manifolds.
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 structure: A smooth structure on a manifold is a mathematical framework that allows for the definition of differentiable functions, meaning that the manifold can be treated like a smooth space where calculus can be applied. This concept is crucial for understanding how topological properties interact with differentiable structures, influencing various aspects of geometry and topology.
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 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.
Triviality condition: The triviality condition is a requirement in the context of h-cobordism that ensures two manifolds are considered equivalent if their structure can be smoothly transformed into one another through a series of simple steps. This condition implies that if two manifolds satisfy the triviality condition, they share the same homotopy type and can be regarded as 'the same' for certain topological purposes. Understanding this concept is crucial for applying the h-cobordism theorem to establish relationships between different manifolds.
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.
Whitney Stratification: Whitney stratification is a way of decomposing a smooth manifold into simpler pieces, called strata, that are easier to analyze and understand. Each stratum is a smooth submanifold of the original manifold, and the stratification respects the topology of the space, which can be very helpful in Morse Theory and other areas of differential topology, particularly when discussing the h-cobordism theorem.