Glossary

6.3 Applications to cobordism theory

4 min read•july 30, 2024

The Thom Isomorphism Theorem plays a crucial role in cobordism theory, connecting topology and algebra. It allows us to study manifolds through homotopy groups of Thom spectra, simplifying calculations and revealing deep connections between different areas of mathematics.

Cobordism theory classifies manifolds based on their boundaries and provides a powerful framework for understanding geometric structures. By applying the Thom Isomorphism, we can relate cobordism groups to more familiar algebraic objects, making abstract concepts more concrete and accessible.

Cobordism Theory Basics

Cobordism as an Equivalence Relation

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  • Cobordism is an equivalence relation on the set of compact manifolds
    • Two manifolds are cobordant if their disjoint union is the of a with boundary (cobordism)
  • The set of equivalence classes of n-dimensional manifolds under the cobordism relation forms an abelian group called the n-th cobordism group, denoted Ωn\Omega_n
    • The group operation is induced by the disjoint union of manifolds (identity element is the empty set, inverse of a manifold is itself with the opposite orientation)

Cobordism as a Generalized Cohomology Theory

  • The direct sum of all cobordism groups, Ω∗=⨁n≥0Ωn\Omega_* = \bigoplus_{n\geq0} \Omega_n, forms a graded ring called the cobordism ring
    • The product is induced by the Cartesian product of manifolds
  • The cobordism ring is a generalized cohomology theory satisfying the Eilenberg-Steenrod axioms except for the dimension axiom
  • The , denoted MO∗MO_*, is the cobordism ring of manifolds without a specified orientation
  • The oriented cobordism ring, denoted MSO∗MSO_*, considers oriented manifolds

Thom Isomorphism for Cobordism

Thom Spaces and the Thom Isomorphism

  • The of a vector bundle ξ\xi over a space XX, denoted Th(ξ)Th(\xi), is the one-point compactification of the total space of ξ\xi
  • The Thom isomorphism states that for a rank nn vector bundle ξ\xi over a space XX, there is an isomorphism between the cohomology of the Thom space Th(ξ)Th(\xi) and the cohomology of the base space XX, shifted by the rank of the bundle: Hk+n(Th(ξ))≅Hk(X)H^{k+n}(Th(\xi)) \cong H^k(X)

Constructing the Cobordism Ring using the Thom Isomorphism

  • The MOMO is constructed by taking the Thom spaces of the universal vector bundles over the classifying spaces BO(n)BO(n) for all n≥0n \geq 0, with the structure maps given by the Thom isomorphism
  • The cobordism ring MO∗MO_* is isomorphic to the homotopy groups of the Thom spectrum MOMO, i.e., MOn≅πn(MO)MO_n \cong \pi_n(MO)
  • The establishes a bijection between the set of cobordism classes of closed nn-manifolds and the nn-th of the Thom spectrum MOMO
    • This allows the computation of cobordism groups using homotopy theory

Low-Dimensional Cobordism Groups

Unoriented Cobordism Groups

  • The unoriented cobordism group MO0MO_0 is isomorphic to Z/2Z\mathbb{Z}/2\mathbb{Z}, generated by a point
    • Every compact 0-manifold is cobordant to either the empty set or a single point
  • The unoriented cobordism group MO1MO_1 is trivial
    • Every compact 1-manifold is the boundary of a compact 2-manifold (disk)
  • The unoriented cobordism group MO2MO_2 is isomorphic to Z/2Z\mathbb{Z}/2\mathbb{Z}, generated by the real projective plane RP2\mathbb{RP}^2
    • Every compact surface is cobordant to either the empty set or a disjoint union of RP2\mathbb{RP}^2s

Oriented Cobordism Groups

  • The oriented cobordism group MSO0MSO_0 is isomorphic to Z\mathbb{Z}, generated by a point with a chosen orientation
  • The oriented cobordism group MSO1MSO_1 is trivial
    • Every oriented compact 1-manifold is the boundary of an oriented compact 2-manifold (oriented disk)
  • The oriented cobordism group MSO2MSO_2 is trivial
    • Every oriented compact surface is cobordant to the empty set (boundary of a 3-ball)

Cobordism vs Characteristic Classes

Characteristic Classes and Vector Bundles

  • , such as and Pontryagin classes, are cohomology classes associated with vector bundles that provide information about the topology of the base space
  • The Stiefel-Whitney classes wi(ξ)w_i(\xi) of a vector bundle ξ\xi over a space XX are elements of the cohomology ring H∗(X;Z/2Z)H^*(X; \mathbb{Z}/2\mathbb{Z})
    • They are related to the unoriented cobordism ring MO∗MO_* via the Thom isomorphism
  • The Pontryagin classes pi(ξ)p_i(\xi) of a real vector bundle ξ\xi over a space XX are elements of the cohomology ring H∗(X;Z)H^*(X; \mathbb{Z})
    • They are related to the oriented cobordism ring MSO∗MSO_* via the Thom isomorphism

Applications of Characteristic Classes to Cobordism

  • The expresses the signature of a compact oriented 4k-dimensional manifold in terms of its Pontryagin numbers
    • Pontryagin numbers are certain characteristic numbers derived from the Pontryagin classes
  • The of a compact spin manifold, defined using the Pontryagin classes, is an invariant that determines the spin cobordism class of the manifold
    • It is related to the index of the Dirac operator via the Atiyah-Singer index theorem
  • The of a complex manifold, defined using the Chern classes, is an invariant that determines the complex cobordism class of the manifold
    • It is related to the Riemann-Roch theorem

Key Terms to Review (24)

â-genus: The â-genus is a topological invariant that generalizes the notion of genus in algebraic topology and is defined as the maximal number of disjoint, non-contractible loops that can be drawn on a surface without separating it. It plays a crucial role in understanding the topology of manifolds and their classification in both differential geometry and cobordism theory, connecting various geometric and algebraic concepts.
Boundary: In the context of cobordism theory, a boundary refers to the topological or geometrical boundary of a manifold, which plays a crucial role in defining relationships between different manifolds and their dimensions. Boundaries help establish connections between various cobordism classes and allow for the study of manifolds through their boundaries, enabling mathematicians to understand how manifolds can be decomposed or constructed. This concept is fundamental in linking the algebraic and geometric aspects of topology.
Characteristic classes: Characteristic classes are a way to associate algebraic invariants to vector bundles, which help in understanding their geometric and topological properties. They are crucial in many areas of mathematics, such as differential geometry and topology, providing insights into the structure of bundles and leading to significant results like the Atiyah-Singer index theorem.
Classification Theorem: The classification theorem is a fundamental result in mathematics that categorizes objects into distinct classes based on shared properties. In the context of cobordism theory, this theorem provides a way to classify manifolds up to cobordism, meaning that it helps to understand how different manifolds can be transformed into one another through smooth deformations and boundaries.
Compact manifold: A compact manifold is a type of manifold that is both compact and differentiable, meaning it is a space that is closed and bounded, allowing for every open cover to have a finite subcover. This property ensures that it has no edges or boundaries, making it a crucial concept in various mathematical fields, particularly in topology and geometry. Compact manifolds often have nice geometric structures and properties, which are significant in cobordism theory, as they relate to the classification of manifolds and the study of their boundaries.
Diffeomorphism: A diffeomorphism is a smooth, invertible function between two smooth manifolds that has a smooth inverse. This concept plays a crucial role in understanding the structure and properties of manifolds, as it allows for the comparison and analysis of their geometrical features. Diffeomorphisms enable mathematicians to establish whether two manifolds can be considered 'the same' in a differential sense, which is particularly important when discussing cobordism and other related theories.
Fundamental group: The fundamental group is an algebraic structure that captures the essential features of a topological space in terms of its loops. It consists of equivalence classes of loops based at a point, with the operation of concatenation. This group reflects how loops can be deformed into one another, providing insight into the shape and connectivity of the space, which is crucial in understanding its cobordism properties.
Hirzebruch Signature Theorem: The Hirzebruch Signature Theorem is a fundamental result in topology that relates the signature of a manifold to its Pontryagin classes and its Euler characteristic. It provides a way to compute invariants of manifolds, bridging the gap between differential topology and algebraic topology, especially within the realms of cobordism and K-homology.
Homology Group: A homology group is a mathematical concept that captures the topological features of a space by associating algebraic structures, specifically groups, to it. It allows mathematicians to study shapes and their properties through sequences of algebraic invariants, which are particularly useful in understanding the relationships between different spaces and their dimensional characteristics.
Homotopy group: A homotopy group is a mathematical concept that captures information about the topological properties of spaces by associating to each space a sequence of groups, which describe the different ways loops can be continuously transformed into each other. These groups are crucial for understanding the structure of manifolds and their relationships in algebraic topology, providing insights into how spaces can be deformed without tearing or gluing.
Intersection form: The intersection form is a bilinear form defined on the middle-dimensional homology of a manifold that measures how submanifolds intersect within that manifold. It plays a crucial role in cobordism theory by providing a way to understand how different manifolds relate to each other through their boundaries and intersections, ultimately influencing the classification of manifolds up to cobordism.
John Milnor: John Milnor is a prominent American mathematician known for his groundbreaking contributions to differential topology, K-theory, and other areas of mathematics. His work has significantly advanced the understanding of vector bundles, cobordism, and the interplay between algebraic and geometric structures in topology.
Michael Atiyah: Michael Atiyah was a prominent British mathematician known for his significant contributions to topology, geometry, and K-Theory. His work laid the groundwork for several important theories and concepts that link abstract mathematics to physical applications, especially in areas like quantum field theory and differential geometry.
Pontryagin-Thom Construction: The Pontryagin-Thom construction is a topological method used to relate stable homotopy theory to cobordism theory by transforming maps from a manifold into a stable homotopy class. It allows the construction of a correspondence between the cobordism classes of manifolds and stable homotopy classes of maps, effectively bridging the gap between topology and algebraic structures in K-theory.
Self-intersection: Self-intersection refers to the phenomenon where a geometric object, such as a manifold or a curve, intersects itself at certain points. This concept plays an important role in understanding the topology and geometry of manifolds, particularly in cobordism theory, where the behavior of these intersections can help classify manifolds based on their properties.
Smooth cobordism: Smooth cobordism is a concept in differential topology that classifies smooth manifolds up to cobordism, which is a relation between two manifolds indicating that one can be transformed into the other through a smooth process. In this framework, two manifolds are considered equivalent if they are the boundaries of a smooth manifold of higher dimension. This concept plays a crucial role in understanding the properties of manifolds and their relationships in the broader context of cobordism theory.
Stable Cobordism: Stable cobordism is a concept in algebraic topology that refers to a certain equivalence relation on manifolds where two manifolds are considered equivalent if they can be related by a cobordism that allows for the addition of trivial dimensions. This means that stable cobordism focuses on classes of manifolds up to their stable equivalence, which is crucial for understanding how manifolds can be transformed into one another while ignoring certain topological features.
Stiefel-Whitney Classes: Stiefel-Whitney classes are characteristic classes associated with real vector bundles, serving as a topological invariant that provides insight into the geometry of the bundle. These classes help distinguish non-isomorphic vector bundles and are crucial in applications such as vector bundle classification, cobordism theory, and K-Theory. They offer a way to understand how the topology of a manifold interacts with its vector bundles, contributing to deeper insights in algebraic topology and differential geometry.
Thom Space: Thom space refers to a specific type of topological space that arises from the study of vector bundles and cobordism theory. It is constructed by taking a vector bundle over a manifold and considering the space of its sections, which provides a way to relate geometric and algebraic properties of the bundle. Thom spaces play a crucial role in connecting stable homotopy theory with cobordism, helping to classify manifolds based on their cobordism classes.
Thom Spectrum: A Thom spectrum is a specific type of spectrum in stable homotopy theory that arises from the study of smooth manifolds and cobordism. It encodes information about the cobordism classes of smooth manifolds, allowing mathematicians to relate topological properties of manifolds to stable homotopy types through the use of stable cohomology theories. This connection makes Thom spectra vital for understanding various applications in cobordism theory and linking geometry to topology.
Todd genus: The Todd genus is an important topological invariant associated with a smooth manifold that provides information about the manifold's characteristic classes and its cobordism classes. This genus plays a crucial role in linking the concepts of differential geometry and algebraic topology, particularly through its relationship with the index theory and cobordism theory, highlighting how these areas intersect in understanding manifolds' properties.
Topological invariant: A topological invariant is a property of a topological space that remains unchanged under continuous transformations, such as stretching or bending, but not tearing or gluing. These invariants provide critical information about the structure and classification of spaces, allowing mathematicians to distinguish between different types of topological spaces. In the context of cobordism theory, topological invariants are essential for understanding the relationships between manifolds and their boundaries.
Topological quantum field theory: Topological quantum field theory (TQFT) is a type of quantum field theory that focuses on topological properties of manifolds, ignoring metric-dependent aspects like distances and angles. In this framework, physical quantities are associated with the topology of the underlying space, allowing for a rich interplay between physics and geometry, particularly in understanding how different dimensions and structures relate to each other.
Unoriented cobordism ring: The unoriented cobordism ring is a mathematical construct that classifies manifolds up to a relation called cobordism, without considering the orientation of the manifolds. It captures essential topological features and allows for the algebraic manipulation of classes of manifolds, making it an important tool in topology and algebraic geometry, particularly in applications involving characteristic classes and duality theorems.
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