11.5 Cobordism theory

Cobordism theory classifies manifolds based on their boundaries, providing a powerful tool for studying topological spaces. It connects different areas of mathematics, from differential topology to algebraic geometry, by defining equivalence relations between manifolds.

Cobordism groups, constructed from these equivalence relations, offer a framework for analyzing manifolds algebraically. These groups have applications in various fields, including physics and low-dimensional topology, and continue to be an active area of research in modern mathematics.

Definition of cobordism

• Cobordism is a fundamental equivalence relation in differential topology that captures the notion of two manifolds being "bordant" or "cobordant"
• Provides a way to classify manifolds up to cobordism, leading to the construction of cobordism groups which are important invariants in algebraic topology

Manifolds with boundary

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• A manifold with boundary is a topological space locally modeled on Euclidean half-space $\mathbb{R}^n_+ = \{(x_1, \ldots, x_n) \in \mathbb{R}^n \mid x_n \geq 0\}$
• The boundary $\partial M$ of a manifold with boundary $M$ is the set of points locally modeled on the hyperplane $\{x_n = 0\}$
  • Example: The closed unit disk $D^2 = \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 \leq 1\}$ is a manifold with boundary $S^1 = \{(x, y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1\}$

Cobordant manifolds

• Two closed $n$-dimensional manifolds $M_1$ and $M_2$ are cobordant if there exists an $(n+1)$-dimensional manifold with boundary $W$ such that $\partial W = M_1 \sqcup M_2$
• The manifold $W$ is called a cobordism between $M_1$ and $M_2$
  • Example: The 2-sphere $S^2$ and the torus $T^2$ are cobordant, with a cobordism given by the "pants" surface (a sphere with two holes)

Cobordism as equivalence relation

• Cobordism defines an equivalence relation on the set of closed $n$-dimensional manifolds
  • Reflexivity: Every manifold is cobordant to itself via the "trivial" cobordism (cylinder)
  • Symmetry: If $M_1$ is cobordant to $M_2$, then $M_2$ is cobordant to $M_1$ by reversing the cobordism
  • Transitivity: If $M_1$ is cobordant to $M_2$ and $M_2$ is cobordant to $M_3$, then $M_1$ is cobordant to $M_3$ by "gluing" the cobordisms along $M_2$

Construction of cobordism groups

• The cobordism equivalence relation leads to the construction of cobordism groups, which are graded abelian groups encoding cobordism classes of manifolds
• Cobordism groups provide a powerful algebraic framework for studying manifolds up to cobordism

Oriented vs unoriented cobordism

• There are two main flavors of cobordism: oriented and unoriented
  • Oriented cobordism considers oriented manifolds and cobordisms respecting the orientations
  • Unoriented cobordism considers manifolds without orientations
• The oriented cobordism groups are denoted $\Omega_n^{SO}$, while the unoriented cobordism groups are denoted $\Omega_n^O$

Graded abelian groups

• The cobordism groups form a graded abelian group, with the grading given by the dimension of the manifolds
• The group operation is induced by the disjoint union of manifolds
  • $[M_1] + [M_2] = [M_1 \sqcup M_2]$, where $[M]$ denotes the cobordism class of $M$
• The identity element is the cobordism class of the empty manifold

Disjoint union as addition

• The disjoint union of manifolds corresponds to addition in the cobordism groups
• If $M_1$ and $M_2$ are cobordant to $M_1'$ and $M_2'$ respectively, then $M_1 \sqcup M_2$ is cobordant to $M_1' \sqcup M_2'$
  • This ensures that the group operation is well-defined on cobordism classes

Cobordism class of empty manifold

• The empty manifold $\emptyset$ represents the identity element in the cobordism groups
• For any manifold $M$, the disjoint union $M \sqcup \emptyset$ is cobordant to $M$
  • The cobordism is given by $M \times [0, 1]$ with $M \times \{0\}$ identified with $M$ and $M \times \{1\}$ identified with $M \sqcup \emptyset$

Computation of cobordism groups

• Computing the cobordism groups is a central problem in algebraic topology
• Various techniques, such as Thom's cobordism theorem and the Pontryagin-Thom construction, have been developed to calculate cobordism groups

Low-dimensional examples

• In low dimensions, the cobordism groups can be computed directly
  • $\Omega_0^{SO} \cong \mathbb{Z}$, generated by the cobordism class of a point
  • $\Omega_1^{SO} \cong 0$, since every closed 1-manifold is cobordant to the empty manifold
  • $\Omega_2^{SO} \cong \mathbb{Z}$, generated by the cobordism class of the 2-sphere $S^2$

Thom's cobordism theorem

• Thom's cobordism theorem relates the oriented cobordism groups to the homotopy groups of certain spaces called Thom spaces
• The theorem states that $\Omega_n^{SO} \cong \pi_n(MO)$, where $MO$ is the Thom space of the universal oriented vector bundle
  • This result allows for the computation of cobordism groups using homotopy-theoretic techniques

Pontryagin-Thom construction

• The Pontryagin-Thom construction is a fundamental tool in the study of cobordism groups
• It establishes a correspondence between cobordism classes of manifolds and homotopy classes of maps into certain classifying spaces
  • For oriented cobordism, the classifying space is the Thom space $MO$
• The construction involves the "collapse map" that sends the complement of a tubular neighborhood of a submanifold to a point

Thom spectra and generalized cohomology

• Thom spectra, constructed from Thom spaces, play a crucial role in the study of cobordism groups
• The oriented cobordism spectrum $MSO$ is obtained by taking the suspension spectrum of the Thom space $MO$
  • The homotopy groups of $MSO$ are precisely the oriented cobordism groups
• Cobordism theories can be viewed as generalized cohomology theories, with the cobordism groups serving as the cohomology groups of a space

Relations to other theories

• Cobordism theory is intimately connected to various other areas of algebraic topology and geometry
• These connections provide deeper insights into the structure and properties of cobordism groups

Cobordism vs homotopy groups of Thom spectra

• Thom's cobordism theorem establishes a direct link between cobordism groups and homotopy groups of Thom spectra
  • $\Omega_n^{SO} \cong \pi_n(MSO)$, where $MSO$ is the oriented cobordism spectrum
• This relationship allows for the application of homotopy-theoretic techniques to the study of cobordism groups

Cobordism invariants and characteristic classes

• Cobordism groups give rise to important invariants and characteristic classes for manifolds
• The Stiefel-Whitney classes, which are cohomology classes associated to real vector bundles, can be viewed as cobordism invariants
  • Two manifolds are unoriented cobordant if and only if they have the same Stiefel-Whitney numbers
• The Pontryagin classes, which are cohomology classes associated to oriented real vector bundles, are also related to cobordism
  • The signature of a 4k-dimensional oriented manifold is a cobordism invariant that can be expressed in terms of Pontryagin numbers

Cobordism and stable homotopy theory

• Cobordism theory is closely tied to stable homotopy theory, which studies the behavior of topological spaces and maps under suspension
• The Thom spectrum $MSO$ is a ring spectrum, with the multiplication given by the Cartesian product of manifolds
  • The Thom spectrum $MO$ for unoriented cobordism is also a ring spectrum
• Stable homotopy groups of spheres can be studied using cobordism theory, via the J-homomorphism which relates stable homotopy groups to cobordism groups

Applications in differential topology

• Cobordism theory has numerous applications in differential topology, particularly in the study of exotic structures on manifolds
• Milnor's exotic 7-spheres, which are homeomorphic but not diffeomorphic to the standard 7-sphere, can be distinguished using cobordism invariants
  • The 28 distinct oriented diffeomorphism classes of exotic 7-spheres correspond to the 28 elements of the oriented cobordism group $\Omega_7^{SO} \cong \mathbb{Z}/28$
• Cobordism theory also plays a role in the classification of manifolds, surgery theory, and the study of diffeomorphism groups

Further developments

• Cobordism theory has been generalized and extended in various directions, leading to new insights and applications
• These developments showcase the richness and versatility of cobordism as a fundamental concept in topology and geometry

Equivariant cobordism theories

• Equivariant cobordism theories consider manifolds equipped with group actions and cobordisms that respect these actions
• The equivariant cobordism groups encode information about the symmetries of manifolds
  • For example, the $G$-equivariant cobordism groups $\Omega_*^G$ study manifolds with an action of a group $G$
• Equivariant cobordism theories have connections to representation theory and equivariant stable homotopy theory

Cobordism categories and TQFTs

• Cobordism categories provide a categorical framework for studying manifolds and cobordisms
  • Objects are closed manifolds and morphisms are cobordisms between them
• Topological quantum field theories (TQFTs) can be viewed as symmetric monoidal functors from a cobordism category to a linear category (e.g., vector spaces)
  • TQFTs assign algebraic data to manifolds and cobordisms, providing a bridge between topology and algebra
• The study of cobordism categories and TQFTs has led to significant developments in low-dimensional topology and mathematical physics

Motivic and algebraic cobordism

• Motivic cobordism is a generalization of cobordism theory to the setting of algebraic geometry
  • It studies cobordism classes of algebraic varieties over a base field
• The motivic cobordism groups, denoted $MGL^{*,*}$, form a bigraded ring that serves as a universal oriented cohomology theory on smooth schemes
  • Motivic cobordism has connections to motivic homotopy theory and algebraic K-theory
• Algebraic cobordism, introduced by Levine and Morel, is a related theory that studies cobordism classes of smooth projective varieties
  • It has applications in enumerative geometry and the study of algebraic cycles

Open problems and current research

• Despite significant progress, many questions in cobordism theory remain open and are actively investigated
• Some current research directions include:
  • Computing cobordism groups in higher dimensions and for various generalized cohomology theories
  • Studying the structure of the cobordism ring and its connections to stable homotopy theory
  • Exploring the relationships between cobordism, surgery theory, and the classification of manifolds
  • Developing equivariant and motivic versions of cobordism theories and their applications in geometry and physics
• As a fundamental and multifaceted theory, cobordism continues to inspire new developments and insights in algebraic topology and related fields