10.4 K-Theory in bordism and cobordism theory

K-Theory in bordism and cobordism theory connects algebraic structures to geometric objects. It explores how manifolds can be equivalent if their union forms a boundary, and how this relates to complex structures and stable normal bundles.

The complex cobordism ring MU_* is isomorphic to the coefficient ring of complex K-Theory. This connection allows for powerful computational techniques, like spectral sequences, to study bordism groups and their relationship to K-Theory and stable homotopy theory.

Bordism and Cobordism Groups

Definition and Equivalence Relations

• Bordism and cobordism are equivalence relations on the set of compact manifolds
  • Two manifolds are equivalent if their disjoint union forms the boundary of a compact manifold with boundary
• The bordism group Ω_n(X) consists of bordism classes of n-dimensional compact manifolds with a map to a space X
  • The group operation is given by the disjoint union of manifolds
• The cobordism group MU_n(X) is defined similarly, but with stably complex manifolds
  • Stably complex manifolds have a complex structure on their stable normal bundle (e.g., complex projective spaces)

Relation to K-Theory

• K-Theory, particularly complex K-Theory, is closely related to cobordism theory
• The complex cobordism ring MU_* is isomorphic to the coefficient ring of complex K-Theory
  • This isomorphism allows for the application of K-theoretic techniques to study cobordism groups
• The Atiyah-Hirzebruch spectral sequence relates the complex K-Theory of a space X to its ordinary cohomology
  • Provides a method to compute K_(X) and, consequently, the complex cobordism ring MU_(X)

K-Theory in Bordism and Cobordism Rings

Generators and Polynomial Structure

• The complex cobordism ring MU_* is a polynomial ring over Z generated by classes of complex projective spaces CP^n
  • CP^n are also generators of the complex K-Theory ring K_*(pt)
• The polynomial structure of MU_* allows for the computation of cobordism rings using algebraic techniques
  • For example, the Landweber exact functor theorem relates the cobordism ring to formal group laws

Computation via Spectral Sequences

• The Atiyah-Hirzebruch spectral sequence relates the complex K-Theory of a space X to its ordinary cohomology
  • Provides a method to compute K_(X) and, consequently, the complex cobordism ring MU_(X)
• The Adams spectral sequence, which is based on the extraordinary cohomology theory K(n), can be used to compute the stable homotopy groups of spheres
  • Consequently, it can also be used to compute bordism and cobordism rings

Adams Spectral Sequence for Bordism

Overview and E_2 Term

• The Adams spectral sequence is a tool for computing stable homotopy groups of a space X using a generalized cohomology theory E (e.g., complex K-Theory or Morava K-Theory K(n))
• The E_2 term of the Adams spectral sequence is given by the Ext groups Ext_A^{s,t}(H_*(X;Z/p),Z/p)
  • A is the Steenrod algebra
  • H_*(X;Z/p) is the mod p cohomology of X

Differentials and Convergence

• The differentials in the Adams spectral sequence are related to the structure of the Steenrod algebra and the cohomology of the Eilenberg-MacLane spaces K(Z/p,n)
• The Adams spectral sequence converges to the p-completed stable homotopy groups of X
  • These groups can be used to determine the bordism and cobordism groups of X
• The convergence of the Adams spectral sequence is a powerful tool for computing bordism and cobordism groups
  • It allows for the application of algebraic techniques to study these geometric objects

K-Theory vs Bordism vs Stable Homotopy

Stable Homotopy Theory

• Stable homotopy theory studies the stable homotopy category
  • Obtained by inverting the suspension functor on the category of pointed topological spaces
• The stable homotopy groups of spheres, denoted by π_*^S, form a graded ring
  • Closely related to the bordism and cobordism rings
• Techniques from stable homotopy theory can be applied to study cobordism and K-Theory

Pontryagin-Thom Construction

• The Pontryagin-Thom construction establishes an isomorphism between the bordism group Ω_n(X) and the n-th stable homotopy group of the Thom space of the universal vector bundle over X
  • Provides a connection between bordism and stable homotopy theory
• The Thom space construction is a key tool in the study of cobordism and stable homotopy theory
  • It allows for the application of homotopy-theoretic techniques to study bordism groups

Complex Cobordism Spectrum

• The complex cobordism spectrum MU is a ring spectrum in stable homotopy theory
  • Its homotopy groups are isomorphic to the complex cobordism ring MU_*
• The ring spectrum structure of MU allows for the application of techniques from stable homotopy theory to study cobordism and K-Theory
  • For example, the Adams-Novikov spectral sequence, which is based on MU, can be used to compute stable homotopy groups and cobordism rings