8.1 The Conner-Floyd Chern character

The Conner-Floyd Chern character extends the classical Chern character, linking complex K-theory to complex cobordism. It's a key tool for understanding the relationship between these important topological invariants, offering deeper insights into geometric structures.

This section dives into the definition, properties, and calculations of the Conner-Floyd Chern character. We'll explore its applications in K-theory and its connections to other mathematical concepts, setting the stage for further developments in algebraic topology.

Conner-Floyd Chern Character

Definition and Properties

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• Conner-Floyd Chern character generalizes classical Chern character as a homomorphism from complex K-theory to complex cobordism
• Defined as $ch_{CF}: K(X) \rightarrow MU^*(X) \otimes \mathbb{Q}$
  • K(X) represents K-theory of space X
  • MU*(X) denotes complex cobordism ring of X
• Ring homomorphism preserves addition and multiplication of K-theory classes
• Satisfies naturality property commuting with pullbacks induced by continuous maps between spaces
• Becomes bijective when tensored with rational numbers (rationally an isomorphism)
• Incorporates complex orientation of cobordism theory
• Expressed using formal group laws associated with complex K-theory and complex cobordism

Mathematical Foundations

• Builds upon classical Chern character by incorporating additional geometric information
• Utilizes complex cobordism theory to capture finer topological invariants
• Employs formal group law structure to relate K-theory and complex cobordism
• Connects to Hirzebruch-Riemann-Roch theorem in context of complex cobordism
• Relates to index theorems for elliptic operators on manifolds
• Links to cohomology operations in complex cobordism (Landweber-Novikov operations)

Theoretical Significance

• Generalizes Atiyah-Bott-Shapiro orientation in K-theory to complex cobordism
• Connects to Pontrjagin character in real K-theory through complexification
• Provides framework for studying relationship between K-theory and other generalized cohomology theories (elliptic cohomology)
• Facilitates investigation of formal group laws in complex-oriented cohomology theories
• Enables analysis of K-theory for classifying spaces of compact Lie groups

Computing the Conner-Floyd Chern Character

Calculations for Specific Spaces

• Line bundles over complex projective spaces
  • Express result in terms of complex cobordism ring
  • Utilize formal group law of complex cobordism
• Vector bundles over spheres
  • Apply reduced K-theory and suspension isomorphisms
  • Relate to complex cobordism classes of spheres
• Bott periodicity elements in K-theory
  • Connect to specific elements in complex cobordism
  • Demonstrate periodicity in cobordism context
• K-theory classes from representations of finite groups
  • Express result using cobordism classes
  • Relate to character theory in complex cobordism
• Thom complexes
  • Relate to Thom isomorphism in complex cobordism
  • Demonstrate connection to cobordism fundamental classes

Advanced Computations

• Exterior powers of vector bundles
  • Express results using operations in complex cobordism
  • Relate to exterior power operations in K-theory
• Symmetric powers of vector bundles
  • Connect to symmetric power operations in complex cobordism
  • Demonstrate relationship to Adams operations
• K-theory classes of flag varieties
  • Express in terms of Schubert classes in complex cobordism
  • Relate to Grothendieck ring of representations
• Conner-Floyd Chern character for equivariant K-theory
  • Compute for group actions on spaces
  • Relate to equivariant complex cobordism

Conner-Floyd Chern Character in K-theory

Relationships to Other Invariants

• Classical Chern character
  • Highlight additional geometric information captured by Conner-Floyd version
  • Demonstrate how it refines classical character
• Todd genus and Â-genus
  • Connect through Hirzebruch-Riemann-Roch theorem in complex cobordism context
  • Show how these genera arise from Conner-Floyd Chern character
• Atiyah-Bott-Shapiro orientation
  • Explain generalization to complex cobordism
  • Demonstrate connection to Thom isomorphism
• Pontrjagin character in real K-theory
  • Emphasize role of complexification
  • Relate to real and quaternionic structures
• Index theorems
  • Describe application to elliptic operators on manifolds
  • Show connection to Atiyah-Singer index theorem

Theoretical Connections

• Landweber-Novikov operations
  • Discuss relationship to cohomology operations in complex cobordism
  • Show how these operations arise from Conner-Floyd Chern character
• Formal group laws
  • Explain connection to complex-oriented cohomology theories
  • Demonstrate how Conner-Floyd Chern character encodes formal group law information
• Chromatic phenomena in stable homotopy theory
  • Relate to height filtration in complex cobordism
  • Show how Conner-Floyd Chern character detects chromatic information

Applications of the Conner-Floyd Chern Character

Problem-Solving in K-theory

• Compute K-theory groups of specific spaces
  • Relate to complex cobordism groups
  • Utilize rational isomorphism property
• Prove multiplicative properties of K-theory classes
  • Use ring structure of complex cobordism
  • Demonstrate power of Conner-Floyd Chern character in algebraic manipulations
• Study K-theory behavior under geometric constructions
  • Analyze fiber bundles and bordisms
  • Relate to corresponding constructions in complex cobordism
• Investigate relationship between K-theory and other cohomology theories
  • Connect to elliptic cohomology
  • Explore Brown-Peterson theory connections

Advanced Applications

• Analyze fixed point theorems in K-theory
  • Relate to analogous results in complex cobordism
  • Apply to equivariant situations
• Study formal group laws of complex-oriented cohomology theories
  • Use Conner-Floyd Chern character to analyze structure
  • Relate to Quillen's work on formal group laws
• Investigate K-theory of classifying spaces of compact Lie groups
  • Relate to complex cobordism of these spaces
  • Connect to representation theory and character formulas