nLab The Relation of Cobordism to K-Theories

This page compiles material related to the book

• Pierre Conner, Edwin Floyd,

  The Relation of Cobordism to K-Theories,

  Lecture Notes in Mathematics 28 Springer 1966

  doi:10.1007/BFb0071091,

  MR216511

on cobordism theory and topological K-theory, meeting in the notion of the e-invariant.

The Thom Isomorphism in K-theory

• Thom isomorphism

• topological K-theory

1. Exterior algebra

• exterior algebra

2. Tensor products of exterior algebras

3. Application to bundles

4. Thom classes of line bundles

• projective spaces, in particular quaternionic projective space

• tautological line bundle

• Thom class

• Whitehead-generalized cohomology theory

• multiplicative cohomology theory

5. Cobordism and homomorphism into K-theory

• cobordism cohomology theory

• MFr$\to$MSp$\to$MSU$\to$MU

• Conner-Floyd orientation

6. The homomorphism $\mu_c$

• Thom isomorphism

• rational Todd class is Chern character of Thom class

Cobordism Characteristic Classes

7. A theorem of Dold

• Leray-Hirsch theorem

• quaternionic oriented cohomology, complex oriented cohomology

• Conner-Floyd Chern classes

8. Characteristic classes on cobordism

• MSp

• Conner-Floyd Chern class

9. Characteristic classes in K-theory

• Chern classes

• Chern character

• Pontrjagin character

10. A cobordism interpretation for $K^\ast(X)$

• Conner-Floyd isomorphism

  (Landweber exact functor theorem for KU and KSp, see also cobordism theory determining homology theory)

11. Mappings into spheres

$U$-Manifolds with Framed Boundaries

12. The $U$-bordism groups $\Omega^U_\bullet$

• MU

13. Characteristic numbers from K-theory

• Todd class

• Todd genus

14. The theorem of Stong and Hattori

15. $U$-Manifolds with stably framed boundaries

16. The bordism groups $\Omega^{U,fr}_\bullet$

• manifold with boundary

• MUFr, (U,fr)-manifolds

• rational Todd class is Chern character of Thom class

• e-invariant

• e-invariant is Todd class of cobounding (U,fr)-manifold

17. The groups $\Omega^{\mathrm{U}, SU}_\bullet$

18. The image of $\Omega^{fr}_{\bullet}$ in $\Omega^{SU}_\bullet$

category: reference