nLab
cobordism theory determining homology theory

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Stable Homotopy theory

Higher algebra

Contents

Idea

Given a multiplicative cohomology theory, hence an E-∞ ring EE, equipped with a multiplicative universal orientation for manifolds with G-structure, hence a homomorphism of E E_\infty-rings MGEM G \longrightarrow E from the GG-Thom spectrum MGM G, then EE canonically becomes an MGM G-∞-module and so one may consider the functor

XMG (X) MG E . X \mapsto M G_\bullet(X) \otimes_{M G_\bullet} E_\bullet \,.

If this is first of all a generalized homology theory itself, hence represented by a spectrum, then one may ask if this spectrum coincides with the original EE, hence if there is an natural equivalence

MG () MG E E (). M G_\bullet(-) \otimes_{M G_\bullet} E_\bullet \simeq E_\bullet(-) \,.

If so one often says that the cobordism theory determines the homology theory (e.g. Hopkins-Hovey 92).

Examples

Originally this was shown to be the case by (Conner-Floyd 66) for E=E= KU with its canonical complex orientation MUKUMU \to KU (they also showed the case for KO with MSpKOM Sp \to KO). Later the Landweber exact functor theorem (Landweber 76) generalized this to all complex oriented cohomology theories MUEM U \to E.

The generalization to the actual Atiyah-Bott-Shapiro orientations of topological K-theory, namely MSpin cKUM Spin^c \to KU and MSpinKOM Spin \to KO is due to (Hopkins-Hovey 92).

For elliptic cohomology with the SO orientation of elliptic cohomology MSOEllM SO \to Ell the statement is due to (Landweber-Ravenel-Stong 93). For the refinement to the spin orientation of elliptic cohomology of (Kreck-Stolz 93) a statement is due to (Hovey 95).

Refereces

  • P. Conner, E. Floyd, The relation of cobordism to K-theories, Lecture Notes in Mathematics 28, 1966 (pdf)

  • Peter Landweber, Homological properties of comodules over MU *(MU)MU^\ast(MU) and BP *(BP)BP^\ast(BP), American Journal of Mathematics (1976): 591-610.

  • Michael Hopkins, Mark Hovey, Spin cobordism determines real K-theory, Mathematische Zeitschrift 210.1 (1992): 181-196. (pdf)

  • Peter Landweber, Douglas Ravenel, Robert Stong, Periodic cohomology theories defined by elliptic curves, in Haynes Miller et. al. (eds.), The Cech centennial: A conference on homotopy theory, June 1993, AMS (1995) (pdf)

  • Matthias Kreck, Stefan Stolz, HP 2HP^2-bundles and elliptic homology, Acta Math, 171 (1993) 231-261 (pdf)

  • Mark Hovey, Spin Bordism and Elliptic Homology, Mathematische Zeitschrift 219, 163-170 1995 (web)

Last revised on December 14, 2016 at 07:55:23. See the history of this page for a list of all contributions to it.