# nLab cobordism theory determining homology theory

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

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

$X \mapsto M G_\bullet(X) \otimes_{M G_\bullet} E_\bullet \,.$

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 $E$, hence if there is an natural equivalence

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

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 $E=$KU with its canonical complex orientation $MU \to KU$ (they also showed the case for KO with $M Sp \to KO$). Later the Landweber exact functor theorem (Landweber 76) generalized this to all complex oriented cohomology theories $M U \to E$.

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

For elliptic cohomology with the SO orientation of elliptic cohomology $M SO \to Ell$ 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 $MU^\ast(MU)$ and $BP^\ast(BP)$, 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, $HP^2$-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.