# nLab cobordism theory determining homology theory

Contents

### Context

#### 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, see Conner-Floyd isomorphism) for $E=$KU with its canonical complex orientation $MU \to KU$ (they also showed the case for KO with MSp $\to$KO). Later the Landweber exact functor theorem (Landweber 76) generalized this to all complex oriented cohomology theories MU$\to E$.

The generalization to the actual Atiyah-Bott-Shapiro orientations of topological K-theory, namely MSpinc $\to$ KU and MSpin $\to$ 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

Last revised on February 18, 2021 at 10:00:17. See the history of this page for a list of all contributions to it.