nLab cobordism theory determining homology theory

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 1992).

Examples

Originally this was shown to be the case by Conner & Floyd 1966, 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 MSpin^c $\to$ 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

• Pierre Conner, Edwin Floyd, The Relation of Cobordism to K-Theories, Lecture Notes in Mathematics 28 Springer 1966 (doi:10.1007/BFb0071091, MR216511)

• 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)