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cobordism theory determining homology theory

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Context

Algebraic topology

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

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