Some remarks on Real and algebraic cobordism.

*(English)*Zbl 1032.55003This paper explores analogies between Real cobordism (previously considered by the authors, as well as Atiyah, Landweber and Araki) and the algebraic cobordism of Voevodsky (over a base field \(k)\). Formal similarities between these two areas lead the authors to new results and conjectures in both.

In Real cobordism, complex conjugation on complex vector bundles is exploited in constructing a \(\mathbb Z/2\)-equivariant spectrum \(M\mathbb R\) (with an associated equivariant cohomology theory). In algebraic cobordism, a similar construction appears making use of the \(\mathbb A^1\)-sphere in place of the compactification of \(\mathbb C\) and this gives rise to an analogue of a Thom spectrum in this context, denoted \(MGL\). Orientations for \(MGL\) in multiplicative cohomology theories on \(\mathbb A^1\)-spaces are studied and shown to correspond to algebraic orientations on associated spectra. Some conjectures are made on transversality and \(MGL\) (similar to ones for \(M\mathbb R\)).

Perhaps the most interesting and deepest part of the paper is related to the Rost motive in studying the Real case; this leads to connections with Real Morava \(K\)-theories. In the other direction, an analogue of the Hopf invariant \(1\) question (at the prime \(2\)) in algebraic cobordism appears; when the base field \(k\) is a subfield of \(\mathbb R\) this reduces to the classical question but for more general fields this is not resolved. The authors remark that the odd prime case is a striking open field.

To sum up, this paper is an intriguing and stimulating attempt at establishing connections between the brave new world of \(\mathbb A^1\)-homotopy theory and more classical homotopy theory.

In Real cobordism, complex conjugation on complex vector bundles is exploited in constructing a \(\mathbb Z/2\)-equivariant spectrum \(M\mathbb R\) (with an associated equivariant cohomology theory). In algebraic cobordism, a similar construction appears making use of the \(\mathbb A^1\)-sphere in place of the compactification of \(\mathbb C\) and this gives rise to an analogue of a Thom spectrum in this context, denoted \(MGL\). Orientations for \(MGL\) in multiplicative cohomology theories on \(\mathbb A^1\)-spaces are studied and shown to correspond to algebraic orientations on associated spectra. Some conjectures are made on transversality and \(MGL\) (similar to ones for \(M\mathbb R\)).

Perhaps the most interesting and deepest part of the paper is related to the Rost motive in studying the Real case; this leads to connections with Real Morava \(K\)-theories. In the other direction, an analogue of the Hopf invariant \(1\) question (at the prime \(2\)) in algebraic cobordism appears; when the base field \(k\) is a subfield of \(\mathbb R\) this reduces to the classical question but for more general fields this is not resolved. The authors remark that the odd prime case is a striking open field.

To sum up, this paper is an intriguing and stimulating attempt at establishing connections between the brave new world of \(\mathbb A^1\)-homotopy theory and more classical homotopy theory.

Reviewer: A.J.Baker (Glasgow)