Summary: The motivic Thom spectrum MG and the algebraic cobordism
This is a report on a "work in progress" of F. Morel and M. J. Hopkins.
Their work is a step toward the identification of the motivically defined theory
(-) with the geometrically defined one
(-). Namely, they prove:
Theorem 1. For any smooth k-variety X the natural graded homomorphism
(X) is surjective.
The plan of the lecture was:
(1) Some basic properties of MG .
(2) The computation of MG 2,
(3) Proof of the main theorem.
From now one, the base field k is fixed and all our varieties will be k-varieties.
For simplicity we shall assume k to be of characteristic zero.
1. Some basic properties of MG