Since June 1999 I am mainly interested in oriented cohomology theories in the setting of algebraic varieties. This concept is well known in the algebraic topology and is very powerful. Beginning with Voevodsky's work on the Milnor conjecture and on motives this concept was appeared in not clearly stated form in the algebraic geometry setting.
Now there are several variations on the topic. One of the definition is given in the joint preprint of I.Panin and A.Smirnov " Push-forward in oriented cohomology theories of algebraic varieties" ( http://www.math.uiuc.edu/K-theory/0459).
Another one is very closed to the concept of M.Levine and F.Morel and is presented in the preprint of I.Panin " Riemann-Roch theorem for oriented cohomology" ( http://www.math.uiuc.edu/K-theory/0552).
Among the oriented cohomology theories in the since of the very first mentioned concept are usual singular cohomology (if the groun field is the field of complex numbers), etale cohomology, K-theory, motivic cohomology, algebraic cobordism and many others.
An oriented cohomology theory $A$ is a ring cohomology theory (in the since of the preprint "Push-forwards in oriented cohomology theories of algebraic varieties" ( http://www.math.uiuc.edu/K-theory/0459) equipped with an Euler structure (another term is Chern structure and aposteriory with an integration).
An Euler structure on $A$ is a rule $L mapsto e(L)$ which assigns to every smooth variety X and every line bundle L over X an element e(L) in A(X) which satisfies certain properties (like first Chern class).
An integration is nothing else as a family of operators f_A: A(Y)--> A(X) (one for each projective morphism of smooth varieties) satisfying certain short list of natural properties. In particular they respect the composition and the operator id_A is the identity operator id_A(X).
A main result of the preprint "Push-forwards in oriented cohomology theories of algebraic varieties" states that given an Euler structure $L mapsto e(L)$ on A there exists a unique integration $f mapsto f_A$ normalized such that for every smooth divisor $i: D on X$ the following relation holds $ i_A(1)=e(L(D)) $, where $L(D)$ is the line bundle over X associated with the divisor D.
It turns out that given an integration on A $f mapsto f_A$ there exists a unique Euler structure $L mapsto e(L)$ such that for every smooth divisor $i: D on X$ one has the relation $ i_A(1)=e(L(D))$ and moreover this Euler structure is given by the following rule (here z: X to L is the zero section of a line bundle L) $ e(L)= z^A (z_A(1)) in A(X)$
Finally these TWO correspondences between Euler structures on A and integrations on A are inverse of each other !!!
If varphi: A to B is a ring morphism of oriented cohomology theories then one can define (via an explicit formulae) an inverse Todd genus of itd(E) of a vector bundle E and for a closed imbedding $i: Y to X $ with the normal bundle N one has the relation in B(X) i_B[varphi(a).itd(N)]=varphi(i_A(a)) which holds for every element a in A(Y).
This is the RIEMANN-ROCH THEOREM FOR CLOSED IMBEDDING If by a case the inverse Todd genus is invertible then define td(E)=itd(E)^{-1} With this definition for every projective morphism f: Y --> X of smooth varieties and for tangent bundles T_Y and T_X of the varieties and every element a in A(Y) one has the relation f_B[varphi(a).td(T_Y)]=varphi(f_A(a)).td(T_X)
This is the THE RIEMANN-ROCH THEOREM FOR PROJECTIVE MORPHISM. Particular cases of this result proved in the preprint "Riemann-Roch theorem for oriented cohomologies" are the classical Riemann-Roch theorem of Hirzebruch in the form of Grothendieck, the Adams Riemann-Roch theorem and many, many others. A lot of examples as classical so new ones one can find in the preprint "Riemann-Roch theorem for oriented cohomologies".
This is a list of main references
on 03.02.2002
