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# Oriented theories and symplectic cobordism-Ivan Panin

## Introductory Lecture

### Riemann-Roch theorem for oriented cohomology theories

Thursday, June 12 (Oberseminar) 16:45-17:45

**Abstract**. This talk is about a joint work with A.Smirnov. A oriented ring cohomology theory on smooth algebraic varieties over a field F is a functor Sm/F → Rings (commutative) equipped with Gysin homomorphisms subjecting some expected properties. A ring morphism φ: A → B between two oriented cohomology theories is just a ring morphism between two ring theories. It does not respect (in general) the Gysin maps. In this setting we are able to formulate a general Riemann-Roch type theorem, and to prove it. MOST of the classical examples of Riemann-Roch type are partial cases of our one There are many new interesting cases which we will present as examples.

## Lecture series: Three Conner-Floyd type theorems in the motivic context

### Lecture 1: Survey of main results

Monday, June 16, 16-18 Uhr, N-U-4.04

**Abstract.** The first theorem relates Voevodsky's algebraic cobordism to higher Quillen's K-groups, the second theorem relates symplectic algebraic cobordism `MSp` to higher hermitian K-groups, the third theorem relates special linear cobordism `MSL` to derived Balmer-Witt groups. The first theorem is due to O.Roendigs, K.Pimenov and the speaker (2009), the second one is due to Ch.Walter and the speaker (2010), the third one is due to A.Ananyevskiy (2012).

These results require motivic versions of the following notions: generalised Chern classes of algebraic vector bundles, Borel classes of symplectic bundles, Pontriagin classes of algebraic vector bundles, Euler classes of `SL`-oriented vector bundles.

In the series of lectures we will introduce step by step all these notions, state key technical results (and prove some of them), state the three theorems mentioned above and give sketch of their proofs.

### Lecture 2: Introduction to oriented cohomology theories (on algebraic varieties).

Tuesday, June 17, 14-16 Uhr, N-U-4.04

**Abstract.** An oriented cohomology theory on algebraic varieties is a ring cohomology theory equipped with a family of Thom isomorphisms. Such theories are algebraic versions of complex oriented cohomology theories: singular cohomology, complex K-theory, complex cobordism and so on. The Thom isomorphisms should satisfy certain natural properties. There are in total 4 different ways to orient a ring cohomology theory and all of them are equivalent to each the other. If there is one orientation then there are plenty essentially different orientations on the same cohomology theory. Once an orientation is choosen and fixed we say that the theory is oriented.

In this lecture we will describe all those 4 ways to orient a ring cohomology theory. Then we will construct Gysin homomorphisms provided we are given just one element `th ∈ A(Th(O(-1))`, whose restrictiction to `A(T)` is a free genetator of the `A(pt)`-module `A(T)`, where `T` is the Voevodsky sphere **A**^{1}/(**A**^{1}- 0). Finally we give a construction of Gysin homomorphisms provided we are given just one element `c ∈ A( P^{∞})` whose restriction to

`A(`vanishes and whose restriction to

**P**^{0})`A(`together with the element

**P**^{1})`1 ∈ A(`form a free

**P**^{1})`A(pt)`-basis of

`A(`.

**P**^{1})### Lecture 3: On a relation of algebraic cobordism `MGL` to higher Quillen K-theory

Wednesday, June 18, 14-16 Uhr, N-U-3.01

**Abstract.** An oriented cohomology theory on algebraic varieties is a ring cohomology theory equipped with a family of Thom isomorphisms for vector bundles. Examples of such theories are motivic cohomology, higher K-theory, Voevodsky cobordism theory `MGL`. Each vector bundle has its Chern classes with values in a given oriented cohomology theory. In particularly, such classes exists with values in the K-theory, represented by the `T`-spectrum `KGL`. These classes help to compute `KGL(BGL _{n})` as the formal power series on Chern classes over the ring

`KGL(pt)`. Thom classes in turn help to identify

`KGL(Th`with

_{n})`KGL(BGL`, where

_{n})`Th`is the Thom space of the rank

_{n}`n`tautological vector bundle over the space

`BGL`. In the same fashion one can compute

_{n}`MGL(BGL`and

_{n})`MGL(Th`. These leads eventually to a canonical ring isomorphism

_{n})`MGL`for arbitrary small space

^{x,x}(Y)⊗_{MGL2x,x(pt)}K_{0}(pt) = K^{x}(Y)`Y`. In particulaly, for all smooth varieties

`Y`and for all spaces of the form

`Y/U`and

`Y/Z`. This result is not tautology even for the case

`Y=pt`.

### Lecture 4. Symplectically oriented cohomology theories on algebraic varieties

Friday, June 20, 14-16 Uhr, N-U-3.04 (note change of schedule)

** Abstract.** A symplectically oriented cohomology theory on algebraic varieties is a ring cohomology theory equipped with a family of Thom isomorphisms for symplectic vector bundles. Examples of such theories are higher hermitian K-theory, derived Balmer-Witt theory, symplectic cobordism theory `MSp`. Each symplectic vector bundle has its Borel classes with values in a given symplectically oriented cohomology theory. In particularly, such classes exists with values in the hermitian K-theory, represented by the `T`-spectrum `BO`. These classes help to express, say, `BO(BSp _{2n})` as the formal power series on Borel classes over the ring

`BO(pt)`. Symplectic Thom classes in turn help to identify

`BO(Th`with

_{2n})`BO(BSp`, where

_{2n})`Th`is the Thom space of the rank

_{2n}`2n`tautological symplectic vector bundle over the cpace

`BSp`. In the same fashion one can compute

_{2n}`MSp(BSp`and

_{2n})`MSp(Th`. These leads eventually to a canonical ring isomorphism

_{2n})`MSp(X)⊗`for arbitrary small space

_{MSp(pt)}BO(pt) = BO(X)`X`. In particulaly, for all smooth varieties

`X`and for all spaces of the form

`X/U`and

`X/Z`.

### Lecture 5: Special linear oriented cohomology theories

Friday, June 20 16-18 Uhr, N-U-3.04 (note change of schedule). This lecture will be given by Alexey Ananyevskiy

**Abstract.** A special linear oriented cohomology theory on algebraic varieties is a ring cohomology theory equipped with a family of Thom isomorphisms for special linear vector bundles. Examples of such theories are derived Balmer-Witt theory and special linear cobordism theory `MSL`. Each vector bundle has its Pontriagin classes with values in a given specially oriented cohomology theory. In particularly, such classes exists with values in derived Balmer-Witt-theory `W[t, t ^{-1}]`, represented by the

`T`-spectrum

`BO[η`, where η is the stable Hopf map. For ODD integer

^{-1}]`m`these classes help to express, say,

`W[t, t`as the formal power series on Borel classes over the ring

^{-1}](BSL_{m})`W[t, t`. Special linear Thom classes in turn help to identify

^{-1}](pt)`W[t, t`with

^{-1}](Th_{m})`W[t, t`, where

^{-1}](BSL_{m})`Th`is the Thom space of the rank

_{m}`m`tautological special linear vector bundle over the cpace

`BSL`. In the same fashion one can compute

_{m}`MSL[η`and

^{-1}](BSL_{m})`MSL[η`. These leads eventually to a canonical ring isomorphism

^{-1}](Th_{m})`MSL`for arbitrary small space

_{η}(X)⊗_{MSL(pt)}W(pt) = W(X)`X`. In particulaly, this equality hold for all smooth varieties

`X`and for all spaces of the form

`X/U`and

`X/Z`. This result is not a consequence of the main result of lecture 4, since

`MSL`probably is not equal to

_{η}`MSp`.

_{η}