We construct algebraic cobordism spectra MSL and MSp. They are commutative monoids in the category of symmetric T^{2}- spectra. The spectrum MSp comes with a natural symplectic orientation given either by a tautological Thom class th^{MSp} in MSp^{4,2}(MSp_{2}), a tautological Borel class b_{1}^{MSp} in MSp^{4,2}(HP^{\infty}) or any of six other equivalent structures. For a commutative monoid E in the category SH(S) we prove that assignment g -> g(th^{MSp}) identifies the set of homomorphisms of monoids g : MSp -> E in the motivic stable homotopy category SH(S) with the set of tautological Thom elements of symplectic orientations of E. A weaker universality result is obtained for MSL and special linear orientations.
A dozen years ago Voevodsky [15] constructed the algebraic cobordism spectrum MGL in the motivic stable homotopy category SH(S). This gave a new cohomology theory MGL * , * on smooth schemes and on motivic spaces. Later Vezzosi [14] put a commutative monoid structure on MGL. This gave a product to MGL * , * . The commutative monoid structure can even be constructed in the symmetric monoidal model category of symmetric T -spectra, with T = A 1 /(A 1 -0) the Morel-Voevodsky object (Panin, Pimenov and Röndigs [10]).
In this paper we construct the algebraic special linear and symplectic cobordism spectra MSL and MSp. The construction of MSL is straightforward although there is one slightly subtle point. We equip each space BSL n and MSL n with an action of GL n which is compatible with the monoid structure BSL m × BSL n → BSL m+n induced by the direct sum of subbundles. This gives an action of the subgroup Σ n ⊂ GL n of permutation matrices. But to define the unit of the monoid structure we need the action on BSL n to have fixed points. The natural action of SL n has fixed points, but the natural action of GL n does not. So we use an embedding Σ n ⊂ Sp 2n ⊂ SL 2n . This means that our MSL is a commutative monoid in the category of symmetric T ∧2 -spectra. The categories of symmetric
The first author gratefully acknowledge excellent working conditions and support provided by Laboratoire J.-A. Dieudonné, UMR 6621 du CNRS, Universite de Nice -Sophia-Antipolis, and by the RCN Frontier Research Group Project no. 250399 Motivic Hopf equations" at University of Oslo.
T -spectra and of symmetric T ∧2 -spectra are both symmetrical monoidal, and their homotopy categories are equivalent symmetric monoidal categories (Theorem 3.2). So a symmetric T ∧2 -spectrum structure is quite satisfactory, and it seems to be a natural structure for this spectrum.
Cobordism spectra and the cohomology theories they define are expected to have some universal properties among certain classes of cohomology theories. For instance Voevodsky’s and Levine and Morel’s algebraic cobordism theories are universal among oriented cohomology theories [6,10,14]. We should therefore expect MSL to have some degree of universality for special linearly oriented theories. Recall that a special linear bundle (E, λ) over X is a pair consisting of a vector bundle E and an isomorphism of line bundles λ : O X ∼ = det E. A special linear orientation on a cohomology theory A * , * is an assignment to every special linear bundle of a Thom class th(E, λ) ∈ A 2n,n (E, E -X) = A 2n,n X (E) with n = rk E which is functorial, multiplicative, and such that the multiplication maps -∪ th(E, λ) : A * , * (X) → A * +2n, * +n (E, E -X) are isomorphisms. In the motivic context we generally also require that the Thom class of the trivial line bundle over a point be Σ T 1 A ∈ A 2,1 (T ) = A 2,1 (A 1 , A 1 -0). Hermitian K-theory and Balmer’s derived Witt groups are examples of special linearly oriented theories which are not oriented.
The universality properties we show for MSL are as follows. A morphism of commutative monoids ϕ : (MSL, µ SL , e SL ) → (A, µ, e) in SH(S) determines naturally a special linear orientation on A * , * with Thom classes written th ϕ (E, λ). The compatibility of ϕ with the monoid structure ensures the multiplicativity of the Thom classes (Theorem 5.5).
Conversely, a special linear orientation on A * , * with Thom classes th(E, λ) determines a morphism ϕ : MSL → A in SH(S) with th ϕ (E, λ) = th(E, λ) for all (E, λ). This ϕ is unique modulo a certain subgroup lim ← -1 A 2n-1,n (M SL n n ) ⊂ Hom SH(S) (MSL, A). The obstruction ϕ•µ SL -µ A •(ϕ∧ϕ) to having a morphism of monoids lies in a similarly defined subgroup of Hom SH(S) (MSL ∧ MSL, A) (Theorem 5.9).
It would be interesting to know if these obstruction subgroups vanish for Witt groups and hermitian K-theory. The necessary calculations are likely very close to Balmer and Calmès’s computation of Witt groups of Grassmannians [1].
Our MSp is defined similarly with an action of Sp 2n on the spaces BSp 2n and MSp 2n . The actions of the subgroups Σ n ⊂ Sp 2n make MSp a commutative monoid in the category of symmetric T ∧2 -spectra. For MSp we can do much more than for MSL because we have the quaternionic projective bundle theorem [13,Theorem 8.2] for symplectically oriented cohomology theories. Therefore for any symplectically oriented cohomology theory A * , * we have Borel classes for symplectic bundles, and we can compute the cohomology of quaternionic Grassmannians [13, §11] and of the spaces BSp 2r and MSp 2r ( § §8-9). Our main result is the following theorem.
Theorem 1.1. Let (A, µ, e) be a commutative monoid in SH(S). Then the following sets are in canonical bijection:
(a) symplectic Thom structures on the bigraded ǫ-commutative ring cohomology theory (A * , * , ∂, ×, 1 A ) such that for the trivial rank 2 bundle A 2 → pt we have th(A 2 , ω 2 ) = Σ 2 T 1 A in A 4,2 (T ∧2 ), (b) Borel stru
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