Universality of MGL. Scribe notes from a talk by Ben Knudsen. 20 Mar We will discuss work of Panin, Pimenov, Röndigs, and Smirnov.

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1 Universality of MGL Scribe notes from a talk by Ben Knudsen 20 Mar 2014 We will discuss work of anin, imenov, Röndigs, and Smirnov. Orientability One major feature of orientability (for manifolds) is the notion of a tangent bundle. This gives us an Euler class and a Thom class, related by the formula e(m) = i (Th(M)). Orientability also gives us oincaré duality, which gives us interesting wrongway maps, proper pushforwards: f : H (M) = H m (M + ) H m (N + ) = H +n m (N). Similarly, we have a notion of E-orientability. Example. A manifold is KO-orientable iff it is spin. Question. Which theories orient all manifolds? Answer (Thom). Basically just HF 2. There is a universal such cohomology theory MO, and MO α Σ iα HF 2. So instead we just ask about certain classes of manifolds: Example. The universal theory M G orients all manifolds with G-structure. For instance we have MU, the start of chromatic homotopy theory. So now let k be a field, and E a (homotopy) commutative 1 -spectrum. Write = colim n. Definition. An orientation for E is a class c E 2,1 ( ) such that c 1 = Σ 1(1). Remark. This is the same as a natural choice of Chern class for line bundles. It s a theorem of Voevodsky, which we haven t proven, that represents the icard group. Examples. 1

2 Betti cohomology (ordinary cohomolgy is complex orientable). Motivic cohomology. We have H 2,1 mot(x, Z) = CH 1 (X), L div L. Algebraic K-theory, L Σ 1([1] [L]). Algebraic cobordism MGL n = Th(γ n ), where γ n Gr n is the tautological bundle. Then we have Σ 1 1 Σ 1 Th(γ 1 ) MGL. 1 Theorem (rojective bundle theorem). Let X be a smooth variety, V X a rank n vector bundle, and E oriented. Then E, ((V )) = E, (X)[t]/(t n ) as E, (X)-modules, where t = c 1 (O V ( 1)). roof. Using Mayer Vietoris, reduce to n. The definition of orientation is the case 1. Remark. We get Chern classes for all bundles: there exist c i (V ) E 2i,i (X) such that t n c 1 (V )t n ( 1) n c n (V ) = 0. We get Thom classes as follows. Construction. Consider the square V X (V 1) (1) V (V 1) By excision on this square, and applying the purity theorem, E, (Th(V )) = E, ((V 1), (V 1) (1)), Th(V ) c n (O V 1 (1) π (V )). 2

3 Applying the projective bundle theorem, we get a diagram E, (Th(V )) 0 E, ((V 1), (V 1) (1)) E, ((V 1))E, ((V )) 0 c n(o(1) π (V )) 0 E, (X) E, (X) E, (X) n E, (X) n 0 By Mayer Vietoris, it suffices to check the case where V is trivial. Then So we get a Thom isomorphism! c n (O(1) π (V )) = c n (O(1) n ) = t n. We get pushforwards for projective maps. Recall that f : X Y is projective if it factors as: X f Y N with i a closed immersion. We will construct pushforwards for i and p, and then show that choices don t matter. For i : X Y, we define the pushforward by Thom E, (X) E +2k, +k (Th(N X/Y )) p Y purity i E +2k, +k (Y, Y X) E +2k, +k (Y ) Theorem (anin imenov Röndigs). roof. To go one way, we assign {orientations of E} = [MGL, E] ring. φ th MGL φ. We have MGL n = Th(γ n ). Note that hocolim Th(γ n ) = MGL. So we have a short exact sequence 0 lim 1 E +,n 1, +n (Th(γ n )) E, (MGL) lim E +2n, +n (Th(γ n )) 0 3

4 Claim. This lim 1 term vanishes. Assuming this: [MGL, E] = E 0,0 (MGL) = lim E 2n,n (Th(γ n )). So {th(γ n )} defines a map φ : MGL E. Σ n Th(γ 1 n ) Σ n Th(γ 1 n ) Σ 2n Th(γ 1 2n ) MGL MGL E E MGL E So these classes are multiplicative. Theorem. where c i = c i (γ n ). roof of claim. It suffices to show that E, (Gr n ) = E, (k) c 1,..., c n, E +2n, +n (Th(γ n )) E +2n 2, +n 1 is surjective. By the Thom isomorphism, it suffices that Gr n 1 Gr n induces a surjection. roof of theorem. Let Fl n (m) be the flag variety of flags 0 = V 0 V 1... V n, dim V i = i, and let Fl n = colim Fl n (m). We have a map π : Fl n Gr n. There is a filtration 0 = γ 0 n γ 1 n... γ n n = π (γ n ), with associated graded L i n = γ i n/γ i 1 n. Now, E, (Fl n ) = E, (k) t 1,..., t n, t i = c 1 (L i n). We proceed by induction. For n = 1, we have Fl 1 =. Apply the projective bundle theorem. For the inductive step, there s a map Fl n Fl n 1 which is a projective bundle with tautological bundle L n n. Now: π is injective, as Fl n Gr n decomposes as a sequence of projective bundles; π (c i ) = σ i (t 1,..., t n ), by the Whitney sum formula; 4

5 im π = E, (Fl n ) Σn, by replacing Fl n by M n, which parameterizes flags together with a splitting. Then M n Gr n is Σ n -equivariant and factors through Fl n, and this map M n Fl n is an A 1 -homotopy equivalence because it is a sequence of projections of vector bundles. Note that we have E, ( ) = E, (k) t, E, ( ) = E, (k) x, y. Now the classifying map for the tensor product of line bundles induces F E (x, y) t, with F E (x, y) a formal group law. 5