Graduate algebraic K-theory seminar

Transcription

1 Seminar notes Graduate algebraic K-theory seminar notes taken by JL University of Illinois at Chicago February 1, 2017 Contents 1 Model categories Definitions Properties Introduction to stable homotopy theory Motivation Definitions and examples References 5

2 1 Model categories Özgür Bayindir 1.1 Definitions Let C be a category. In a model category, we have distinguished classes of morphisms: 1. weak equivalences 2. cofibrations 3. fibrations If f is a weak equivalence and a cofibration, we call it a trivial or acyclic fibration. The following axioms are satisfied: MC1. C has finite limits and colimits MC2. weak equivalences satisfy the 2-out-of-3 proposition (regarding f, g, g f) MC3. weak equivalences, cofibrations, and fibrations are closed under retracts. That is, if Z f g f Y T Y commutes, then f is a retract of g. MC4. The diagram i A B f g Y p lifts if either i is a trivial cofibration and p is a fibration, or i is a cofibration and p is a trivial fibration. For the diagram to lift, the diagonal map must exist so that the two triangles commute. MC5. Every morphism can be factored as a cofibration followed by a trivial fibration ( T Y ), or a trivial cofibration followed by a fibration ( T Y ). Definition 1 is cofibrant if is a cofibration. is fibrant if is a fibration. Note that any object has a cofibrant replacement or a fibrant replacement. These are objects so that the respective diagrams below commute. Example 1 Top is a model category. - weak equivalences are homotopy equivalences - cofibrations are maps f : Y where Y is obtained from by attaching cells - fibrations are Serre fibrations (maps that have the lifting property with respect to inclusion of cylinders) Example 2 Ch R with a negative grading is a model category. A map f : M N is a - weak equivalence if f is an isomorphism (that is, f is a quasi-isomorphism) - cofibration if f is a monomorphism and coker(f ) is projective for k 0 - fibration if f is an epimorphism for k 1 2

3 The above is a projective model structure for Ch R. It is also possible to define an injective model structure analogously. Proposition 1 Cofibrations are the maps which have the left lifting property with respect to acyclic fibrations. That is, A i B has the left lifting property with respect to p Y (or p Y has the right lifting property with respect to A i B) means if i A B with the sol lines commutes, the diagonal map exists. Note that fibrations are the maps which have the right lifting property with respect to trivial cofibrations. Definition 2 A cylinder object for A is an object A I with a diagram A A A I A factoring into A + A. It is a good cylinder object if the first map is a cofibration ( ), and a very good cylinder object if it is good and the second map is a fibration ( ). Definition 3 Two maps f, g : A are left-homotopic if the dotted map in the diagram Y p A A A I f + g exists, making it commute. 1.2 Properties Proposition 2 Weak equivalences between bifibrant (fibrant and cofibrant) objects are homotopy equivalences. Another way of saying the above is Hom(R 1 Q, R 1 QY )/ = Hom Ho(C) (, Y ), where Q is the cofibrant functor. The homotopy category Ho(C) has the same objects as C, and the morphisms are defined by a functor (f : Y ) (f : R 1 Q R 1 QY ). Definition 4 A pair of adjoint functors F : C D and G : D C is called a Quillen adjunction if F preserves cofibrations, and G preserves fibrations. They induce another pair of adjoint functors LF : Ho(C) Ho(D) and RG : Ho(D) Ho(C). For A cofibrant in C and B fibrant in D, if f : A G(B) is a qeak equivalence iff its adjoint f : F (A) B is a weak equivalence, then LF and RG are equivalences of categories. References: [DS95] 2 Introduction to stable homotopy theory Maximilien Péroux 2.1 Motivation Stable homotopy theory is the study of spectra as linear approximations of spaces. Stable means we are consering the spaces after inverting the suspension functor. 3

4 Theorem 1 (Freudenthal suspension theorem) Let, Y be finite dimensional pointed CW -complexes. Then the sequence [, Y ] Σ [Σ, ΣY ] Σ Σ [Σ n, Σ n Y ] Σ eventually stabilizes. If is an (n 1)-connected space, then π k () Σ π k (Σ) Σ stabilizes as well. When the latter sequence stabilizes, the groups are called the stable homotopy groups of, and we write π s k () = π k+r(σ r ), where r is big enough. Note that π s is much easier to compute than π (we will see why later). Theorem 2 (Brown representability theorem) Given a reduced cohomology theory h on CW (the category of pointed CW -complexes), there exists a sequence of pointed spaces E 0, E 1,..., such that h n () = [, E n ] There exists a unique map (up to homotopy) σ n : E n ΩEn+1, and by adjointness, we get another map σ n : ΣE n E n+1. Together with the suspension axiom, we get the commutative diagram below. h n () Σ hn+1 (Σ) = = [, E n ] [ΣS, E n+1 ] The dotted map exists and is unique by the Yoneda lemma. 2.2 Definitions and examples = [, ΩE n+1 ] Definition 5 A spectrum is E = (E n, σ n ) n 0, where (E n ) n 0 is sequence of pointed spaces, and σ n : ΣE n E n+1 are pointed maps. A map of spectra f : E F is a collection of level-wise pointed maps f n : E n F n such that ΣE n E n Σf n f n+1 ΣF n F n commutes. This creates a category Sp of spectra. To construct Ho(Sp), different definitions of Sp may be used. Example 3 Given any pointed space, define the spectrum Σ by (Σ ) n = Σ n and σ n : Σ(Σ ) n (Σ ) n+1. This gives a functor Σ : Top Sp. When = S 0, we get Σ S 0 = S, the sphere spectrum. Example 4 Another example is Eilenberg Maclane spectra. Let G be an abelian group. Set (MG) n = K(G, n). Structure maps are induced by K(G, n) ΩK(G, n + 1). Then π n () = π n+1 (Ω). Definition 6 E is an Ω-spectrum if σ n : E n ΩEn+1 is weak homotopy equivalence. Example 5 Some more examples of spectra include: - Cobordism spectra: (MO) n = Thom space of universal real k-vector bundles - Topological K-theory: KO, KU,..., where (KU) 0,2,... = Z BU and (KU) 1,3,... = U - Algebraic K-theory: For R a ring, (KR) n = K 0 (Σ n R) BGL(Σ n R) 4

5 Definition 7 Let E be a specturm. The stable homotopy groups of E are defined as π n (E) = colim k π n+k (E k ) References [DS95] W. G. Dwyer and J. Spaliński. Homotopy theories and model categories. In: Handbook of algebraic topology. North-Holland, Amsterdam, 1995, pp