Zoran Skoda

Lecture 2, Wed 26. 11. 2011. at 16 00 (exactly!) Math, room 002; lec 1, lec 3. UNDER CONSTRUCTION

This lecture will continue about the cocycle description of locally trivial principal bundles and fiber bundles. We will also spend a bit of time on the special case of covering spaces. We sketch the concept, theorem of existence and construction of classifying spaces. Best references are Husemoller, Fibre bundles, but I have used also Postnikov and Switzer.

All bundles given in this lecture will be locally trivial.

From morphisms of principal bundles to cocycles

Let X,XX,X' be two topological spaces and 𝒰={U α} αA,𝒰={U γ} γΓ\mathcal{U} = \{U_\alpha\}_{\alpha\in A},\mathcal{U}' = \{U'_\gamma\}_{\gamma\in\Gamma} be coverings of X,XX,X' respectively and fC 1(𝒰,G)fC 1(𝒰,G)\mathbf{f}\in C^1(\mathcal{U},G)\mathbf{f}'\in C^1(\mathcal{U}',G) 1-cocycles with values in GG. A morphism of cocycles r=(r,ϕ 0):ff\mathbf{r} = (r,\phi_0) : \mathbf{f}\to\mathbf{f}' is a a pair of a map ϕ 0:XX\phi_0:X\to X' and a rule rr which to each pair (α,γ)A×Γ(\alpha,\gamma)\in A\times\Gamma assigns a map r γα:U αϕ 0 1(U γ)Gr_{\gamma\alpha} : U_\alpha\cap\phi_0^{-1}(U'_\gamma)\to G satisfying

f γδ(ϕ 0(x))r δβ(x)=f γα(x)f αβ(x) f'_{\gamma\delta}(\phi_0(x))r_{\delta\beta}(x) = f_{\gamma\alpha}(x)f_{\alpha\beta}(x)

for all xU αU βϕ 1U γϕ 1U δx\in U_\alpha\cap U_\beta \cap \phi^{-1}U'_{\gamma}\cap \phi^{-1}U'_\delta.

(Need also the composition of cocycles)

Let π:PX\pi:P\to X and π:PX\pi':P\to X' be two (locally trivial) principal GG-bundles corresponding to cocycles f,f\mathbf{f},\mathbf{f}' respectively. Recall that a map ϕ:(P,π,X)(P,π,X)\phi : (P,\pi,X)\to (P,\pi',X') is a pair (ϕ,ϕ 0)(\phi,\phi_0) of maps of topological spaces ϕ:PP\phi : P\to P' and ϕ 0:XX\phi_0:X\to X', such that ϕ\phi is GG-equivariant and ϕ 0π=πϕ\phi_0\circ \pi = \pi'\circ\phi.

PP is viewed as the space of equivalence classes p=[x,g p] α:=[(x p,g p) α]p = [x,g_p]_\alpha := [(x_p,g_p)_\alpha].

We define

ϕ[x,g p] α=[ϕ 0(x),r γα(x)g p] γ \phi [x,g_p]_\alpha = [\phi_0(x), r_{\gamma\alpha}(x) g_p]_\gamma

Then ϕ([x,1] β)=ϕ([x,1] αf αβ(x))ϕ([x,f αβ(x)] α)=[ϕ 0(x),r γα(x)f αβ(x)] γ\phi([x,1]_\beta) = \phi([x,1]_\alpha f_{\alpha\beta}(x)) \phi([x,f_{\alpha\beta}(x)]_\alpha) = [\phi_0(x), r_{\gamma\alpha}(x) f_{\alpha\beta}(x)]_\gamma. On the other hand, ϕ([x,1] β)=[ϕ 0(x),r δβ(x)] δ=[ϕ 0(x),f γδ(x)r δβ(x)] γ\phi([x,1]_\beta)= [\phi_0(x), r_{\delta\beta}(x)]_\delta = [\phi_0(x),f'_{\gamma\delta}(x) r_{\delta\beta}(x)]_\gamma. Therefore, f γδ(x)r δβ(x)=r γα(x)f αβ(x)f'_{\gamma\delta}(x) r_{\delta\beta}(x) = r_{\gamma\alpha}(x) f_{\alpha\beta}(x) for all xx, i.e. r\mathbf{r} is a morphism of cocycles.

Toward classifying functors

Za zadanu kategoriju CC, i objekt cc u CC, h c=Hom(,c):dHom(d,c)h_c = Hom(-,c): d\mapsto Hom(d,c) je kontravarijantan funktor, tj. funktor h c:C opSeth_c:C^{op}\to Set. Funktore oblika h ch_c, i njima izomorfne nazivamo reprezentabilni funktori. Ako znamo reprezentabilan funktor h ch_c, onda znamo i cc, do na izomorfizam. Preslikavanje h:ch ch:c\mapsto h_c koje šalje objekt u njemu pridruženi reprezentabilni funktor se može proširiti do kovartijantnog funktora h:CC^h:C\to\hat{C}, iz kategorije CC u kategoriju predsnopova, tj. u C^:=Fun(C op,Set)\hat{C} := Fun(C^{op},Set) kojeg nazivamo Yonedino ulaganje (Yoneda embedding). Slaba Yonedina lema kaže da je taj funktor vjeran i potpun (drugim riječima, ulaganje kategorija).

Funktori koji su reprezentabilni obično su posebno važni. U slučaju kad se radi o nekim tipičnim klasama primjera, npr. o homotopskoj kategoriji neke kategorije prostora CC, tada reprezentirajući objekt cc nekog funktora HH često nazivamo klasificirajući objekt/prostor, tj. ako je Hh cH\cong h_c onda je cc klasificirajući objekt za HH. Za nas je važan slučaj kad je H=k GH=k_G funktor koji nekom parakompaktnom Hausdorffovom prostoru XX zadaje skup k G(X)k_G(X) klasa izomorfizama lokalno trivijalnih glavnih GG-svežnjeva nad XX. Pokazuje se da taj funktor ima klasificirajući prostor. Slično imamo i za slučaj vektorskih svežnjeva.

– first generalities on representing functors, universal elements etc.

Then existence of classifying spaces for principal bundles and associated bundles.

General existence: from E. Brown’s representability theorem.

More specific construction: Milnor’s construction via infinite join.

Even more specific construction (in a special case) via Grassman and Stiefel manifolds and the idea of Gauss map.

Prerequisites to cover: crash course on CW-complexes.

Mention Dold’s theorem. As a consequence any locally trivial fiber bundle over a paracompact Hausdorff space is a Serre fibration (the direct proof is easier than the Dold’s theorem). Over general spaces one has only a Hurewicz fibration.

Brown’s representability theorem:

Let FF be a contravariant functor from the homotopy category of connected pointed CW-complexes connCW *connCW_* to the category Set *Set_* of pointed sets which sends small colimits to small limits, or equivalently, satisfies

(i) wedge (sum) axiom: let {X α}\{X_\alpha\} be a family of objects and i β:X β αX αi_\beta: X_\beta\to \vee_\alpha X_\alpha, be the canonical inclusions; then F(i β):F( αX α)F(X β)F(i_\beta):F(\vee_\alpha X_\alpha)\to F(X_\beta) induce a map F( αX α γF(X γ)F(\wedge_\alpha X_\alpha\to \prod_\gamma F(X_\gamma). The wedge axiom requires that this map is an isomorphism (here this means bijection).

(ii) Mayer-Vietoris axiom: Let A,BXA,B\subset X be CW-subcomplexes, and ABA\cap B a CW-subcomplex of XX and AB=XA\cup B = X (we say that (X,A,B)(X,A,B) is a CW-triad). Let i A,i Bi_A,i_B be the natural inclusions. Let u AF(A)u_A\in F(A) and u BF(B)u_B\in F(B) be such that u A| AB=u B| ABu_A|_{A\cap B} = u_B|_{A\cap B}. Then there exist uF(X)u\in F(X) such that u|A=u Au|A = u_A and u|B=u Bu|B=u_B.

Then FF is representable.

One can apply this to the functor k Gk_G which assigns the principal GG-bundles to a CW-complex.

Last revised on February 29, 2012 at 22:53:33. See the history of this page for a list of all contributions to it.