Zoran Skoda hom11lec2

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 $X,X'$ be two topological spaces and $\mathcal{U} = \{U_\alpha\}_{\alpha\in A},\mathcal{U}' = \{U'_\gamma\}_{\gamma\in\Gamma}$ be coverings of $X,X'$ respectively and $\mathbf{f}\in C^1(\mathcal{U},G)\mathbf{f}'\in C^1(\mathcal{U}',G)$ 1-cocycles with values in $G$. A morphism of cocycles $\mathbf{r} = (r,\phi_0) : \mathbf{f}\to\mathbf{f}'$ is a a pair of a map $\phi_0:X\to X'$ and a rule $r$ which to each pair $(\alpha,\gamma)\in A\times\Gamma$ assigns a map $r_{\gamma\alpha} : U_\alpha\cap\phi_0^{-1}(U'_\gamma)\to G$ satisfying

$$f'_{\gamma\delta}(\phi_0(x))r_{\delta\beta}(x) = f_{\gamma\alpha}(x)f_{\alpha\beta}(x)$$

for all $x\in U_\alpha\cap U_\beta \cap \phi^{-1}U'_{\gamma}\cap \phi^{-1}U'_\delta$.

(Need also the composition of cocycles)

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

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

We define

$$\phi [x,g_p]_\alpha = [\phi_0(x), r_{\gamma\alpha}(x) g_p]_\gamma$$

Then $\phi([x,1]_\beta) = \phi([x,1]_\alpha f_{\alpha\beta}(x)) \phi([x,f_{\alpha\beta}(x)]_\alpha) = [\phi_0(x), r_{\gamma\alpha}(x) f_{\alpha\beta}(x)]_\gamma$. On the other hand, $\phi([x,1]_\beta)= [\phi_0(x), r_{\delta\beta}(x)]_\delta = [\phi_0(x),f'_{\gamma\delta}(x) r_{\delta\beta}(x)]_\gamma$. Therefore, $f'_{\gamma\delta}(x) r_{\delta\beta}(x) = r_{\gamma\alpha}(x) f_{\alpha\beta}(x)$ for all $x$, i.e. $\mathbf{r}$ is a morphism of cocycles.

Toward classifying functors

Za zadanu kategoriju $C$, i objekt $c$ u $C$, $h_c = Hom(-,c): d\mapsto Hom(d,c)$ je kontravarijantan funktor, tj. funktor $h_c:C^{op}\to Set$. Funktore oblika $h_c$, i njima izomorfne nazivamo reprezentabilni funktori. Ako znamo reprezentabilan funktor $h_c$, onda znamo i $c$, do na izomorfizam. Preslikavanje $h:c\mapsto h_c$ koje šalje objekt u njemu pridruženi reprezentabilni funktor se može proširiti do kovartijantnog funktora $h:C\to\hat{C}$, iz kategorije $C$ u kategoriju predsnopova, tj. u $\hat{C} := Fun(C^{op},Set)$ 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 $C$, tada reprezentirajući objekt $c$ nekog funktora $H$ često nazivamo klasificirajući objekt/prostor, tj. ako je $H\cong h_c$ onda je $c$ klasificirajući objekt za $H$. Za nas je važan slučaj kad je $H=k_G$ funktor koji nekom parakompaktnom Hausdorffovom prostoru $X$ zadaje skup $k_G(X)$ klasa izomorfizama lokalno trivijalnih glavnih $G$-svežnjeva nad $X$. 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 $F$ be a contravariant functor from the homotopy category of connected pointed CW-complexes $connCW_*$ to the category $Set_*$ of pointed sets which sends small colimits to small limits, or equivalently, satisfies

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

(ii) Mayer-Vietoris axiom: Let $A,B\subset X$ be CW-subcomplexes, and $A\cap B$ a CW-subcomplex of $X$ and $A\cup B = X$ (we say that $(X,A,B)$ is a CW-triad). Let $i_A,i_B$ be the natural inclusions. Let $u_A\in F(A)$ and $u_B\in F(B)$ be such that $u_A|_{A\cap B} = u_B|_{A\cap B}$. Then there exist $u\in F(X)$ such that $u|A = u_A$ and $u|B=u_B$.

Then $F$ is representable.

One can apply this to the functor $k_G$ which assigns the principal $G$-bundles to a CW-complex.