# Tuesday, September 07 :::{.remark} Recall that given a $B\in \Top$ and $\mcu \covers B$, we defined $\Hc_1(\mcu; G)$ which classified isomorphism classes of fiber bundles $E \mapsvia{\pi} B$ with fiber $F$, $G \subseteq \Homeo(F)$, and structure group $G$, given by clutching data using $\mcu$. The cochains were given by the following: \[ \Cc^0(\mcu; G) &= \ts{\ts{ \gamma_i: U_i \to G}_{i\in I}}\\ \Cc^1(\mcu; G) &= \ts{\ts{ \phi_{ij}: U_i \intersect U_j \to G }_{i, j \in I}}\\ \Cc^2(\mcu; G) &= \ts{\ts{ \eta_{ijk}: U_i \intersect U_j \intersect U_k \to G }_{i,j,k\in I}} \] with boundary maps $\delta_i: \Cc^{i-1} \to \Cc^i$: \[ (\delta_1 \gamma)_{ij} &= \gamma_i \gamma_j\inv \\ (\delta_2 \phi)_{ijk} &= \phi_{ij} \phi_{jk} \phi_{ik}\inv .\] Note that - \( \delta_2 \circ \delta_1 = 0\) - \( \ker \delta_2 = Z^1(\mcu; G) \) yields clutching data, i.e. a fiber bundle with fiber $F$, - \( \im \delta_1 \) yields trivial bundles, - \( \Hc^1(\mcu; G) \da Z^1(\mcu; G) / \im(\Cc^0(\mcu; G) \to Z^1(\mcu; G)) \). We'll see that \( (\gamma \varphi)_{ij} = \gamma_i \varphi_{ij} \gamma_j\inv \), and by a lemma this will prove the above claim about classifying isomorphism classes. ::: :::{.definition title="Refinement of covers"} We say a cover $\mcv \da \ts{V_j}_{j\in J}$ is a **refinement** of $\mcu \da \ts{U_i}_{i\in I}$ iff there exists a function $f:J\to I$ between the index sets where $V_j \subseteq U_{f(j)}$ for all $j$. > DZG: I'll write $\mcv \leq \mcu$ if $\mcv$ is a refinement of $\mcu$. ::: :::{.remark} Since any two covers have a common refinement, we'll assume $\mcv \leq \mcu$ is always a refinement. We can then restrict clutching data from $\mcu$ to $\mcv$: given $\ts{\phi_{ij}}_{i,j\in I}$, we can set $\psi_{ij} \da \ro{ \phi_{f(i), f(j)}} {V_i \intersect V_j}$, noting that if $V_j \subseteq U_{f(j)}$ and $V_i \subseteq U_{f(i)}$ then $V_i \intersect V_j \subseteq U_{f(i)} \intersect U_{f(j)}$. These yield maps $\psi_{ij}: V_i \intersect V_j \to G$ satisfying the cocycle condition, so $\psi_{ij} \in Z^1(\mcv; G)$. This means that we have map $Z^1(\mcu; G)\to Z^1(\mcv; G)$ which respects the actions of $\Cc^0(\mcu; G), \Cc^0(\mcv; G)$ respectively. Since the category of covers with morphisms given by refinements come from a preorder, we can take a colimit to define \[ \Hc^1(B; G) \da \colim_{\mcu \covers B} \Hc^1(\mcu; G) .\] ::: :::{.lemma title="?"} There is a bijection \[ \correspond{ \text{Fiber bundles with fiber }F \\ \text{and structure group }G }\slice{\sim} &\mapstofrom \Hc^1(B; G) \] In particular, these classes are independent of $F$. ::: :::{.corollary title="?"} There is an equivalence of categories \[ \correspond{ \text{Fiber bundles with fiber $F$} \\ \text{and structure group }G \\ \text{over }B }\slice{\sim} &\mapstofrom \Prin\Bun(G) \slice{B} ,\] where the right-hand side are principal $G\dash$bundles. ::: :::{.definition title="$G\dash$structures"} Given a map $G \to \Homeo(F)$, a **$G\dash$structure** on an $F\dash$bundle $E \mapsvia{\pi} B$ is the following data: given clutching data $\phi_{ij}$, lifts of the following form that again satisfy the cocycle condition: \begin{tikzcd} && G \\ \\ {U_i\intersect U_j} && {\Homeo(F)} \arrow["{\phi_{ij}}"', from=3-1, to=3-3] \arrow[from=1-3, to=3-3] \arrow["{\tilde \phi_{ij}}", dashed, from=3-1, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJVX2lcXGludGVyc2VjdCBVX2oiXSxbMiwwLCJHIl0sWzIsMiwiXFxIb21lbyhGKSJdLFswLDIsIlxccGhpX3tpan0iLDJdLFsxLDJdLFswLDEsIlxcdGlsZGUgXFxwaGlfe2lqfSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==) ::: :::{.remark} Note that we need to impose the cocycle condition, since lifts may not be unique and some choices may not glue correctly! ::: :::{.example title="$\Spin_n\dash$structures"} Using the known $\Spin$ double covers, we can form the composition \[ \Spin_n(\RR) \mapsvia{\times 2} \SO_n(\RR) \injects \Homeo(\RR^n) .\] Then a $\Spin_n\dash$structure on any $\RR^n\dash$bundle is a lift of transition functions from $\Homeo(\RR^n)$ to $\Spin_n$ satisfying the cocycle condition. ::: :::{.definition title="Fiber products"} We can fill in a commutative square in the following way: \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{X\fiberprod{B}Z} && Z \\ \\ X && B \arrow["\pi", from=1-3, to=3-3] \arrow["f"', from=3-1, to=3-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=3-1] \arrow[color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, color={rgb,255:red,92;green,92;blue,214}, draw=none, from=1-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJYIl0sWzIsMiwiQiJdLFsyLDAsIloiXSxbMCwwLCJYXFxmaWJlcnByb2R7Qn1aIixbMjQwLDYwLDYwLDFdXSxbMiwxLCJcXHBpIl0sWzAsMSwiZiIsMl0sWzMsMCwiIiwyLHsiY29sb3VyIjpbMjQwLDYwLDYwXSwic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzMsMiwiIiwwLHsiY29sb3VyIjpbMjQwLDYwLDYwXSwic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzMsMSwiIiwxLHsiY29sb3VyIjpbMjQwLDYwLDYwXSwic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d) Here we can construct the fiber product as \[ X\fiberprod{B} Z = \ts{(x, e) \st \pi(e) = f(x)} .\] It satisfies the following universal property: \begin{tikzcd} \textcolor{rgb,255:red,214;green,92;blue,92}{W} \\ \\ && {X\fiberprod{B}Z} && Z \\ \\ && X && B \arrow["\pi", from=3-5, to=5-5] \arrow["f"', from=5-3, to=5-5] \arrow[from=3-3, to=5-3] \arrow[from=3-3, to=3-5] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=3-3, to=5-5] \arrow[color={rgb,255:red,214;green,92;blue,92}, from=1-1, to=3-5] \arrow[color={rgb,255:red,214;green,92;blue,92}, from=1-1, to=5-3] \arrow["{\exists!}", color={rgb,255:red,214;green,92;blue,92}, dashed, from=1-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiw0LCJYIl0sWzQsNCwiQiJdLFs0LDIsIloiXSxbMiwyLCJYXFxmaWJlcnByb2R7Qn1aIl0sWzAsMCwiVyIsWzAsNjAsNjAsMV1dLFsyLDEsIlxccGkiXSxbMCwxLCJmIiwyXSxbMywwXSxbMywyXSxbMywxLCIiLDEseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XSxbNCwyLCIiLDAseyJjb2xvdXIiOlswLDYwLDYwXX1dLFs0LDAsIiIsMCx7ImNvbG91ciI6WzAsNjAsNjBdfV0sWzQsMywiXFxleGlzdHMhIiwwLHsiY29sb3VyIjpbMCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX0sWzAsNjAsNjAsMV1dXQ==) ::: :::{.lemma title="?"} If $\pi: P\to X$ is a principal $G\dash$bundle and $f:Y\to X$ is a continuous map, then the following highlighted portion of the pullback is again a principal $G\dash$bundle: \begin{tikzcd} \textcolor{rgb,255:red,92;green,92;blue,214}{f^* p \da Y\fiberprod{X} P} && P \\ \\ \textcolor{rgb,255:red,92;green,92;blue,214}{Y} && X \arrow["{\pr_1}", color={rgb,255:red,92;green,92;blue,214}, from=1-1, to=3-1] \arrow["\pi", from=1-3, to=3-3] \arrow["f"', from=3-1, to=3-3] \arrow["{\exists \tilde f}", dashed, from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJmXiogcCBcXGRhIFlcXGZpYmVycHJvZHtYfSBQIixbMjQwLDYwLDYwLDFdXSxbMCwyLCJZIixbMjQwLDYwLDYwLDFdXSxbMiwyLCJYIl0sWzIsMCwiUCJdLFswLDEsIlxccHJfMSIsMCx7ImNvbG91ciI6WzI0MCw2MCw2MF19LFsyNDAsNjAsNjAsMV1dLFszLDIsIlxccGkiXSxbMSwyLCJmIiwyXSxbMCwzLCJcXGV4aXN0cyBcXHRpbGRlIGYiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbMCwyLCIiLDAseyJzdHlsZSI6eyJuYW1lIjoiY29ybmVyIn19XV0=) We in fact obtain $\pr_1\inv(y) = \pi\inv(f(y))\cong G$, and there will be a right $G\dash$action on each fiber. Behold this gnarly diagram: \begin{tikzcd} {f\inv(U)\times G} \\ {\pr_1\inv(f\inv(U))} & {f^* P} && P & {\pi\inv(U)} && {U\times G} \\ \\ & Y && X & U \\ {f\inv(U)} \arrow["\phi", from=2-5, to=2-7] \arrow[dotted, hook', from=2-5, to=2-4] \arrow[dotted, hook', from=4-5, to=4-4] \arrow["\pi"', from=2-4, to=4-4] \arrow["f"', curve={height=30pt}, dotted, maps to, from=5-1, to=4-5] \arrow["f"', from=4-2, to=4-4] \arrow["{\pr_1}"', from=2-7, to=4-5] \arrow["\pi"', dotted, from=2-5, to=4-5] \arrow[from=2-2, to=2-4] \arrow["{\pr_1}"', from=2-2, to=4-2] \arrow[curve={height=-30pt}, dotted, from=1-1, to=2-7] \arrow[dotted, hook, from=1-1, to=2-2] \arrow[dotted, hook, from=5-1, to=4-2] \arrow["{\pr_1}"', dotted, from=2-1, to=5-1] \arrow[dotted, hook, from=2-1, to=1-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) If $P\to X$ is trivial, this says the pullback will be trivial and $U\times G\mapsto f\inv(U)\times G$ will be a homeomorphism. ::: :::{.remark} So the functor $X\mapsto \Prin_G(X)$ is contravariant functor. ::: :::{.theorem title="Bundle homotopy lemma"} Suppose $B$ is paracompact and Hausdorff, then there is a principal $G\dash$bundle $P \mapsvia{\pi} I \times B$. Consider the fiber bundle $P_0 \da \ro{P}{\ts{0}\times B} \to B$, then there is a diagram: \begin{tikzcd} {P_0} && {\ro{P}{0\times B}} \\ \\ B && 0\times B \arrow["\id", from=1-1, to=1-3] \arrow["\pi"', from=1-1, to=3-1] \arrow["\id"', from=3-1, to=3-3] \arrow[from=1-3, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJQXzAiXSxbMiwwLCJcXHJve1B9ezBcXHRpbWVzIEJ9Il0sWzAsMiwiQiJdLFsyLDIsIjBcXHRpbWVzQiJdLFswLDEsIlxcaWQiXSxbMCwyLCJcXHBpIiwyXSxbMiwzLCJcXGlkIiwyXSxbMSwzXV0=) This extends to an isomorphism $I\times P_0 \to I\times B$ and $P\to I\times B$: \begin{tikzcd} & {P_0\times I} && P \\ & {} \\ & {B\times I} && I\times B \\ {P_0} && {\ro{P}{0\times B}} \\ \\ B && 0\times B \arrow["\id", from=4-1, to=4-3] \arrow["\pi"', from=4-1, to=6-1] \arrow["\id"', from=6-1, to=6-3] \arrow[from=4-3, to=6-3] \arrow["\cong"', from=1-2, to=1-4] \arrow[from=3-2, to=3-4] \arrow[from=1-2, to=3-2] \arrow["\pi"', from=1-4, to=3-4] \arrow[dashed, from=4-1, to=1-2] \arrow[dashed, from=6-1, to=3-2] \arrow[dashed, from=4-3, to=1-4] \arrow[dashed, from=6-3, to=3-4] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) ::: :::{.corollary title="?"} $P_1 = \ro{P}{1\times B} \cong P_0$. ::: :::{.corollary title="?"} If $f_0 \sim f_1: Y\to X$ are homotopic and $P\to X$, then $f_0^*P \cong f_1^* P$. ::: :::{.proof title="?"} Use the homotopy lifting property to get a map $h$: \begin{tikzcd} {h^*P} && P \\ \\ {I\times Y} && Y \arrow["h", from=3-1, to=3-3] \arrow[from=1-1, to=3-1] \arrow["\pi"', from=1-3, to=3-3] \arrow[from=1-1, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125}, draw=none, from=1-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJoXipQIl0sWzAsMiwiSVxcdGltZXMgWSJdLFsyLDAsIlAiXSxbMiwyLCJZIl0sWzEsMywiaCJdLFswLDFdLFsyLDMsIlxccGkiLDJdLFswLDJdLFswLDMsIiIsMSx7InN0eWxlIjp7Im5hbWUiOiJjb3JuZXIifX1dXQ==) Then $\ro{h^* P}{0\times Y} \homotopic \ro{h^*P}{1\times Y} \cong f_1^*P$. :::