# Wednesday, February 10 :::{.theorem title="Stable cohomology of BOn"} As $n\to \infty$, we have \[ H^*(BO_n, \ZZ/2\ZZ) = \ZZ/2\ZZ[w_1, w_2, \cdots] && w_i \in H^i .\] ::: :::{.definition title="Stiefel-Whitney class"} Given any principal $O_n\dash$bundle $P\to X$, there is an induced map $X \mapsvia{f} BO_n$, so we can pull back the above generators to define the **Stiefel-Whitney classes** $f^* w_i$. ::: :::{.remark} If $P \da \OFrame TX$, then $f^* w_1$ measures whether $X$ has an orientation, i.e. $f^* w_1 = 0 \iff X$ can be oriented. We also have $f^* w_i(P) = w_i( \bundle{E} )$ where $P = \OFrame( \bundle{E} )$. In general, we'll just write $w_i$ for Stiefel-Whitney classes and $c_i$ for Chern classes. ::: :::{.definition title="Pontryagin Classes"} The **Pontryagin classes** of a real vector bundle \( \bundle{E} \) are defined as \[ p_i( \bundle{E} ) = (-1)^i c_{2i}( \bundle{E} \tensor_\RR \CC) .\] Note that the complexified bundle above is a complex vector bundle with the same transition functions as \( \bundle{E} \), but has a reduction of structure group from $\GL_n(\CC)$ to $\GL_n(\RR)$. ::: :::{.observation} $\RP^{\infty }$ and $\CP^{\infty }$ are examples of $K(\pi, n)$ spaces, which are the unique-up-to-homotopy spaces defined by \[ \pi_k K (\pi, n) = \begin{cases} \pi & k=n \\ 0 & \text{else}. \end{cases} \] ::: :::{.theorem title="Brown Representability"} \[ H^n(X; \pi) \cong [X, K( \pi, n) ] .\] ::: :::{.example title="?"} \[ [X, \RP^{\infty } ] &\cong H^1(X; \ZZ/2\ZZ) \\ [X, \CP^{\infty } ] &\cong H^2(X; \ZZ) .\] ::: :::{.proposition title="Classification of complex line bundles"} There is a correspondence \[ \correspond{ \text{Complex line bundles} } \mapstofrom [X, \CP^{\infty }] = [X, BC\units] \mapstofrom H^2(X; \ZZ) \] Importantly, note that for $X \in \Mfd_\CC$, $H^2(X; \ZZ)$ measures *smooth* complex line bundles and not holomorphic bundles. ::: :::{.proof title="of proposition"} We'll take an alternate direct proof. Consider the exponential exact sequence on $X$: \[ 0 \to \constantsheaf{Z} \to \OO \mapsvia{\exp} \OO\units .\] Note that $\constantsheaf{\ZZ}$ consists of locally constant $\ZZ\dash$valued functions, $\OO$ consists of smooth functions, and $\OO\units$ are ???. \todo[inline]{Can't read screenshot! :(} This yields a LES in homology: \begin{tikzcd} {H^0(X; \constantsheaf{\ZZ})} && {H^0(X; \OO)} && {H^0(X; \OO\units)} \\ \\ {H^1(X; \constantsheaf{\ZZ})} && \textcolor{rgb,255:red,214;green,92;blue,92}{H^1(X; \OO)} && {H^1(X; \OO\units)} \\ \\ {H^2(X; \constantsheaf{\ZZ})} && \textcolor{rgb,255:red,214;green,92;blue,92}{H^2(X; \OO)} && {H^2(X; \OO\units)} \arrow[from=1-1, to=1-3] \arrow[from=1-3, to=1-5] \arrow[from=1-5, to=3-1, out=0, in=180] \arrow[from=3-1, to=3-3] \arrow[from=3-3, to=3-5] \arrow[from=3-5, to=5-1, out=0, in=180] \arrow[from=5-1, to=5-3] \arrow[from=5-3, to=5-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOSxbMCwwLCJIXjAoWDsgXFxjb25zdGFudHtcXFpafSkiXSxbMCwyLCJIXjEoWDsgXFxjb25zdGFudHtcXFpafSkiXSxbMCw0LCJIXjIoWDsgXFxjb25zdGFudHtcXFpafSkiXSxbMiwwLCJIXjAoWDsgXFxPTykiXSxbMiwyLCJIXjEoWDsgXFxPTykiLFswLDYwLDYwLDFdXSxbMiw0LCJIXjIoWDsgXFxPTykiLFswLDYwLDYwLDFdXSxbNCwwLCJIXjAoWDsgXFxPT1xcdW5pdHMpIl0sWzQsMiwiSF4xKFg7IFxcT09cXHVuaXRzKSJdLFs0LDQsIkheMihYOyBcXE9PXFx1bml0cykiXSxbMCwzXSxbMyw2XSxbNiwxXSxbMSw0XSxbNCw3XSxbNywyXSxbMiw1XSxbNSw4XV0=) Since $\OO$ admits a partition of unity, $H^{>0}(X; \OO) = 0$ and all of the red terms vanish. For complex line bundles $L$, $H^1(X, \OO\units) \cong H^2(X; \ZZ)$. Taking a local trivialization $\ro{L}{U} \cong U \cross \CC$, we obtain transition functions \[ t_{UV} \in C^{\infty }(U \intersect V, \GL_1(\CC) ) \] where we can identify $\GL_1(\CC) \cong \CC\units$. We then have \[ (t_{U_{ij}}) \in \prod_{i < j} \OO\units(U_i \intersect U_j) = C^1(X; \OO\units) .\] Moreover, \[ \qty{ t_{U_{ij}} t_{U_{ik}} \inv t_{U_{jk}} }_{i,j,k} = \bd ( t_{U_{ij} } ) _{i, j} = 0 ,\] since transitions functions satisfy the cocycle condition. So in fact $(t_{U_{ij}}) \in Z^1(X; \OO\units) = \ker \bd^1$, and we can take its equivalence class \( [ ( t_{U_{ij} } ) ] \in H^1(X; \OO\units) = \ker \bd^1 / \im \bd^0 \). Changing trivializations by some $s_i \in \prod_i \OO\units(U_i)$ yields a composition which is a different trivialization of the same bundle: \begin{tikzcd} {\ro{L}{U_i}} && {U_i \cross \CC} &&& {U_i \cross \CC} \arrow["{h_i}", from=1-1, to=1-3] \arrow["{\cdot s_i}", from=1-3, to=1-6] \arrow[curve={height=30pt}, from=1-1, to=1-6] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwwLCJcXHJve0x9e1VfaX0iXSxbMiwwLCJVX2kgXFxjcm9zcyBcXENDIl0sWzUsMCwiVV9pIFxcY3Jvc3MgXFxDQyJdLFswLDEsImhfaSJdLFsxLDIsIlxcY2RvdCBzX2kiXSxbMCwyLCIiLDIseyJjdXJ2ZSI6NX1dXQ==) So the $(t_{ U_{ij}}$ change *exactly* by an $\bd^0( s_i)$. Thus the following map is well-defined: \[ L \mapsto [ (t_{U_{ij}} ) ] \in H^1(X; \OO\units) .\] There is another construction of the map \[ \ts{L} &\to H^2(X; \ZZ) \\ L &\mapsto c_1(L) .\] Take a smooth section of $L$ and $s\in H^0(X; L)$ that intersects an $\OO\dash$section of $L$ transversely. Then \[ V(s) \da \ts{ x\in X \st s(x) = 0 } \] is a submanifold of real codimension 2 in $X$, and $c_1(L) = [ V(s) ] \in H^2(X; \ZZ)$. ::: :::{.theorem title="Splitting Principle for Complex Vector Bundles"} \envlist 1. Suppose that $\bundle{E} = \bigoplus_{i=1}^r L_i$ and let $c(\bundle{E}) \da \sum c_i(\bundle{E}$. Then \[ c(\bundle{E}) = \prod_{i=1}^r \qty{ 1 + c_i (L_i) } .\] 2. Given any vector bundle \( \bundle{E} \to X \), there exists some $Y$ and a map $Y\to X$ such that $f^*: H^k(X; \ZZ) \injects H^k(Y; \ZZ)$ is injective and $f^* \bundle{E} = \bigoplus_{i=1}^r L_i$. ::: :::{.slogan} To verify any identities on characteristic classes, it suffices to prove them in the case where \( \bundle{E} \) splits into a direct sum of line bundles. ::: :::{.example title="?"} \[ c( \bundle{E} \oplus \bundle{F}) = c( \bundle{E} ) c( \bundle{F} ) .\] To prove this, apply the splitting principle. Choose $Y, Y'$ splitting $\bundle{E}, \bundle{E}'$ respectively, this produces a space $Z$ and a map $f:Z\to X$ where both split. We can write \[ f^* \bundle{E} &= \bigoplus L_i && c(f^* \bundle{E} ) = \prod \qty{ 1 + c_1(L_i) } \\ f^* \bundle{F} &= \bigoplus M_j && c(f^* \bundle{E} ) = \prod \qty{ 1 + c_1(M_j) } .\] We thus have \[ c( f^* \bundle{E} \oplus f^* \bundle{F} ) &= \prod \qty{1 + c_1(L_i) } \qty{1 + c_1(M_j)} \\ &= c(f^* \bundle{E} ) c(f^* \bundle{F} ) ,\] and $f^* (c( \bundle{E} \oplus \bundle{F} ) = f^* (c (\bundle{E}) c( \bundle{F}))$. Since $f^*$ is injective, this yields the desired identity. ::: :::{.example title="?"} We can compute $c(\Sym^2 \bundle{E})$, and really any tensorial combination involving \( \bundle{E} \), and it will always yield some formula in the $c_i( \bundle{E} )$. :::