
\input amstex

\documentstyle{amsppt}

\input arrow.tex


\define\GL{\operatorname{GL}}
\define\ORTH{\operatorname{O}}
\define\U{\operatorname{U}}
\define\Lie{\operatorname{Lie}}

\define\Hom{\operatorname{Hom}}
\define\Iso{\operatorname{Iso}}
\define\End{\operatorname{End}}
\define\Aut{\operatorname{Aut}}
\define\Ker{\operatorname{Ker}}
\define\Coker{\operatorname{Coker}}
\redefine\Im{\operatorname{Im}}

\define\Rank{\operatorname{Rank}}
\define\Tr{\operatorname{Tr}}
\define\Mat{\operatorname{Mat}}
\define\Vect{\operatorname{Vect}}

\define\rp{\Bbb{RP}}
\define\cp{\Bbb{CP}}

\define\R{\Bbb R}
\define\C{\Bbb C}
\define\Z{\Bbb Z}
\redefine\P{\Bbb P}



\hsize  6.5 true in
\hoffset  0.0 true in
\vsize 9.0 true in
\voffset 0.0 true in
\NoRunningHeads
\parindent 16 pt

\NoBlackBoxes

\title       The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 32
\endtitle
\author      Professor Steven Bradlow \\ Class Notes From Math 433
\endauthor
\affil       University of Illinois at Urbana-Champaign
\endaffil
\address     273 Altgeld Hall, 1409 W. Green Street, Urbana, IL  61801
\endaddress
\email       bradlow\@math.uiuc.edu
\endemail
\date        April 17, 1998
\enddate

\endtopmatter

\document

\vskip.25in
\leftline{{\bf Characteristic Classes for Real Bundles}}
\vskip.25in


Before we return to {\it Characteristic Classes} for {\it real} bundles, we go
back to {\it complex} bundles\,: we had 

\noindent
{\it Chern Classes} defined by the generating function
$$c(A)=\det(I+\frac{i}{2\pi}A)=\prod^n_{i=1}(1+x_i)\quad (\text{where $x_i$
are the eigenvalues of $\frac{i}{2\pi}A$})\ .$$

\noindent
{\it Chern Characters} defined by the generating function
$$ch(A)=\Tr\left(\exp(\frac{i}{2\pi}A)\right)=\sum^n_{i=1}\exp(x_i)\ .$$

Another important characteristic class is the {\bf Todd class}
$$Td(A)=\prod^n_{j=1}\frac{x_j}{1-e^{-x_j}}\ .$$

\remark{Note}
 \itemitem{$\bullet$} When applied to $A=F=\text{curvature}$, 
   series terminates at terms of $\deg=\frac{\text{dim} B}{2}$. 
\itemitem{$\bullet$} For a complex bundle $E$, 
   $Td(E)=1+\frac{1}{2}c_1(E)+\frac{1}{12}(c_1^2(E)+c_2(E))+\cdots$\ .
\itemitem{$\bullet$} Appears in Riemann-Roch Theorem.
\endremark

\vskip.12in
For a holomorphic bundle (complex bundle such that $E$ is a complex
manifold and $\pi$ is holomorphic) over a complex manifold 
$E\overset\pi\to\to X$, we have an integer
$$
\split \chi(E)
&=\text{bundle version of {\bf Euler characteristic} of a manifold.}\\
&=\text{{\bf index} of a differential operator.} 
\endsplit
$$ 
The Riemann-Roch Theorem relates this (global, topological) quantity
$\chi(E)$ to characteristic classes:
$$\chi(E)=\int_X\,Td(T'X)\wedge ch(E)\ .$$ 
This is an example of an {\bf Index Theorem} which relates the
(integer valued) index of a differential operator
$\Omega^0(E_1)\to\Omega^0(E_2)$ to $\int_B(\text{characteristic classes
of $TB$ and bundles})$.

If $X$ is a {\it Riemann surface} ($\dim_{\C}X=1$), then $\Rank(T'X)=1$
and $Td(T'X)=1+\frac{1}{2}c_1(T'X)$. 
So $$Td(T'X)\wedge ch(E)=(1+\frac{1}{2}c_1(T'X))\wedge (r+c_1(E))\ .$$ Thus
$$
\split \int_X\, Td(T'X)\wedge ch(E)
&=\int_X\,(1+\frac{1}{2}c_1(T'X))\wedge (r+c_1(E))\\
&=\int_X\, c_1(E)+\frac{r}{2}\,\int_X\, c_1(T'X)\ .
\endsplit
$$
\example{Fact\,/\,Definition} 
\vskip.08in
\itemitem{$\bullet$} $\int_X\, c_1(E)\equiv\deg(E)$\,.
\vskip.05in
\itemitem{$\bullet$} $\deg(T'X)=2-2g$, where $g=\text{genus of Riemann
surface $X$}$.
\endexample

So the Riemann-Roch Theorem:
$$\chi(E)=\deg(E)+\Rank(E)(1-g)\ .$$

\vskip.25in
\leftline{{\bf Back to the Characteristic Classes for Real Bundles}}
\vskip.25in

\noindent
Recall the \underbar{First Approach} from last time:

Given a real bundle $E_{\R}$ with fiber $\R^n$ and $\GL(n,\R)$, we can
define a complex bundle $E_{\C}=E_{\R}\otimes\C$ with fiber $\C^n$ and
$\GL(n,\C)(\hookleftarrow\GL(n,\R))$.
\remark{Exercise 1} Show that $E_{\C}=E_{\R}\oplus E_{\R}$ as real
bundle.
\endremark

{\bf Hint}: Look in detail at 
$\GL(n,\R)\hookrightarrow\GL(n,\C)\hookrightarrow\GL(2n,\R)$.\qed
\remark{Note} Given any complex bundle, we can look at underlying 
{\it real} bundle uning $\C^n\cong\R^{2n}$ to realize fibers as real
and $\GL(n,\C)\hookrightarrow\GL(2n,\R)$ to obtain real transition
functions.
\endremark

Therefore we can define 
$$c_k(E_{\R})\equiv c_k(E_{\C})\ .$$

\remark{Note} For $k$ odd, $c_k(E_{\R})=0$.
\endremark

{\bf Proof}: Given a complex bundle $E_{\C}$, we can define 
$\overline{E_{\C}}$. 
If $g_{\alpha\beta}\in\GL(n,\C)$ are transition functions for
$E_{\C}$, 
then $\overline{g_{\alpha\beta}}\in\GL(n,\C)$ are transition
functions for $\overline{E_{\C}}$. We now use the following key facts:

\proclaim{Key Facts} 

(1) If $E_{\C}=E_{\R}\otimes\C$, then 
                         $\overline{E_{\C}}=E_{\C}$.

(2) $c_k(E_{\C})=(-1)^kc_k(\overline{E_{\C}})$, 
    for any complex bundle $E_{\C}$.
\endproclaim

{\bf Hint}: Given a connection $D$ on $E_{\C}$, we can get a connection
    $\overline{D}$ on $E_{\C}$ such that
    $F_{\overline{D}}=\overline{F_D}$, 
    then $i\,F_{\overline{D}}=-(\overline{i\,F_D})$.\qed

\vskip.08in
Thus $c_k(E_{\C})=(-1)^kc_k(\overline{E_{\C}})=(-1)^kc_k(E_{\C})$.\qed

\vskip.12in
So we can get $c_{2k}(E_{\R})$ in $H^{4k}(B,\R)$.

\vskip.1in
\noindent
\underbar{Second Approach}: Look at $\GL(n,\R)$ invariant polynomials
    on $\Mat_n(\R)$.
\vskip.05in 
\underbar{Easier}: Look at $\ORTH(n)$-invariant polynomials on 
$\Lie(\ORTH(n))=\{\,A|\,A^t+A=0\,\}$.

\underbar{Why} can we always do this: Fix a Riemann metric on
$E_{\R}$ and use orthonormal frames and orthogonal
connections, then the curvature $F$ is $\Lie(\ORTH(n))$-valued and the
transition functions are $\ORTH(n)$-valued. Hence give any
$\ORTH(n)$-invariant $P:\Lie(\ORTH(n))\to\R$, we can get 
$[P(\frac{1}{2\pi}F)]\in H^*(B,\Z)$.

\vskip.05in
So what can we say about such $P$\,?
\example{Fact} $A+A^t=0$, $A$ cannot be diagonalized:
\endexample
Since 
$$\left\{\aligned A&+A^t=0\\
                  A&=\bmatrix \lambda_1&&\\ 
                               &\ddots&\\
                               &&\lambda_n\endbmatrix
         \endaligned\right.\qquad\Rightarrow\quad\lambda_i=0\ ! $$
But $A$ is conjugate (via $T\in\ORTH(n)$) to 
$$
\aligned
&\bmatrix 
0&\lambda_1&&&\\
-\lambda_1&0&&&\\
&&\ddots&&\\
&&&0&\lambda_k\\
&&&-\lambda_k&0
\endbmatrix\\
&\\
&\quad\qquad\qquad n=2k
\endaligned
\quad\text{or}\quad
\aligned
&\bmatrix 
0&\lambda_1&&&&\\
-\lambda_1&0&&&&\\
&&\ddots&&&\\
&&&0&\lambda_k&\\
&&&-\lambda_k&0&\\
&&&&&0
\endbmatrix\\
&\\
&\qquad\qquad\quad\,n=2k+1
\endaligned
\ .$$
Thus we only need to consider $P$'s on such $A$. Write
$$P\bmatrix 
0&\lambda_1&&&&\\
-\lambda_1&0&&&&\\
&&0&\lambda_2&&\\
&&-\lambda_2&0&&\\
&&&&\ddots&\\
&&&&&\ddots
\endbmatrix
=P(\lambda_1,\cdots,\lambda_n)\ .$$ 



\enddocument
\end





