
\input amstex

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\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}



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\parindent 16 pt

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\title       The Geometry of Vector Bundles and an Introduction to Gauge
Theory \\
Lecture 31
\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 15, 1998
\enddate

\endtopmatter

\document

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

We saw last time: for complex bundles $E_i\to B$, $i=1,2$, 
$$\align
  c(E_1\oplus E_2)&=c(E_1)\cdot c(E_2)\\
  ch(E_1\oplus E_2)&=ch(E_1)+ch(E_2)
  \endalign$$
where $c(E)=\left[\,\det(I+\frac{i}{2\pi}F)\,\right]$
       ($F$ is a curvature of any connection) and 
      $ch(E)=\left[\,\Tr(\,\exp(\frac{i}{2\pi}F)\,)\,\right]$

\vskip.25in
\noindent 
\leftline{{\bf Chern Classes for Dual Bundles}}
\vskip.12in

Say $E^*$= dual bundle to $E$. Recall: given a connection $D$ on $E$, we
can define a connection $D^*$ on $E^*$ such that $A^*=-A^t$. Thus 
$$\split
  F^*&=dA^*+A^*\wedge A^*\\
     &=-dA^t+A^t\wedge A^t.
  \endsplit$$
But $A^t\wedge A^t=-(A\wedge A)^t$, if $A$ is a matrix of $1$ forms.
So the curvature $F^*=-F^t$. Then 
$$
\split
c(E^*)
&=\det(I-\frac{i}{2\pi}F)^t\\
&=\det(I-\frac{i}{2\pi}F)\\
&=\sum_k c_k(E^*)\\
&=\sum_k c_k(-\frac{i}{2\pi}F)\\
&=\sum_k (-1)^kc_k(\frac{i}{2\pi}F).
\endsplit
$$ 
So $$c_k(E^*)=(-1)^kc_k(E).$$
For example, $c_1(E^*)=-c_1(E)$, etc.

\vskip.25in
\noindent
{\bf Relations between Chern Classes $c_k$ and Chern Characters $ch_k$} 
\vskip.12in
We know $ch_k(A)=\Tr(A^k)$, so
$$
\align
ch_0(E)&=\Tr(\frac{i}{2\pi}F)^0=\Tr I=\Rank E\\
ch_1(E)&=\Tr(\frac{i}{2\pi}F)^1=c_1(E)\\
ch_2(E)&=\Tr(\frac{i}{2\pi}F)^2\\
\vdots&
\endalign
$$
If the eigenvalues of $\frac{i}{2\pi}F$ are $x_1,\cdots,x_n$, then 
$$
\split
ch_2(E)
&=\frac{1}{2}\sum^n_{i=1}x_i^2\\
&=\frac{1}{2}\left(\,\sum^n_{i=1}x_i\,\right)^2
  -\left(\,\sum_{i<j}x_ix_j\,\right)\\
&=\frac{1}{2}c_1(E)^2-c_2(E)
\endsplit
$$
Thus $$ch_2(E)=\frac{1}{2}c_1(E)^2-c_2(E)$$
etc.


\vskip.25in
\noindent
{\bf Chern Characters as a Map between Bundles and Cohomology}
\vskip.12in
\proclaim{Property} $ch(E_1\otimes E_2)=ch(E_1)\cdot ch(E_2)$.
\endproclaim
{\bf Hint}: Given connections $D_i$ on $E_i$, $i=1,2$, we can define
$D=D_1\otimes I_2+I_1\otimes D_2$ with $F_d=F_1\otimes I_2+I_1\otimes
F_2$ on $E_1\otimes E_2$. Then use $\Tr(A\otimes B)=\Tr A\cdot\Tr B$.
\qed


\remark{Exercise 1} Prove the above claim.
\endremark

\example{Remark} Given $B$, if we define
$\Vect(B):=\{\text{Isomorphism classes of complex bundle on $B$}\}$, 
then we have two ``ring operations'' on $\Vect(B)$\,:
$$
\alignat2
&+&&\ :\qquad E_1\oplus E_2\\
&\,\cdot&&\ :\qquad E_1\otimes E_2.
\endalignat
$$
$\Vect(B)$ can be made into a ring under these operations (cf. a
course on $K$-theory\,!) in which case
$$ch:\Vect(B)\to H^*(B,\R)$$ defines a ring isomorphism.
\endexample

\remark{Note} For a complex line bundle $L\to B$, we only have 
$$
\align
c_0(L)&=1\\
c_1(L)&\in H^2(B,\R).
\endalign
$$
(will see later how to compute in the case of $L$ over complex
manifold\,!) 

So the total Chern class $$c(L)=(1+c_1(L)).$$

For $E=L_1\oplus L_2\oplus\cdots\oplus L_n$ (sum of $n$ line bundles),
we can get 
$$
\split
c(E)
&=c(L_1)c(L_2)\cdots c(L_n)\\
&=\prod^n_{i=1}(1+c_1(L_i))
\endsplit
$$
For example, 
$$
\align
c_1(E)&=\sum^n_{i=1}c_1(L_1)\\
c_2(E)&=\sum_{i\neq j}c_1(L_i)c_1(L_j)\\
\vdots&\\
c_n(E)&=\prod^n_{i=1}c_1(L_i)
\endalign
$$
\endremark

\remark{Exercise 2} Suppose $E=L_1\oplus L_2\oplus\cdots\oplus L_n$.
If $c_1(L_i)=0$, for some $i$, then $c_n(E)=0$
\endremark

\remark{Exercise 3} Suppose $E=L_1\oplus L_2\oplus\cdots\oplus L_n$. 
If $c_1(L_1)=c_1(L_2)=\cdots=c_1(L_k)=0$, then $c_j(E)=0$, for $j>n-k$.
\endremark

\vskip.25in
\leftline{{\bf Characteristic Classes for Real Bundles I}}
\vskip.12in

We want to define the {\it characteristic classes} for real bundles, 
say $E_{\R}\to B$ of rank $n$ ($E_{\R}|_b\cong\R^n$).

\vskip.1in
\noindent
\underbar{First Approach}: Complexify $E_{\R}$ and use the previous
discussions about complex bundles.

\vskip.08in 
Let's recall some facts. We complexify a real vector space $V_{\R}$ by
taking $V_{\R}\otimes\C\equiv V_{\C}$. 
If $\dim_{\R}(V_{\R})=n$, then $\dim_{\C}(V_{\C})=n$.
(If $V_{\R}=\text{Span}_{\R}\{v_1,\cdots,v_n\}$, 
 then $V_{\C}=\text{Span}_{\C}\{v_1,\cdots,v_n\}$.)

For a real bundle $E_{\R}$ of rank $n$, we can form a {\it complex
bundle} $E_{\C}:=E_{\R}\otimes\C$ as follows\,: 

If $$E_{\R}=\coprod_{\alpha}U_{\alpha}\times\R^n/g_{\alpha\beta}\qquad\quad 
\text{where \ $g_{\alpha\beta}:U_{\alpha}\cap U_{\beta}\to\GL(n,\R)$},$$

then 
$$E_{\C}=E_{\R}\otimes\C=\coprod_{\alpha}U_{\alpha}\times\C^n/g_{\alpha\beta}
\qquad\quad\text{where \ $g_{\alpha\beta}:U_{\alpha}\cap U_{\beta}
                       \to\GL(n,\R)\hookrightarrow\GL(n,\C)$}.$$ 
Therefore we can define the {\bf characteristic classes} for $E_{\R}$
by $$P(E_{\R})\equiv P(E_{\C})\qquad\quad \text{for any
$\GL(n,\C)$-invariant polynomial $P$}.$$
For example, $c^{\R}_k(E_{\R})\equiv c_k(E_{\C})$.

\remark{Note} We will see that for $k$ odd, $c_k(E_{\R})=0$.
\endremark



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