Let $E$ be a complex vector bundle over a compact smooth manifold $M$. Let $\\nabla^E$ be a (C-linear) connection on $E$ and let $R^E$ denote its curvature.<\/p>\n
The Chern character form associated to $\\nabla^E$ is defined by
\n\\[
\n ch\\left(E,\\nabla^E\\right)=tr\\left[ \\exp\\left(\\frac{\\sqrt{-1}}{2\\pi}R^E\\right)\\right]\\in\\Omega^{even}(M).
\n\\]
\nObviously, $ch\\left(E,\\nabla^E\\right)$ is a closed form. We denote the association cohomology class by $ch(E)$ which is called the \\emph{Chern character} of $E$.
\n\\begin{prop}
\nLet $E$, $F$ are complex vector bundles on manifold $M$. Then the following propositions hold:
\n\\begin{enumerate}
\n \\item $ch(E\\oplus F)= ch(E)+ch(F)$; \\qquad ($\\oplus$ is Whithney direct sum)
\n \\item $ch(E\\otimes F)=ch(E)\\times ch(F)$;
\n \\item If $E\\cong F$, then $ch(E)=ch(F)$.
\n\\end{enumerate}
\n\\end{prop}
\nThe crux of the proof is that the calculate does not depend on the representative element.<\/p>\n
Denote by Vector($M$) the set of all complex vector bundles over $M$, then under the Whitney direct sum operation, Vect($M$) becomes a semi-abelian group. And $(Vect(M)$, $\\otimes$, $\\oplus)$ is a semi-ring. Now we introduce an equivalence relation ‘$\\sim$’ in Vect($M$) such that
\n\\[
\n E \\sim F\\Leftrightarrow E\\cong F.
\n\\]
\nSo the following map
\n\\[
\n ch : Vect(M)\\rightarrow H^{even}_dR(M,C)
\n\\]
\nis a homomorphism between semi-groups. there is a nature method for extending a semi-abelian group to a abelian group. The fundamental ideal: let $N$ be the natural number($0 \\not\\in N$). Now we extend semi-abelian group $(N,+)$ to a abelian group.<\/p>\n
i) Take
\n\\[
\n N\\times N=\\set{(m,n)|m,n\\in N}.
\n\\]
\nIn $N\\times N$, there have a natural sum ‘$+$’ as follows:
\n\\[
\n\\bigg( \\forall m_1,m_2,n_1,n_2 \\in N \\bigg) \\quad (m_1,n_1)+(m_2,n_2)=(m_1+m_2, n_1+n_2).
\n\\]
\nOne introduces an equivalence relation ‘$\\sim$’ in $N\\times N$ as follows:
\n\\[
\n(m_1,n_1)\\sim (m_2,n_2) \\Leftrightarrow m_1+n_2=m_2+n_1.
\n\\]
\nPlease note there is not subtract. Let $Z=N\\times N \/\\sim$, it is easily proved that $Z$ is a abelian group. The zero element is $[(m,m)],m\\in N$. And $[(m,n)]^{-1}=[(n,m)]$.<\/p>\n
ii)Make a map
\n\\[
\n\\phi : N \\rightarrow Z, \\qquad \\phi(m)=[(m+1,1)].
\n\\]
\nObviously, the map $\\phi$ is isomorphism from $N$ to a semi-subgroup $\\set{[(m+1,1)]}$. So $Z$ is a dilation of $N$. then we can denote
\n\\[
\n m=[(m+1,1)],\\quad 0=[(m,m)],\\quad -m=[(1,m+1)],\\quad\\forall m \\in N.
\n\\]
\nOne can prove this dilation is the smallest.<\/p>\n
Now we can extend semi-abelian group $Vect(M)\/\\sim$ to a group $K(M)$ which is called the \\emph{K-group} of $M$. Naturally, the map $ch$ is extended a group homomorphism,
\n\\[
\n ch: K(M) \\rightarrow H^{even}_dR(M,\\C).
\n\\]
\nAtiyah and Hirzebruch prove this homomorphism is an isomorphism if one ignores the torsion elements in $K(M)$. This theory belongs the \\emph{K-theory}.
\n\\begin{examp}
\nLet $M$ be a closed manifold.
\n\\[
\n \\langle \\hat{A}(TM)ch(E),[M] \\rangle =\\int_M \\hat{A}(TM,\\nabla^{TM})ch(E,\\nabla^E)\\in \\C.
\n\\]
\nIf $M$ is an even dimensional oriented spin closed manifold(see[milnor]), the characteristic number
\n\\[
\n \\langle \\hat{A}(TM)ch(E),[M] \\rangle
\n\\]
\nis a integer.(Atiyah, Hirzebruch, Borel theorem)
\n\\end{examp}<\/p>\n","protected":false},"excerpt":{"rendered":"
Let $E$ be a complex vector bundle over …<\/p>\n