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AS: Ex da assinatura AS: Símbolo na K-teoria AS: Demonstração com grupóides Outros teoremas do índice: L2, Connes-Moscovici

Produto interno e calc vetorial ${\displaystyle }$

Roe algebras

If $(X,{\mathcal {E}})$ is a coarse space, we can associate to it a C*-algebra $C^{*}X$, called its Roe algebra. If $a\in {\mathcal {B}}({\mathcal {l}}_{2}(X))$, its support is defined as

$\operatorname {supp} (a)=\{(y,x)\in X\times X;a_{xy}\neq 0\}$

Then we define $C^{*}X$ as the completion of the locally compact operators of controlled support in ${\mathcal {B}}({\mathcal {l}}_{2}(X))$.

A great source for elements of K-homology is given by the Dirac operators on a compact manifold. As an example, consider the Dirac operator

$D=-i{\frac {d}{d\theta }}$

defined on the circle. This is an unbounded operator, but (properly defining its domain) also self-adjoint, so that, through the spectral theorem, we may define its signal $F={\frac {D}{|D|}}$ NO ORTOGONAL DE C. This operator acts as follows on the basis $e_{n}=e^{2pin\theta }$:

$F(e_{n})=$

IndTop

In this construction, two facts are fundamental: firstly, that $TN$ can always be given a complex structure. This can be seen by noting that $TN=\pi ^{*}N\oplus \pi ^{*}N=N\otimes \mathbb {C}$

second, it is an open subspace of $K(T\mathbb {R} ^{k})$, and by the realization of K-theory through compactly supported triples, we have an extension homomorphism the topological index is given by the composition $i_{*}\circ \phi :K(TX)\rightarrow K(T\mathbb {R} ^{k})=\mathbb {Z}$, where

• $\phi :K(TX)\rightarrow K(TN)$ is the Thom isomorphism in complex K-theory;
• $i_{*}:K(TN)\rightarrow K(T\mathbb {R} ^{k})$ is the extension pushforward homomorphism given by the inclusion.

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AS=

Seja $M$ uma variedade compacta orientável de dimensão $n=2r$. Se $\Lambda ^{even}$ representar a soma dos produtos exteriores de grau par do fibrado cotangente, e $\Lambda ^{odd}$ a soma dos de grau ímpar, defina $D=d+d^{*}$, considerado como uma aplicação de $\Lambda ^{even}$ a $\Lambda ^{odd}$. Então o índice analítico de $D$ é a característica de Euler de $\chi (M)$, e o índice analítico é a integral da classe de Euler sobre a variedade. Essa é a versão "topológica" do teorema de Chern-Gauss-Bonnet.

Mais concretamente, segundo uma variação do splitting principle, se $E$ é um fibrado vetorial real de dimensão $n=2r$, para provarmos fórmulas envolvendo classes características, é possível supor que existem fibrados de linha complexos $l_{1},...l_{r}$ tais que $E\otimes \mathbb {C} =l_{1}\oplus {\overline {l_{1}}}\oplus ...l_{r}\oplus {\overline {l_{r}}}$. Logo, podemos tratar das raízes de Chern $x_{i}(E\otimes \mathbb {C} )=c_{1}(l_{i})$, $x_{r+i}(E\otimes \mathbb {C} )=c_{1}({\overline {l_{i}}})=-x_{i}(E\otimes \mathbb {C} )$, $i=1,...,r$.

Usando raízes de Chern como acima e aplicando as propriedades básicas da classe de Euler, temos que $e(TM)=\prod _{i}^{r}x_{i}(TM\otimes \mathbb {C} )$. Em relação ao caráter de Chern e à classe de Todd,

$${\begin{aligned}\operatorname {ch} (\Lambda ^{even}-\Lambda ^{odd})&=1-\operatorname {ch} (T^{*}M\otimes \mathbb {C} )+\operatorname {ch} (\Lambda ^{2}T^{*}M\otimes \mathbb {C} )-...+(-1)^{n}\operatorname {ch} (\Lambda ^{n}T^{*}M\otimes \mathbb {C} )\\&=1-\sum _{i}^{n}e^{-x_{i}}(TM\otimes \mathbb {C} )+\sum _{i<j}e^{-x_{i}}e^{-x_{j}}(TM\otimes \mathbb {C} )+...+(-1)^{n}e^{-x_{1}}...e^{-x_{n}}(TM\otimes \mathbb {C} )\\&=\prod _{i}^{n}(1-e^{-x_{i}}(TM\otimes \mathbb {C} ))\end{aligned}}$$

$${\begin{aligned}\operatorname {Td} (TM\otimes \mathbb {C} )=\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}(TM\otimes \mathbb {C} )\end{aligned}}$$

Aplicando o teorema do índice,

$\chi (M)=(-1)^{r}\int _{M}{\frac {\prod _{i}^{n}(1-e^{-x_{i}})}{\prod _{i}^{r}x_{i}}}\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}(TM\otimes \mathbb {C} )=(-1)^{r}\int _{M}(-1)^{r}\prod _{i}^{r}x_{i}(TM\otimes \mathbb {C} )=\int _{M}e(TM)$,

que é a versão topológica do teorema de Chern-Gauss-Bonnet (a geométrica sendo obtida ao aplicarmos o homomorfismo de Chern-Weil)

DIMENSIONALIDADE, ELEMENTO DE DIFERENÇA

Suppose that $M$ is a compact oriented manifold of dimension $n=2r$. If we take $\Lambda ^{even}$ to be the sum of the even exterior powers of the cotangent bundle, and $\Lambda ^{odd}$ to be the sum of the odd powers, define $D=d+d^{*}$, considered as a map from $\Lambda ^{even}$ to $\Lambda ^{odd}$. Then the topological index of $D$ is the Euler characteristic $\chi (M)$ of the Hodge cohomology of $M$, and the analytical index is the Euler class of the manifold. The index formula for this operator yields the Chern–Gauss–Bonnet theorem.

The concrete computation goes as follows: according to one variation of the splitting principle, if $E$ is a real vector bundle of dimension $n=2r$, in order to prove assertions involving characteristic classes, we may suppose that there are complex line bundles $l_{1},...l_{r}$ such that $E\otimes \mathbb {C} =l_{1}\oplus {\overline {l_{1}}}\oplus ...l_{r}\oplus {\overline {l_{r}}}$. Therefore, we can consider the Chern roots $x_{i}(E\otimes \mathbb {C} )=c_{1}(l_{i})$, $x_{r+i}(E\otimes \mathbb {C} )=c_{1}({\overline {l_{i}}})=-x_{i}(E\otimes \mathbb {C} )$, $i=1,...,r$.

Using Chern roots as above and the standard properties of the Euler class, we have that $e(TM)=\prod _{i}^{r}x_{i}(TM\otimes \mathbb {C} )$. As for the Chern and Todd classes,

$${\begin{aligned}\operatorname {ch} (\Lambda ^{even}-\Lambda ^{odd})&=1-\operatorname {ch} (T^{*}M\otimes \mathbb {C} )+\operatorname {ch} (\Lambda ^{2}T^{*}M\otimes \mathbb {C} )-...+(-1)^{n}\operatorname {ch} (\Lambda ^{n}T^{*}M\otimes \mathbb {C} )\\&=1-\sum _{i}^{n}e^{-x_{i}}(TM\otimes \mathbb {C} )+\sum _{i<j}e^{-x_{i}}e^{-x_{j}}(TM\otimes \mathbb {C} )+...+(-1)^{n}e^{-x_{1}}...e^{-x_{n}}(TM\otimes \mathbb {C} )\\&=\prod _{i}^{n}(1-e^{-x_{i}}(TM\otimes \mathbb {C} ))\end{aligned}}$$

$${\begin{aligned}\operatorname {Td} (TM\otimes \mathbb {C} )=\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}(TM\otimes \mathbb {C} )\end{aligned}}$$

and so the index theorem applies to show that

$\chi (M)=(-1)^{r}\int _{M}{\frac {\prod _{i}^{n}(1-e^{-x_{i}})}{\prod _{i}^{r}x_{i}}}\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}(TM\otimes \mathbb {C} )=(-1)^{r}\int _{M}(-1)^{r}\prod _{i}^{r}x_{i}(TM\otimes \mathbb {C} )=\int _{M}e(TM)$

which is the "topological" version of the Gauss-Bonnet-Chern theorem (the geometric one being obtained by applying the Chern-Weil homomorphism).

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Since we are dealing with complex bundles, the computation of the topological index is simpler. Using Chern roots and doing similar computations as in the previous example, the Euler class is given by $e(TX)=\prod _{i}^{n}x_{i}(TX)$ and

$$\operatorname {ch} (\sum _{j}^{n}(-1)^{j}V\otimes \Lambda ^{0,j}{\overline {T^{*}X}})=\operatorname {ch} (V)\prod _{j}^{n}(1-e^{x_{j}})(TX)$$

$$\operatorname {Td} (TX\otimes \mathbb {C} )=\operatorname {Td} (TX)\operatorname {Td} ({\overline {TX}})=\prod _{i}^{n}{\frac {x_{i}}{1-e^{-x_{i}}}}\prod _{j}^{n}{\frac {-x_{j}}{1-e^{x_{j}}}}(TX)$$

Applying the index theorem, we obtain the Hirzebruch-Riemann-Roch theorem:

$\chi (X,V)=\int _{X}\operatorname {ch} (V)\operatorname {Td} (TX)$

Let $BX$ and $SX$ be the unit ball and sphere bundles of $X$, respectively. The symbol of an elliptic operator is associated to an element of the K-theory group $K(BX,SX)$ by the following construction. Let $L(X,Y)$ be the group of triples $(E,F,\sigma )$, where $E$ and $F$ are bundles over $X$ and $\sigma$ is an isomorphism, except inside a compact subspace of $X$ MAIS A EQUIVALENCIA. Then there is an isomorphism $\chi :L(X,Y)\rightarrow K(X,Y):={\widetilde {K}}(X/Y)$, equal to $\chi ([E,F])=[E]-[F]$ when $Y=\emptyset$. If $\sigma$ is the symbol of an elliptic differential operator from $E$ to $F$, then $\sigma (D):\pi ^{*}E\rightarrow \pi ^{*}F$ is an isomorphism away from the zero section, and so defines an element $\sigma (D)\in K(BX,SX)$.

As an example of the above construction, take the bundle

In some situations, it is possible to simplify the above formula for computational purposes. In particular, if $X$ is a $2m$-dimensional orientable (compact) manifold with non-zero Euler class $e(TX)$, then applying the Thom isomorphism and dividing by the Euler class, the topological index may be expressed as

$(-1)^{m}\int _{X}{\frac {\operatorname {ch} (E)-\operatorname {ch} (F)}{e(TX)}}\operatorname {Td} (X)$

where division makes sense by pulling $e(TX)^{-1}$ back from the cohomology ring of the classifying space $BSO$.

在特别的情况下，上方的方程可以被简单化。设$X$为一个 $2m$-维、可定向、紧的流行，还设它的欧拉示性数不等于零。引用托姆同构，我们可以将拓朴指标写为

在此处，除以欧拉示性数是允许的因为我们可以拉回从同调环

PDIV

Theorem—Let $\xi \colon E\rightarrow X$ be a vector bundle of rank $n$ over a paracompact space $X$. There exists a space $Y=Fl(E)$, called the flag bundle associated to $E$, and a map $p\colon Y\rightarrow X$ such that

1. the induced cohomology homomorphism $p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)$ is injective, and
2. the pullback bundle $p^{*}\xi \colon p^{*}E\rightarrow Y$ breaks up as a direct sum of line bundles: $p^{*}(E)=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{n}.$

Teorema—Seja $\xi \colon E\rightarrow X$ um fibrado vetorial de dimensão $n$ sobre um espaço paracompacto $X$. Então existe uma variedade $Y=Fl(E)$ e uma aplicação $p\colon Y\rightarrow X$ tal que

1. o homomorfismo induzido na cohomologia $p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)$ é injetivo e
2. o pullback $p^{*}\xi \colon p^{*}E\rightarrow Y$ se divide como a soma direta de fibrados de linha: $p^{*}(E)=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{n}.$

As classes de Chern $c_{1}(L_{1}),...,c_{n}(L_{n})$ são ditas as raízes de Chern de $E$. O ponto é que, como $p^{*}$ é injetiva, toda fórmula envolvendo classes de Chern em $Y$ vale também em $X$. Para provarmos fórmulas do tipo, portanto, podemos considerar somente somas diretas de fibrados de linha.

O princípio da divisão possui várias variações. A seguinte, em particular, trata de fibrados vetoriais reais e suas complexificações:

Teorema—Seja $\xi \colon E\rightarrow X$ um fibrado vetorial real de dimensão $2n$ sobre um espaço paracompacto $X$. Então existe um espaço $Y$ e uma aplicação $p\colon Y\rightarrow X$ tal que

1. o homomorfismo induzido na cohomologia $p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)$ e injetivo e
2. o pullback $p^{*}\xi \colon p^{*}E\rightarrow Y$ se divide como a soma de fibrados de linha: $p^{*}(E\otimes \mathbb {C} )=L_{1}\oplus {\overline {L_{1}}}\oplus \cdots \oplus L_{n}\oplus {\overline {L_{n}}}.$

The splitting principle admits many variations. The following, in particular, concerns real vector bundles and their complexifications:

Theorem—Let $\xi \colon E\rightarrow X$ be a real vector bundle of rank $2n$ over a paracompact space $X$. There exists a space $Y$ and a map $p\colon Y\rightarrow X$ such that

1. the induced cohomology homomorphism $p^{*}\colon H^{*}(X)\rightarrow H^{*}(Y)$ is injective, and
2. the pullback bundle $p^{*}\xi \colon p^{*}E\rightarrow Y$ breaks up as a direct sum of line bundles: $p^{*}(E\otimes \mathbb {C} )=L_{1}\oplus {\overline {L_{1}}}\oplus \cdots \oplus L_{n}\oplus {\overline {L_{n}}}.$