Atiyah-Jänich's theorem

The set of Atiyah-Jänich is a theorem from functional analysis . He establishes a connection between Fredholm operators and K-theory .

Fredholm Operator Space and Index Mapping

It is the (up unique to isomorphism) infinite-dimensional separable Hilbert space and the space of bounded Fredholm operators on the operator norm topology. ${\displaystyle {\mathcal {H}}}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle {\mathcal {H}}}$

For a compact space denote its topological K-theory . Elements in are due to formal differences ${\displaystyle X}$ ${\displaystyle K(X)}$ ${\displaystyle K(X)}$

${\displaystyle \left[E_{0}\right]-\left[E_{1}\right]}$,

represented by vector bundles over . We want to assign such an element to a continuous mapping . ${\displaystyle X}$ ${\displaystyle F\colon X\to {\mathcal {F}}}$ ${\displaystyle K(X)}$

For a continuous mapping one has the finite-dimensional vector spaces in every point ${\displaystyle F\colon X\to {\mathcal {F}}}$ ${\displaystyle x\in X}$

${\displaystyle \mathrm {ker} (F(x))}$and , that is, the core and coke of the operator . ${\displaystyle \mathrm {coker} (F(x))}$ ${\displaystyle F(x)}$

In general it is possible that the dimension of these vector spaces is discontinuous at individual points . However, every map is homotopic to a continuous map for which ${\displaystyle x\in X}$ ${\displaystyle F}$ ${\displaystyle F^{\prime }}$

${\displaystyle (\mathrm {ker} (F^{\prime }(x)))_{x\in X}}$ and ${\displaystyle (\mathrm {coker} (F^{\prime }(x)))_{x\in X}}$

have constant dimension and are sub-bundles of , that is, we have an element ${\displaystyle X\times {\mathcal {H}}}$

${\displaystyle \left[(\mathrm {ker} (F^{\prime }(x)))_{x\in X}\right]-\left[(\mathrm {coker} (F^{\prime }(x)))_{x\in X}\right]\in K(X)}$.

Furthermore, this element does not depend on which map to be homotopic is used. ${\displaystyle F}$ ${\displaystyle F^{\prime }}$

Hence, this construction defines a map

${\displaystyle \mathrm {ind} \colon \left[X,{\mathcal {F}}\right]\to K(X)}$

on the set of homotopy classes of mappings from to in . It is called the index mapping and the formal difference is called the index bundle. ${\displaystyle \left[X,{\mathcal {F}}\right]}$ ${\displaystyle X}$ ${\displaystyle {\mathcal {F}}}$ ${\displaystyle K(X)}$ ${\displaystyle \left[(\mathrm {ker} (F^{\prime }(x)))_{x\in X}\right]-\left[(\mathrm {coker} (F^{\prime }(x)))_{x\in X}\right]}$

Atiyah-Jänich's theorem

The proposition assumed by Michael Atiyah and proven by Klaus Jänich states that

${\displaystyle \mathrm {ind} \colon \left[X,{\mathcal {F}}\right]\to K(X)}$

is a bijection.

The space of the Fredholm operators thus realizes the space that classifies the topological K-theory . ${\displaystyle BU}$

If one considers the special case of a one-point space, then on the one hand , and on the other hand, the continuous mappings can be identified with the Fredholm operators . It shows that the homotopy class of a mapping is determined by the Fredholm index of and the above mapping exactly matches the Fredholm index when identifying with . Hence, the index mapping generalizes the Fredholm index. ${\displaystyle X=\{p\}}$ ${\displaystyle K(X)\cong \mathbb {Z} }$ ${\displaystyle X\rightarrow {\mathcal {F}}}$ ${\displaystyle F(p)}$ ${\displaystyle X=\{p\}\rightarrow {\mathcal {F}}}$ ${\displaystyle F(p)}$ ${\displaystyle \mathrm {ind} }$ ${\displaystyle K(X)}$ ${\displaystyle \mathbb {Z} }$

literature

• Klaus Jänich: vector space bundles and the space of the Fredholm operators. Math. Ann. 161 (1965) 129-142.
• Max Karoubi: Espaces classifiants en K-théorie. Trans. Amer. Math. Soc. 147 (1970) 75-115.
• Bernhelm Booss: Topology and Analysis. Introduction to the Atiyah-Singer index formula. University text. Springer-Verlag, Berlin-New York, 1977. ISBN 3-540-08451-7

Web links

Atiyah: Algebraic topology and operators in Hilbert space