nLab Atiyah-Jänich theorem

Idea

The Atiyah-Jänich theorem (Atiyah 1967 Thm A1, Jänich 1965) states that the space of Fredholm operators $Fred(\mathscr{H})$ on a (countably infinite-dimensional separable, complex) Hilbert space $\mathscr{H}$ is a classifying space for topological K-theory $K(-)$:

For $X$ a compact Hausdorff space, the homotopy classes of continuous maps from $X$ (hence the connected components of the mapping space) to $Fred(\mathscr{H})$ are in natural bijection with $K(X)$

$$\pi_0 \, Map\big( X, \, Fred(\mathscr{H}) \big) \xrightarrow[\sim]{ind_X} K(X) \,,$$

where $ind_X$ forms the index bundle, an $X$-parameterized enhancement of the Fredholm index.

This relation generalizes to a definition of twisted K-theory and of equivariant K-theory (hence of twisted equivariant K-theory) as given by homotopy classes of sections of $Fred(\mathscr{H})$-fiber bundles (Atiyah & Segal 2004).

Examples

Example

For discussion of the basic complex line bundle on the 2-sphere realized as a Fredholm index bundle see there.

References

The original articles:

• Michael Atiyah, Thm. A1 in: K-theory, Harvard Lecture 1964 (notes by D. W. Anderson), Benjamin (1967) [pdf, pdf]

• Klaus Jänich: Vektorraumbündel und der Raum der Fredholm-Operatoren, Mathematische Annalen 161 (1965) 129–142 [doi:10.1007/BF01360851]

In monographs:

• Max Karoubi, §7.10 in: K-Theory – An Introduction, Grundlehren der mathematischen Wissenschaften 226, Springer (1978) [pdf, doi:10.1007%2F978-3-540-79890-3]

Lecture notes:

• Arshay Sheth: The Atiyah-Jänich Theorem [pdf, pdf]

As the basis of a definition of twisted K-theory and equivariant K-theory and twisted equivariant K-theory:

• Michael Atiyah, Graeme Segal, Twisted K-theory, Ukrainian Math. Bull. 1 3 (2004) [arXiv:math/0407054, journal page, published pdf]