Editing Atiyah–Singer index theorem (section)

===Connes–Donaldson–Sullivan–Teleman index theorem===
Due to {{harv|Donaldson|Sullivan|1989}}, {{harv|Connes|Sullivan|Teleman|1994}}:

:'''For any quasiconformal manifold there exists a local construction of the Hirzebruch–Thom characteristic classes.'''

This theory is based on a signature  operator ''S'', defined on middle degree differential forms on even-dimensional quasiconformal manifolds (compare {{harv|Donaldson|Sullivan|1989}}).

Using topological cobordism and K-homology one may provide a full statement of an index theorem on quasiconformal manifolds (see page 678 of {{harv|Connes|Sullivan|Teleman|1994}}). The work {{harv|Connes|Sullivan|Teleman|1994}} "provides local constructions for characteristic classes based on higher dimensional relatives of the measurable Riemann mapping in dimension two and the Yang–Mills theory in dimension four."

These results constitute significant advances along the lines of Singer's program ''Prospects in Mathematics'' {{harv|Singer|1971}}. At the same time, they provide, also, an effective construction of the rational Pontrjagin classes on topological manifolds. The paper {{harv|Teleman|1985}} provides a link between Thom's original construction of the rational Pontrjagin classes {{harv|Thom|1956}} and index theory.

It is important to mention that the index formula is  a topological statement. The obstruction theories  due to Milnor, Kervaire, Kirby, Siebenmann, Sullivan, Donaldson show that only a minority of topological manifolds possess differentiable structures and these are not necessarily unique. Sullivan's result on Lipschitz and quasiconformal structures {{harv|Sullivan|1979}} shows that any topological manifold in dimension different from 4 possesses such a structure which is unique (up to isotopy close to identity).

The quasiconformal structures {{harv|Connes|Sullivan|Teleman|1994}} and more generally the ''L''<sup>''p''</sup>-structures, 
''p'' > ''n''(''n''+1)/2, introduced by M. Hilsum {{harv|Hilsum|1999}}, are the weakest analytical structures on topological manifolds of dimension ''n'' for which the index theorem is known to hold.