In 2009, he solved a 55-year-old open problem posed in 1954 by Friedrich Hirzebruch,[3] which asks "which linear combinations of Chern numbers of smooth complex projective varieties are topologically invariant".
[4] He found that only linear combinations of the Euler characteristic and the Pontryagin numbers are invariants of orientation-preserving diffeomorphisms (and thus according to Sergei Novikov also of oriented homeomorphisms) of these varieties.
Kotschick proved that if the condition of orientability is removed, only multiples of the Euler characteristic can be considered among the Chern numbers and their linear combinations as invariants of diffeomorphisms in three and more complex dimensions.
[6][7][8] In the case of the sphere, there is only the standard football (12 black pentagons, 20 white hexagons, with a pattern corresponding to an icosahedral root) provided that "precisely three edges meet at every vertex".
If more than three faces meet at some vertex, then there is a method to generate infinite sequences of different soccer balls by a topological construction called a branched covering.