The Hopf Conjecture and the Singer Conjecture
نویسندگان
چکیده
The conjecture is true in dimension 2 since the only surfaces which have positive Euler characteristic are S2 and RP2 and they are the only two which are not aspherical. In the special case where M2k is a nonpositively curved Riemannian manifold this conjecture is usually attributed to Hopf by topologists and either to Chern or to both Chern and Hopf by differential geometers. When I first heard about this conjecture in 1981, I thought I could come up with a counterexample by using right-angled Coxeter groups. Given a finite simplicial complex L which is a flag complex, there is an associated right-angled Coxeter group W . Its Euler characteristic is given by the formula
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