  In 1997, Liberati and Pollifrone in Phys. Rev. D56 (1997) 6458
(hep-th/9708014) achieved a new formulation of the Bekenstein-Hawking formula,
where the entropy and the Euler characteristic are related by $S=\chi A/8$. In
this work we present a relation between the low-energy absorption cross section
for minimally coupled fermions and the Euler characteristic of
(3+1)-dimensional spherically symmetric black holes, i.e. $\sigma =\chi
g_h^{-1}A$. Based on the relation, using the Gauss--Bonnet--Chern theorem and
the $\phi$-mapping method, an absorption cross section density is introduced to
describe the topology of the absorption cross section. It is shown that the
absorption cross section and its density are determined by the singularities of
the timelike Killing vector field of the spacetime and these singularities
carry the topological numbers, Hopf indices and Brouwer degrees, naturally.
