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We propose a new conjecture on the relation between the species doubling of
lattice fermions and the topology of manifold on which the fermion action is
defined. Our conjecture claims that the maximal number of fermion species on a
finite-volume and finite-spacing lattice defined by discretizing a
$D$-dimensional manifold is equal to the summation of the Betti numbers of the
manifold. We start with reconsidering species doubling of naive fermions on the
lattices whose topologies are torus ($T^{D}$), hyperball ($B^D$) and their
direct-product space ($T^{D} \times B^{d}$). We find that the maximal number of
species is in exact agreement with the sum of Betti numbers $\sum^{D}_{r=0}
\beta_{r}$ for these manifolds. Indeed, the $4D$ lattice fermion on torus has
up to $16$ species while the sum of Betti numbers of $T^4$ is $16$. This
coincidence holds also for the $D$-dimensional hyperball and their
direct-product space $T^{D} \times B^{d}$. We study several examples of lattice
fermions defined on discretized hypersphere ($S^{D}$), and find that it has up
to $2$ species, which is the same number as the sum of Betti numbers of
$S^{D}$. From these facts, we conjecture the equivalence of the maximal number
of fermion species and the summation of Betti numbers. We discuss a program for
proof of the conjecture in terms of Hodge theory and spectral graph theory.