We present a description of the quantum mechanical states of a system of n indistinguishable particles moving on a manifold M by C/S2- valued functions ψ′ defined on the configuration space Mn/Sn. These functions satisfy one of two homotopy conditions, which characterize the particles as bosons or fermions. Any closed curve of Mn/Sn which does not intersect the diagonals can be classified as even or odd according to whether its lifts to Mn have end points in Mn which are even or odd permutations of each other. For bosons, ψ′ must map any such closed curve which also avoids kerψ′ onto an even number of basic loops of (C−{0})/S2. For fermions, ψ′ must map such even loops of Mn/Sn onto an even number, and odd loops onto an odd number, of basic loops of (C−{0})/S2.
We argue that fermion-boson mapping techniques represent a natural tool for studying many-body supersymmetry in fermionic systems with pairing. In particular, using the generalized Dyson mapping of a many-level fermion superalgebra with the SU(2) type of pairing we investigate two kinds of supersymmetry connecting excitations in the systems with even and odd particle numbers: dynamical supersymmetry, which ensures a unified classification of states for both even and odd populations, and invariant supersymmetry with actual degeneracies of states within the same supermultiplet. Dynamical supersymmetries based on the dynamical algebra of the type $\mathrm{U}(K∕2\ensuremath{\Omega})$ (where
K and $2\ensuremath{\Omega}$ denote the number of fermion-pair and single-fermion states, respectively) naturally arise in the bosonized description of the system. Conditions for invariant supersymmetry are determined in a restricted case of bilinear supercharge operators.