The first and second homology groups, H1 and H2, are computed for configuration spaces of framed three-dimensional point particles with annihilation included, when up to two particles and an antiparticle are present, the types of frames considered being S2 and SO(3). Whereas a recent calculation for two-dimensional particles used the Mayer–Vietoris sequence, in the present work Morse theory is used. By constructing a potential function none of whose critical indices is less than four, we find that (for coefficients in an arbitrary field K) the homology groups H1 and H2 reduce to those of the frame space, S2 or SO(3) as the case may be. In the case of SO(3) frames this result implies that H1 (with coefficients in ℤ2) is generated by the cycle corresponding to a 2π rotation of the frame. (This same cycle is homologous to the exchange loop: the spin-statistics correlation.) It also implies that H2 is trivial, which means that there does not exist a topologically nontrivial Wess–Zumino term for SO(3) frames [in contrast to the two-dimensional case, where SO(2) frames do possess such a term]. In the case of S2 frames (with coefficients in ℝ), we conclude H2=ℝ, the generator being in effect the frame space itself. This implies that for S2 frames there does exist a Wess–Zumino term, as indeed is needed for the possibility of half-integer spin and the corresponding Fermi statistics. Taken together, these results for H1 and H2 imply that our configuration space “admits spin 1/2” for either choice of frame, meaning that the spin-statistics theorem previously proved for this space is not vacuous.
Farley–Sabalka’s Morse-Theory Model and the Higher Topological Complexity of Ordered Configuration Spaces on Trees
