MORSE-THEORY AND THE TOPOLOGY OF CONFIGURATION-SPACE

The first and second homology groups, H-1 and H-2, are computed for co nfiguration spaces of framed three-dimensional point particles with an nihilation included, when up to two particles and an antiparticle are present, the types of frames considered being S-2 and SO(3). Whereas a recent calculation for two-dimensional particles used the Mayer-Vieto ris sequence, in the present work Morse theory is used. By constructin g a potential function none of whose critical indices is less than fou r, we find that (for coefficients in an arbitrary field K) the homolog y groups H-1 and H-2 reduce to those of the frame space, S-2 or SO(3) as the case may be. In the case of SO(3) frames this result implies th at H-1 (with coefficients in Z(2)) is generated by the cycle correspon ding to a 2 pi rotation of the frame. (This same cycle is homologous t o the exchange loop: the spin-statistics correlation.) It also implies that H-2 is trivial, which means that there does not exist a topologi cally 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 S-2 frames (with coefficients in R), we conclude H-2 = R, the generator being in effect the frame space itself. This implies tha t for S-2 frames there does exist a Wess-Zumino term, as indeed is nee ded for the possibility of half-integer spin and the corresponding Fer mi statistics. Taken together, these results for H-1 and H-2 imply tha t our configuration space ''admits spin 1/2'' for either choice of fra me, meaning that the spin-statistics theorem previously proved for thi s space is not vacuous.