Generalized Casson Invariants for SO(3),U(2),Spin(4) , and SO(4)

Abstract

We investigate Casson-type invariants corresponding to the low-rank groups $\mathrm{SO}(3), \mathrm{SU}(2) \times S^1, \mathrm{U}(2), \operatorname{Spin}(4)$ and SO(4). The invariants are defined following an approach similar to those of K. Walker and S. Cappell, R. Lee, and E. Miller. We obtain a description for each of the invariants in terms of the SU(2)-invariant. Thus, all of them may be calculated using formulae for the SU(2)-invariant. In defining these invariants, we offer methods which should prove useful for studying the invariants for other non-simply-connected groups once the invariants for the simply-connected covering groups are known.