On the homotopy type of the space of fiberings of S1 × S2 by simple closed curves

For most aspherical Seifert-fibered 3-manifolds [Formula: see text], the space of Seifert fiberings [Formula: see text] is known to have contractible components [McCullough and Soma, The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold, J. Differential Geom. 93(2) (2013) 327–353]. It is also known that the space of Hopf fiberings of the three-sphere is noncontractible [DeTurck et al., Homotopy type of the moduli space of fibrations of the three-sphere by simple closed curves, in progress]. We provide the second example of a non-aspherical 3-manifold [Formula: see text] such that [Formula: see text] has noncontractible components. In particular, we show that [Formula: see text] has two connected components which are both homotopy equivalent to the identity-based loop space [Formula: see text]. We also utilize the reduced product suspension of James [Reduced product spaces, Ann. of Math. [Formula: see text]2 [Formula: see text] 62 (1955) 170–197] to exhibit second homology generators for both connected components of [Formula: see text].