# Surface mapping class group actions on 3-manifolds

with A. al Beaini and L. Chen

### Submitted. arxiv:2311.15508

For a circle bundle X→S over a surface S, there is a natural surjection Homeo(X) → Mod(S). This surjection splits for the unit tangent bundle by a well-known construction. On the other hand, it does not split when the Euler number e(X) is not divisible by the Euler characteristic chi(S) because Mod(X) → Mod(S) does not split. When e(X) is divisible by chi(S), Mod(X) → Mod(S) does split, but we show Homeo(X) → Mod(S) does not split in many cases.