Let $M$ be a closed connected smooth oriented manifold. The following two properties - that $M$ can either enjoy or not - intuitively both mean that $M$ has very little symmetry:
(a) Every self-map $M \to M$ has degree $0,\pm 1$. (This is usually called inflexible.)
(b) Every smooth $S^1$-action on $M$ is trivial.
My question is whether any of (a) or (b) implies the other or, if not, what concrete examples of manifolds are that have one of these properties, but not the other.
Here are some thoughts:
If $M = S^k \times N$ then of course $M$ has neither of both properties.
If $M$ has positive simplicial volume (aka Gromov norm), like it happens for negatively curved manifolds, and in particular surfaces of hgher genus, then $M$ has both properties (a) and (b). For (a), that's rather obvious, for (b) it is a theorem due to Yano, published in 1982.
It is a theorem due to Gromov that simply-connected manifolds always have simplicial volume $0$, but it is known that there are inflexible simply connected manifolds in any big enough dimension, see http://arxiv.org/pdf/1109.0960.pdf. I am not able at all to check whether these manifolds admit non-trivial $S^1$ ations, though...
Very naively, it seems that if $M$ has a nontrivial $S^1$-action, we can use the orbit circles to wrap $M$ around itself with a positive degree, but it is not at all clear to me how this intuitive idea can be made precise. Maybe it is also a fallacy to think about $S^1$-actions in general by having pictures in mind where $M$ is a sphere or torus.
