Suppose that a $d$-dimensional torus $T$ acts smoothly and effectively on an $n$-dimensional closed manifold $M$. What conditions on $d$ and $n$ imply that $\pi_1(M)$ must be infinite?

Consider the case where $d=n-1$. Within Section 4 of this paper by Grove and Ziller, it is shown that if $n\geq 4$ and $T^{n-1}$ acts smoothly and effectively on $M^n$, then $\pi_1(M)$ is infinite. In contrast for $n=2$, there is of course the $S^1$-action on $S^2$, and for $n=3$, there is the $T^2$-actions on $S^3$ and lens spaces.

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up vote 6 down vote accepted

Actions of $T^n$ on simply-connected $n+2$-manifolds were constructed in Theorem 4.7 of this paper. However, I haven't checked that the action is smoothable. The authors are only considering locally smooth actions in the paper (see p. 170 of Bredon for the definition of locally smooth); however, the construction yielding Theorem 4.7 is quite explicit, and seems to be at least piecewise-smooth. All such actions were subsequently classified by McGavran. So presumably one could use this classification to determine if there are smooth actions.

Addendum: Oops, it looks like there's some gaps in McGavran's arguments that were fixed by Hae Soo Oh at least in dimensions 5 and 6. Moreover, Oh's classification is in the smooth category. Moreover, in Remark (4.7) of this paper, Oh constructs examples of $T^n$ actions on simply-connected $n+2$-manifolds in all dimensions. Since he's working in the smooth category, this seems to show that there are smooth examples (although I don't see immediately in his argument where smoothness is proven).

By taking products, it seems clear that something like $d\leq 2n/3$ is possible without infinite fundamental group, and this is, in some metaphysical sense, sharp: see Torus actions on rationally elliptic manifolds, by Galaz-Garcia et al.

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