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](https://www.researchgate.net/publication/227180588_Cohomogeneity_one_manifolds_with_positive_Ricci_curvature), 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.


Are there other results or references related to this question?



Torus action implying infinite fundamental group
Actions of $T^n$ on simply-connected $n+2$-manifolds were constructed in Theorem 4.7 of [this paper](https://msp.org/pjm/1974/53-2/p12.xhtml). 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](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf) 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](http://www.ams.org/journals/tran/1979-251-00/S0002-9947-1979-0531977-5/home.html). 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](https://www.jstor.org/stable/1999313) and [6](https://doi.org/10.1016/0166-8641(82)90016-5). Moreover, Oh's classification is in the smooth category. Moreover, in Remark (4.7) of [this paper](https://www.jstor.org/stable/1999313), 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). 


