Mixing
Torus action implying infinite fundamental group
Clash Royale CLAN TAG#URR8PPP
up vote
8
down vote
favorite
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 $ngeq 4$ 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?
at.algebraic-topology differential-topology group-actions torus-action
asked Aug 14 at 22:47
Lawrence Mouillé
178210
add a comment |Â
up vote
8
down vote
favorite
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 $ngeq 4$ 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?
at.algebraic-topology differential-topology group-actions torus-action
asked Aug 14 at 22:47
Lawrence Mouillé
178210
1
Some relevant references: mathscinet.ams.org/mathscinet-getitem?mr=1255926 mathscinet.ams.org/mathscinet-getitem?mr=2784821
– Ian Agol
Aug 15 at 3:36
add a comment |Â
up vote
8
down vote
favorite
up vote
8
down vote
favorite
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 $ngeq 4$ 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?
at.algebraic-topology differential-topology group-actions torus-action
asked Aug 14 at 22:47
Lawrence Mouillé
178210
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 $ngeq 4$ 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?
at.algebraic-topology differential-topology group-actions torus-action
asked Aug 14 at 22:47
Lawrence Mouillé
178210
asked Aug 14 at 22:47
Lawrence Mouillé
178210
asked Aug 14 at 22:47
Lawrence Mouillé
178210
asked Aug 14 at 22:47
Lawrence Mouillé
178210
178210
1
Some relevant references: mathscinet.ams.org/mathscinet-getitem?mr=1255926 mathscinet.ams.org/mathscinet-getitem?mr=2784821
– Ian Agol
Aug 15 at 3:36
add a comment |Â
1
Some relevant references: mathscinet.ams.org/mathscinet-getitem?mr=1255926 mathscinet.ams.org/mathscinet-getitem?mr=2784821
– Ian Agol
Aug 15 at 3:36
1
1
Some relevant references: mathscinet.ams.org/mathscinet-getitem?mr=1255926 mathscinet.ams.org/mathscinet-getitem?mr=2784821
– Ian Agol
Aug 15 at 3:36
Some relevant references: mathscinet.ams.org/mathscinet-getitem?mr=1255926 mathscinet.ams.org/mathscinet-getitem?mr=2784821
– Ian Agol
Aug 15 at 3:36
add a comment |Â
2 Answers
2
active
oldest
votes
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).
answered Aug 15 at 3:32
Ian Agol
46.6k1119225
add a comment |Â
up vote
3
down vote
By taking products, it seems clear that something like $dleq 2n/3$ 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.
answered Aug 15 at 0:18
Igor Rivin
77.2k8109290
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
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).
answered Aug 15 at 3:32
Ian Agol
46.6k1119225
add a comment |Â
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).
answered Aug 15 at 3:32
Ian Agol
46.6k1119225
add a comment |Â
up vote
6
down vote
accepted
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).
answered Aug 15 at 3:32
Ian Agol
46.6k1119225
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).
answered Aug 15 at 3:32
Ian Agol
46.6k1119225
edited Aug 15 at 4:22
answered Aug 15 at 3:32
Ian Agol
46.6k1119225
answered Aug 15 at 3:32
Ian Agol
46.6k1119225
answered Aug 15 at 3:32
Ian Agol
46.6k1119225
46.6k1119225
add a comment |Â
add a comment |Â
up vote
3
down vote
By taking products, it seems clear that something like $dleq 2n/3$ 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.
answered Aug 15 at 0:18
Igor Rivin
77.2k8109290
add a comment |Â
up vote
3
down vote
By taking products, it seems clear that something like $dleq 2n/3$ 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.
answered Aug 15 at 0:18
Igor Rivin
77.2k8109290
add a comment |Â
up vote
3
down vote
up vote
3
down vote
By taking products, it seems clear that something like $dleq 2n/3$ 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.
answered Aug 15 at 0:18
Igor Rivin
77.2k8109290
By taking products, it seems clear that something like $dleq 2n/3$ 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.
answered Aug 15 at 0:18
Igor Rivin
77.2k8109290
answered Aug 15 at 0:18
Igor Rivin
77.2k8109290
answered Aug 15 at 0:18
Igor Rivin
77.2k8109290
answered Aug 15 at 0:18
Igor Rivin
77.2k8109290
77.2k8109290
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f308318%2ftorus-action-implying-infinite-fundamental-group%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
Some relevant references: mathscinet.ams.org/mathscinet-getitem?mr=1255926 mathscinet.ams.org/mathscinet-getitem?mr=2784821
– Ian Agol
Aug 15 at 3:36