15
votes
Accepted
Automorphisms of $GL_n(\mathbb{Z})$
Hua and Reiner described this in their paper "Automorphisms of the unimodular group" (for them "unimodular" means determinant of absolute value 1, hence it's ${\rm GL}_n(\mathbf Z)$ rather than just ${...
- 50.6k
10
votes
Accepted
Faithful locally free circle actions on a torus must be free?
This follows from Theorem 9.3 (page 216) of the book "Compact tranformation groups" by Bredon (note that it is freely available online).
More is true: Any effective group action of a torus ...
- 6,995
6
votes
Accepted
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. However, I haven't checked that the action is smoothable. The authors are only considering locally ...
- 68.9k
5
votes
Topology of windings on the two-torus
The quotient topology, by a linear flow of irrational slope, is the trivial topology on an uncountable set of the same cardinality as the reals.
When the slope is rational the quotient is much nicer; ...
- 28.2k
5
votes
Accepted
Plus and minus Białynicki-Birula decomposition for normal variety
Yes, \eqref{star} is always a disjoint union (that's obvious). Moreover, each set $X^+(F_k)$ is locally closed and the map $x\mapsto\lim_{t\to0}t\cdot x$ induces an affine morphism $\pi_k:X^+(F_k)\to ...
- 15.5k
4
votes
Accepted
Fixed point stack for a torus action
I hadn't seen this question until Arkadij contacts me directly. The answer is yes: if $G$ is a group scheme of multiplicative type then the fixed point stack is algebraic. This is now here : https://...
- 4,492
3
votes
Accepted
Hamiltonian $S^1$ actions with isolated fixed points
Nick Lindsay and have just proved that such a manifold indeed exists. And surprise, surprise, this is Tolman's manifold. See Theorem 1.3 and Corollary 1.4 of our paper: https://arxiv.org/abs/1912....
3
votes
Torus action implying infinite fundamental group
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 ...
- 96.4k
2
votes
Explicit local normal form symplectomorphism at torus fixed point of a coadjoint orbit
Here is a partial answer.
The map $\mathfrak{t}^{\perp} \to T_{\lambda}\mathcal{O}_{\lambda}$, $X \mapsto ad_X(\lambda)$, endows $\mathfrak{t}^{\perp}$ with a linear symplectic form
$$ \omega_0(X,Y) ...
- 401
2
votes
Accepted
Torus actions with more than one fixed point
If t3suji wants to post an answer, then I will delete this answer. I am just posting this until further notice. There is a stratification of $X$ into locally closed subsets according to the ...
2
votes
Accepted
Linearization of hamiltonian torus action
There is always an equivariant local symplectomorphism with $T_pM$ with its 2-form and linear isotropy action, by the Moser-Weinstein proof. But that constant 2-form then has more possible “...
- 31.5k
1
vote
Linearization of hamiltonian torus action
For fixed submanifolds, there is an analogous statement in Theorem 22.1 of V. Guillemin and S. Sternberg. Symplectic techniques in physics. Cambridge University Press, Cambridge, xi+468 pp, (1984).
...
- 6,995
1
vote
Accepted
Index formula with nonisolated fixed points
Atiyah-Segal-Singer Theorem, Theorem 6.16 in
Berline, Nicole; Getzler, Ezra; Vergne, Michèle, Heat kernels and Dirac operators, Grundlehren der Mathematischen Wissenschaften. 298. Berlin etc.: ...
- 1,628
1
vote
Does there exist smooth circle action on manifolds M^{4n} with exactly three fixed points such that n\neq 1
There is an $S^1$-action on the Quaternionic projective plane $\mathbb{H}\mathbb{P}^2$ with exactly 3 fixed points. They are not hard to construct (done in a similar way to the standard $S^1$-actions ...
- 6,995
Only top scored, non community-wiki answers of a minimum length are eligible